AMCP 706-110

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20315

ENGINEERING DESIGN (HANDBOOK

1 EXPERIMENTAL STATISTICS

SECTION 1

BASIC CONCEPTS

AND ANALYSIS OF

MEASUREMENT DATAReproduced by the

CL F AR IN G HOUSE

tor Federal Scienbific & TechnicalInformeaion Springfield Va. 22151

PE., NTA EIS, U.S. ARMY MATERIEL COMMAND DECEMBER 1969

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,IEADVUAAk'-f Er.UNITED STATES ARMY MATERIEL COMMAND

4.t WASHINGTON, D.C. 20315AMC PAMPilLET 16 December 1969

No. 706-1160ENGINEERING DESIGN HANDBOOK

EXPER1MEN'i AL STATIS rICS (SEC 1)

Paragraph Page

CHAPTER 1

SOME BASIC STATISTICAL CONCcPTSAND PRELIMINARY CONSIDERATIONS

1-1 INTROD UCTION .................................. .... 1-11-2 POPULATIONS, SAMPLES, AND DISTRIBUTIONS..... 1-11-3 STATISTICAL INFERENCES AND SAMPLING ......... 1-31-3.1 Statistical Inferences .................................... 1-31-3.2 Random Sampling ..................................... 141-4 SELECTION OF A RANDOM SAMPLE ................ 1-61-5 SOMP. PROPERTIES OF DISTRIBUTIONS .............. 1-61-6 ESTL.ATION OF m AND o ............................ 1-101-7 CONFIDENCE IN"J'EPV.VALS .......................... 1-111-8 STATISTICAL TOLERANCE LIMITS ................... 1-141-9 USING STATISTICS TO MAKE DECISIONS............ 1-15

S1-9.1 Approach to a Decision Problem ..................... '-151-9.2 Choice of Null and Alternative Hypotheses ................ 1-161-9.3 Two Kinds of Errors .................................... 1-171-94 Significance Level and Operating Characteristi,: (OC) Curve

of a Statistical Test...... .............................. 1-171-9.5 Choice of the Significance Level ...... ................ 1-171-9.6 A W ord of Caution.................................... 1-18

DISCUSSION OF TECHNIQUES IN CHAPTERS 2THROUGH 6 .................. ................... 2-ii

CHAPTER 2

CHARACTERIZING THE MEASURED OERFORMANCE OFA MATREMLA, PRODUCT, OR PROCE59

2-1 ESTIMATING AVERAGE PERFORMANCE FROM ASA M P L E ................ . ............................. 2-1

2-1.1 General ........................................... 2-1

2-1.2 Best Single Eptimate ................... ... ........... 2-12-1.3 Some Remarks on Confidence Inter val Estimates ........... 2-22-1.4 Confidence Interv~da for the Population Mean When

Knwledge of the Variability Cannot Be Assumed ......... 2-22-1.4.1 Two-sided Confidence Intem val .................... 2-22-1.4.2 Onc-uided Confidence Interval ...................... 2-32-1.5 Confldence Interval Estimates W1 en We HIave Previous

4 •, Knowledge of the Variability ............................ 2-4

;This painphmct supcrscdct AM('P 706-110, 31 July 1963.

TABLE OF CONTENTS (CONT) AMCP 706-110

Paragraph Pa~e

CHAPTER 2 (Cont) 4

2-2 ESTIMATING VARIABILITY OF PERFORMANCE FROMA SAM PLE ..................................... ....... 2-6 3

2-2.1 G eneral, .... ... ...................................... 2-62 2.2 Single Estim ates ...................................... 2-62-2.2,1 s8 and s ... ........................................ -62-2.2.2 The Sample Range as an Estimate of the Standard Deviation 2-62-2.3 Cenfidence Interval Estimates ........................... 2-72-2.3.1 Two-sided Confidence Intdrval Estimates ................ 2-72-2.3.2 One-sided Corfidence Interval Estimates ................ 2-72-2.4 Estimating the Standard Deviation When No Sample Data

an A vailable .......................................... 2-82-3 NUMBER OF MEASUREMENTS REQUIRED TO ESTAB-

LISH THE MEAN WITH PRESCRIBED ACCURACY... 2-9A G eneral ............................................... 2-9

2- 3,- Estimation of the Mean of a Population Using a SingleSam ple ................................... ............ 2,10

2-3.3 Estimation Using a Sample Which is Taken Ir. Twc Stages.. 2-102-4 NUMBER OF MEASUREMENTS REQUIRED TO ESTAB-

LISH THE VARIABILITY WITH STATED PRECISION.. 2-122-5 STATISTICAL TOLERANCE LIMITS .................. 2-13.,-5.1 G eneral ................................ .............. 2-132-5.2 Two-sided Tolerance Limits for a Normal Distribution ...... 2-132-5.3 One-sided Tolerance Limits for a Normal Distribution ...... 2-14"2-5.4 Tor!e..-. '. iitq Which are Independent of the Form of the

Distribution ......... .......................... 2-152-5.4.1 Two-sided Tolerance Limits (Distribution-Free) ......... 2-152-5.4.2 One-sided Tolerance Limits (Distribution-Free) ......... 2-15

CHAPTER 3

COMPARING MATERIALS OP. PRODUCTSWITH RESPECT TO AVERAGE PERFORMANCE

8-1 GENERAL REMARKS ON STATISTICAL TESTS ....... 3-18-2 COMPARING THE AVERAGE OF A NEW PRODUCT

WITH THAT OF A STANDARD ........................ 3-38-2.1 To Determine Whether the Average of a New Product Differs

From the Standard ..................................... 34t-2. 1.1 Does the Average of the New Product Dif!fr From the

Stazdard (u Unknown)? ............................... 3-415-2.1.2 Does the Average of the New Product Differ From the

Standard (a Known)? ................................. 8-8;&2.2, To Determine Whether the Average of a Now Product Exceeds

the Standard ............................ . ........... 8-18A-14.2.1 Does the Average of the New Product Exceed the Standard

(v Unknown)?..... . ................ 3-13

it

AIRCP 706-110 TABLE OF CONTENTS (CONT)

Paragraph Pap

CHAPTER 3 (Cant)

8-2.2.2 Does the AN erage of the New Product Exceed the Standard(a K now n)? .............................. ........... 8-16

8-2.3 To Determine Whether the Avernge of a New Product is LessThan the Staidad ................................. .... 8-20

3-2.3.1 Is the Avcrage: of the Now Product Less Than the Standard(o U nknow n)? ......................................... 8-20

3-2.3.2 Is the Average of the New Product Less Than That of theS&andard (a Known)?. ... ..................... 3-21

3-3 COMPARING THE AVERAGES OF TWO MATERIAI.,PRODUCTS. OR PROCESSES ........................... 3-22

3-3.1 Do Products A and B Differ In Average Performance? ...... 3-238-3.1.1 (Case 1) - Variability of A and B Is Unknown, But Can Be

Assumed to be Equal ................................. 3-233-3.1.2 (Case 2) - Variability of A and B is Unknown, Cannot Be

Assumed Equal ............................. ........ 3-263-3.1.3 (Case 3) - Variability in Performance of Each of A and

B is Known from Previous Experience, and the StandardDeviations are oA and vp, Respectively ................ 8-30

3-3.1.4 (Case 4) - -The Observations are Paired ................. 3-318-3.2 Does the Average of Product A Exceed the Average of

"Product B ? ............................................ 3 34"3-3.2.1 (Case 1) - Variability of A and B is Unknown, But Can Be

Assumed to be Equal ................................. 8-43-3.2.2 (Case 2) - Variability of A and B is Unknown, Cannot Be

Assum ed Equal ...................................... -36&. (Casel,= 3) -- V -ability in Performance of Each of A ajd

B is Known from Previous Experience, and the StandardDeviations are a, and as, Respectively ................. -37

6-3.2.4 (Case 4) - The Observations are Paired ............... 3-383-4 COMPARING THE AVERAGES OF SEVERAL PRODUCTS 3-40

CHAPTER 4

COMPARINrG . MAYFR!A1$ OR PRODUCTS WITH RESPECT TO

VARIABILITY OF PERFORMANCE

4-1 COMPARING A NEW MATERIAL OR PRODUCT WITH ASTANDARD WITH RESPECT TO VARIABILITY OFPERFORMANCE ...................................... 4-1

4-1.1 Does the Variability of the New Product Differ From That ofthe Standard? .................................. 4-1

41.2 Doea the Variability of the New Product Exceed That of the8 Standard? ............................... ............. 4-3

S4-1.8 Is the Variability of the New Product Lesi Than That of theStandard?.................. .................. 4-5

TABLE OF CONTENTS (CONTj AMCP 706-1 0

Paragraph Page

CHAPTER 4 (Cont)

4-2 COMPARING TWO MATERIALS Oi PRODUCTS WITHRESPECT TO VARIABILITY OF PERFORMANCE ... 4-8

4-2.1 Loet the Variability of Product A Dlffer From That ofir uct B ... . ......................... ... ...

4-2.2 Does the Variability of Product A Exceed That of Product B?. 4-9

CHAPTER 5

CHARACTERIZING LINEAR RELt TIONSHISBETWEEN TWO VARIABLES

5-1 INTRODUCTION ....................................... 5-1&-2 PLOTTING THE DATA ...... .................... 5-15-3 TWO IMPORTANT SYSTEMS OF ,jINEAR

RELATIONSHIPS ................................ 5-35-3.1 Functional Relationships ................................ 5-35-3.2 Statistical Relationships ................................. 5-5&-4 PROBLEMS AND PROCEDURES FOR FUNCTIONAL

RELATIONSHIPS ....................................... 5-116-4.1 F1 Relationships (General Case) .......................... 5-11 (7'!5-4.1.1 What Is the Bebt Line To Be Used for Estimating y From

Given Values of xT ................................... 5-125-4.1.2 What Are the Coafidence Interval Estimates for: the Line

as a Whole; a Point on the Line; a Future Value of YCorr -- di- . .. liv-em Valli Afr? 5-16

54.1.3 What Is the Confidence Interval Estimate for Oi, the Slopeof the Tru Line y - po + px? ......................... 5-19

5-4.1.4 11 We Observe n' New Values of Y "With Average f?'), HowCan We Use the Fitted RegrEtioi Line to Obtain a:.Interval Estimate of the Value of z that Produced TheseValues of Y? ......... ................................. 5-20

5-4.1.5 Using the Fitted Regression Line, How Can We Choose aValue (z") of x Which We May Expect with Confidence(1 - a) Will Produce a Value of Y Not Less Than SomeSpecified Value Q! .................................... 5-21

6-4.1.6 Is the Assaumption, of Linear Regression Justified? ....... 5-2264.2 F1 Relationships When dhe Intecctopt Is Kxiurw-- To Be Equ.I

to Zero (Lints Through the Origin) ..................... - 245-4.2.1 Line Through Origin, Variance of Y'a Independent of z... 5-2464.2.2 Line Through Origin, Variance Proportional to x (4y,.. - zx') 6-255-4.2.8 Line Through Origin, Standard Deviation Proportional to

z (cr.. x-') ................ ................. 5-265-4.24 Line Through Origin, Erroru of Y'a Cumulative (Cumulative

D ata) .................................... .......... 5 -265-4.8 FII Relationships ...................................... 5-27

-4.8.1 A Simple Method of Fitting tne I.ne In the Gene.-al Case, -27""-4.8,2 An Important Exceptional Caw ....................... 5-29

iv

- I-

AMCP 706-110 1ABLE OF CONTENTS (CONTY

I'arag; aph Page

CHAPTER 5 (Cont)

Some Linearizing Transformations................... 5-305-5 PROBLEMS AND PRItOCEDURES F*9R STATISTICAL

RELATIONSH IPS . ..................................... 5-316-5.1 SI Relationships .. ............................ ........ 5-315-5.1.1 What Is the Best Line To Ile Used fc: Estimating Yx fnr

G iven Values of X ? ................................... 6-35-5.1.2 What Are the Confidence Interval Estimates for: the Line

as a Whole; a PoiLkt on the Line; a Single Y Correspondingto a New Value of X ? ................................. 5-36

5-5.1.3 Give a Confidence Interval Estimate for pi, the Slope of theTrue Regression Line, Yx - go 4- OX? ............... 5-38

5-5.1.4 What Is the Best Line for Predicting .A From GivenValues of Y? -....................................39

5-5.1.5 What Is the Degree of Relationship of the Two Variables Xand Y as Measured by p, the Correlation Coefficient?.. 5-40

5-5.2 SII Relationships ............................... 5....... -405-5.2.1 What Is the Best Line To Be Used for Estimating Yx From '

Given Values of X ? ................................... 5-415-5.2.2 What Are the Confid-nee Interval Estimates for: the Line

Wa a Whole; a Point on the Line; a Single Y Correspondingto a New Value of X ? ................................. 5-42

"5-5.2.3 What Is the Confidence Interval Estimate for #,, the Slopeof the Trne Line Yx - Au + - X? .................. 5-45

CHAPTER 6

POLYNOMIAL AND MULTIVARIABLE RELATIONSHIPSANALYSIS BY THE AE,HOD OF LEAST SQUARES

6-1 INTROD UCTION .................. .................... 6-16-2 LEAST SQUARES THEOREM ...................... 6-36-3 MULTIVARIABI.E FUNCTIONAL RELATIONSHIPS .... 6-46-3.1 Use and Assumptions ................................... 6-46-1.2 Discussion of Procedures and Examples ................... 6-56-3.3 jroed rý a d........... " '..........-.. 6-G6-4 MULTIPLE MEASUREMENTS AT ONE OR MORE

POINTS .......................................... 6-176-4 POLYNOMIAL FITTING ............................... 6-186-6 INEQUALITY OF VARIANCE .......................... 6-196.-6.1 Discussion of Procedures and Examples................ 6-196-6.2 Procedures and Examples ............................... 6-20C-7 CORRELATED MEASUREMENT ERRORS ............ 6-226-7.1 Discussion of Procedures and Examples ................... 6-226-7.2 Procedures and Examples ............................... 6,22

S-% 6-8 UIE OF OhTHOGONAL POLYNOMIALS WITH EQUALLYSPACED x VALUES .................................... 6-26

"6-8.1 Discunion of Procedures pnd Examples ................ 6-26

v

TAILE OF CONTENTS WCONT) AMCP 706-110I

Parag. ýph I ge

CHAPTER 6 (Cont)

6-8.2 Procedures and Examples ............. ................. 6-806-9 MATRIX METHODS ............................. 6-376-9.1 Formuls Using Triangular F.'ctorization of Normal

E qu ations ............................................. 6-376-9.2 Triangularization of M atrices ............................ 6-386-9.3 R em arks .............................................. 6-41

A

LIST OF ILLUSTRATIONS

Fig. No. Tit Paee

1-1 Hstograin representing the distribution of 5,000 Rockweli hardnessrm ad ings ................................................... 1-7

1-2 Norm-al curve fitted to 'he distribution of 5,000 Rockwell hardnessreadings ....... .. ......................................... 1-7! -0 -.equenfcy distribution- or various shapes ........ ........

1-4 Three different normal distributions .......................... 1-81-5 Percentage of the population in various intervals of a normal

distribution ............................................... 1-91-6 Sampling distribution of X for random bamples of size n from a

normall population with mean -m ............... ............. 1-111-7 Sampling distribution of s' for samples of sise n from a normal

population with c - 1 ...................................... 1-111-8 Computed confideince intervals for 100 samples of size 4 drawn at

random from a normal population with m - 50,000 psi, - 5,000psi. Case A shows 50% confidence intervals; Case B shows 90%confidence intervals ......................................... 1-12

1-9 Computed 50% confidence intervals for the population mean mfrom 100 samples of 4, 40 samples of 100, and 4 samples of 1000.. 1-13

1-10 Computed statistical tolerance limits for 9J.7% of the populationfrom 100 samplks of 4, 40 eamples of 100, and 4 samples of 1000... 1-14

2-1 The standard deviation of some simple distributions ............ 2-92-2 Number of degreet of freedom required to estimate the standard (

deviation within P% of its true value with confidence coefficient -,. 2-12

vii

AMCP 706-110 LIST OF ILLUSTRATIONS ICONT)

Fig. No. Titl Page

3-1 OC curves for the two-sided t-test (a - .65)................... 8-63-2 OC curves for the two-sided t-test (a - .01) ................... 3-73-3 OC curves for the two-sided normal test (a - .05) ............. 8-113-4 OC curves for the two-sided nornial test (o - .01) ............ .3-123-o OC curves for the one-sided t-test (a - .05) .................. 3-143-6 OC curves for the one-sided i-test (a M .01) .................... 3-153-7 OC curves for the one-sided normal t st (a - .05) .............. 3-183-8 OC curves for the one-sided normal test (a - .00 .............. 8-198-9 Probability of rejection of hypothesis mA - B when tu ue,

plotted against. 0 .............................................. 3-25

4-1 Operating characteristics of the uue-siueu x'-tktt o oeternimewhether the standard deviation a, of a new product exceeds thestandard dcviahon ao of a standard. (a - .05) ................ 4-4

4-2 Operating characteristics of ihe one-sided it-•esf to determinewhether the standard deviation a, of a new product is less than thestandard deviation ao of a standard. (a - .05) ................ 4-6

4-3 Operating characteristics of the one-sided F-teat to determinewhether the standard dwiation Ca of product A exceeds the stand-ard deviation vB of product B. (a - .05; TIA - n,)............... 4-11

1-4 Operating characteristics of the one-sided F-test to determinewhether the standard dciatio-, T of product A exceeds the stand-ard deviation rE of product B.(a =-.05; nA -Ln, 6aA - 2nD, 2nf - nD) ....................... 4-12

4-5 Operating characteristic. of the one-sided F-test to determinewhether the standard deviation CA of pr oduct A exceeds the stand-ard deviation 'T of product B.(a - .05; nA -= nz,, 2 7tA - 3nB, nA = 2nB) .................... 4-13

5-1 Time required for a drop of dye to travel between distance markers. 5-25-2 Linear functional relationship of Type F1 (only Y affected by

measurement errors) ..................................... 5-45-3 Linear functional relationship of Type FI1 (Both X and Y affected

by m easurement errors) ..................................... 5-55-4 A normal bivariate fre4uency surface ......................... 5-65-5 Contour ellipses for normal bivariate distributions having dii.erent

values of the five parameters, mIx, 7r, 7 ax, ar, Pxr. ............ 5-75-6 Diagram showing effect of restrictions of X or Y on the regression

of Y on X .................................. .............. 5-85-7 Young's modulus of sapphire rods as a function of temperature -

an F1 relationship ....... ................. 5-125-8 Young's modulus of sapphire rods as a function of temperature,

showing computed regre.sion line and confidence interval for thelin e . .. .. .... .. . . .. . .. . . .. .. .. .. . .. . .. ... .. . . .. . . .. . . .. .. .. 5 -14

5-9 Relationship between two methods of determining a chemicalconstituent - an FlI relationship ............................ 5-28

5-10 Relationship between the weight method and the center groovemethod of estimating tread life - an SI relationship ............ 5-32

vii

LIST OF ILLUSTRATIONS (CONT) AMCP 706-110

I IFig. No. T'ite PIsge

5-11 Relationship Ix tweer. weight method and center groove method -1

the line showa" with its conifidence band is for estimating tread lifeby center grooýe method frorn tread life by weighL method ...... 5-35

6-12 Relationship betwtxn weight method and center groove method -showing the two regression lines .............................. 5-39

5-13 Relationship between weight method and eenter groove methodwhen tie range of the weight method has been restricted - a.#SIL relationship ............................................ 5-42

A

LIST OF TABLES

Table No. TiMel Pave

2-1 Table of factors for converting the range of a sa&-ape of n to an N AestimaLe of a, the populaLion standard deviation. Estimate of

- range/d . ............................................. 2-6

3-1 Summawy of techniques for comparing the average of a newproduct 'vith that of a standard I.........................

3-2 Summary of techniques for comparing the average performanceof tw o products ........................................... 8-22

5-1 Summary of four cases of linear relationships .................. 5-95-2 Computational arrangement for Procedure 5-4.1.2.1 ............ 5-175-3 Computational arrangement for test of linearity ............ 5... -225-4 Some linearizing transformations ....................... 5-81(5-5 Coinput.Lional arrangement for Procedure 5-5.1.2.1 ............ 5-875-6 Computational arrangement for Procedure 6-5.2.2.1 ............ 5-44

6-1 Sample table of orthogenal polynomials ...................... 6-28

I

viii

FOREWORD AMCP 706 110

INTRO DUCT IOH niet Ut ttit a St tion 2 prtovide's detniled pra-

I'lis s ol(,of grupof andook eoer (g edo res ior tiit uaulv.ýis and11 int rprcit t ion of

thel is iisoiern inof rnitipof wiallhok qunottverin cittnicralt ivt' and c I-ificatory- data. 8Seetion 3

d1 le ii i c rii i fhe dvijnt deeopmend , coinsltriut e- in to do wih the' pht~miiig and anlalysis of colin-

twio, tilitl test o.' mtilitary vitjuipinict Ado(t a prtivitt experiments. exeitipibjoui if a iititud to

group ) voutistit t the Armty Alateriel Commitanid ndvitoadex. mlfali -famm'r

Engneein DeignHadbok.of iti;mportanlt but as ' ekt non -si a ndard stat ist ~calEuiguccrig DsignIltii~ibok.techniquets, and to distussioti of vitriotis)tther

special topics. A-i idex for the material in alM

PURPOSE OF HANDBOOK four sect ions is placed at. the endI of 'Section 4.

'I'l(-11aidbok il xpeimetalStaiste,;hit,, Sect ion 5 conitainis all theiit' ithemnitit icl tablesTiu' I adbok o 1xptritettti Sttit i's as needed for uipjdicatit ol f tit(th procedures given

beent prepared as, all aid to seit'itist~s a110. eOlili- ill Sect ions I tiuroaguh 4ýnt'es eigttgd inA rny reearc anddcveoi- Alit uiidersttiiding, of a few basic xt at istiesl

mciii progi-awts, and especially its a1 luide autI iýlcjits as grivt'i iii Chapter 1. is neet'sssaryready reference for military anid civ'iliamu person- thrlech fIefisforsectiosi a'el

nthrl %iC hoth have hepuit fors tour jotixn isaarellie1 wtorpliavti o r experimenl ts and test p ralatinig t 11ldp io de'~dnt otf thli other Cx Fmali proctedure, test,

jto cit( I~et f at io aiwo xie of Am yi equa i pm e nts rela the lgait i t chlii i Ut' dost ribe d is illutst rated by m etita s

to tes g Iterfi uNvIlte e n a o t ge ofm e u pm en ut i on. the ' if t w orked vxaitujie. A list of autkiorit Mtive

dcxiii ittl dvclpttcnta stg'sof iroucton. reti'renvets is incluidod, where atpproplriate', at tin't' id of cavieli chpter. S te'p-lm -st .'p ii st ni citi its

SCOPE AND USE OF HANDBOOK ae1'livt'ik for attaining it statted goal, and the

Thi Hadbok i a ollctin o stttitivl oniditionsi utitlet' Nhiih it part itular pro't'dure is

Thiu rs lan"',o t s ah collect ion pofswltatisiallfe strict ly valid are' stated t'xplivit 13'. Ai atittempt isinad to ind''nt. the ex tent tc) which4 reszi!-4 oh.

sectiotis', viz: tainod by P. given proct'dure are valid to it good

A MCli 706-110, Sect iou 1, Baisic Coitetpts apiproximatijolt -when thetse votid Piotis are net.mid Anialysis of Measurement 1)ati (Chtapters fully ili t. Alterniative' lrovedtret tire~at given for1-6) cnudin astes where the mttre staundard pr'oce-

AM~l 70-1 I, Sctioi 2 Anaysi of un- duries cairiot be t rusted to yield rteliable resailts.intrt heandCltusitcatoy l~ta Chapers Thte 1:tnilbook is inteinded for the user with

7-Iativ n))asfctoyDt Catr alt engineering background who, although be has7-10) ni occasioinal need for statistical ttcelnitlite's. does

AAMCLI 706-112, Sectioni 3, Planninttg and not 1tavt' the timte or inicliiiatiton to becomke ali eX-

11-1~~rh H)Tt'landbhook has betet writtten Nwith three

AMCII 706-113, Sec'tio~n 4, Special Top~ies t~liws of iisers in iniiiid, The first ix the person(Chapters 15-23) v.ho bais had a vourse or two inl statisties, and

AMUP t~ti114,Sectoti , Tible~who wnay ever' have had somte prtiwtieal expei'i-ci ice ink ap ply inig st atist ical miit 1101 iii t he just,

Svvt ioni I prox ides anl eletatuitary itit riduv- but who does not have stotist i al idteas and tech-tioti to basic statistival concep~ts aitd furn'lishes utiques titt his, fingertipts. For hiit. the Ilatidbookfull details on stanidard statisticil technitques ,ill prov'ide a rteady referenee source of onicefor the analysis and initerprtetatiotn o' uiit-osure- ftiltii it' ideas flidt techniquijes. Thte sevond( is tile

ix

AN2CP 706-110

person who feels, or hats beeni advised, that sonme *Kmt of the present text is by Mairy G. Na-Iparticular problcin can b,_ solv(:d by means of trella, who had o~erAl1 responsibility for the comn-fairlyý sim-pkc statistieal techniques, and is in need pictiun of the final version of die Handbook.of a book that will enable himi to obtain t~he so- The o~iginal plans for coveragc, a first draft of

lutio'n to his problem with a. minimum of outside the text, and sonme original tables wer,; prepared

assistance. '.i.he Hlandb-ok should enable tiuch a by Paul N. Somerville. Chapter 6 is by Josephperson to become familiar with the statistical M. Cameron; mnobt of Chapter 1 and all of Chap-

ideas, and reasonably adept at the techniques, ters 20 and 24 are by Churchill Eisenhart.; and

that are most fruitful in his partieular line of re- Chapter 10 is basied Oil a nearly-final draft by..earch and development work. Finally, thei-e is Mary L. Epling.the individual who, as the head of, or as a mein- Other memberg of the staff of the. Statistical

ber of a service group, has rcL~ponsibility for ana- Engineering 1.4ab-natory have aided in variouslyzing and interpreting experimental and test 'days through the yeark and the assistance of all

data brought in by geie,6i~ts Hnd engineers en- who helped is gratvf%,ý, acknowledged. I1)artic-gaged in Armiy research and development work. ular mention should W made of Norman. C.

This individual needs a ready source of movdel Severo, for maistance with Section 2, and ofwork sheets tand worked examples corrosi; 3nding Shirley Youing Lehmana for help in the collectionto the more common applications of statistics, to -aiid computation of examples.free him, from the ziied of translating textbook Editorial assistaaee and art preparation were

discu~,sions into step-by-2tep procedures that can provided by John 1. Thompson & Company,bu foliowed by individuals having little or no Washington, D. C. Finial preparation and ar-

previous experience with statistical methods, rangeinent for publicatian of the Hlandbook were

It is with this lest need in mind that some performe-d by the Eugineering Handbook Office,

of the procedureg included in the Handbook have Duke. University.been explainied and illustrated in detail twice: Appreciation is expressed for the generousonce for the case where- the important qiiestionr coeaino ulrhr n uhr ngat

is whether the performance of a new material, ing permis. )n for the use of their gource niateri-

p' oduct, or process exceeds, an establishe"' stan- al. References for tables and other matetial,dard; and. again for the case where the important tahen wholly or in part, from publislied works,question is whether its pvrfwriiqnce is not up to are given on the respcetive first pages.the specified standards. Small but serious errors Elements of the U. S. Army Materiel Comn-

are often made in ehaimg;ng "'greater than" pro- inand havinig need for handbook~s may subtait,cedures into " le--% than " procedures. requisitions or officia requests directly to the

Publications and Reprodudion Agency, Lcttcr-AlUTHORSH11, AND k(CKNOWLEDG.AENTS kenny Armny DeDot. Chambcmshtirg, Pennsyl-

The Handbook on Ex~perimiental Statistics -vania 17201. Contra'ctors should submit suchwas prepared in tim? Statistial Engineering L-ab- requisitions or requests to their contracting of-oratory, National Bureau of Standards, under a ficers.contract with the Department of Arrmy. The Comments and suggestions on thir -handbook

project i-aij under the geiieral guidance of are welcome and should be addressed to ArmyChurchill Eisenhar', Chiei, 3tatistical Engineer- Research Officc-D.-rhani, Box CM, Puke Station,ing Laboratory. Durham, North Carolina 27706.

PREFACE AMCP 706-110

TIhis listing is a guide to the Section anl Chapter subject coverage in all Seetiutts of the Ilaud-book on Experimental Statistics.

Chaptcr TitleIVo.

AMCP 706-110 (SECTION 1)- BASIC STATISTICAL CONCEPTS ANDSTANDARD TECHNIQUES FOR ANALYSIS AND INTERPRETATION OF

MEASUREMENT DATA

I- -Nowe Basic Stalistical Concepts and Preliminary Considerativns2 - Characterizing the Measured Performai iec of a Material, Product, or Process3 - 4,omuparing Materials or Products with Respect to Average Performance4 - Comparing Materials or Products with Respect to Variability of Performance5 - Charaetei'izing Linear Relationships Between Two Variables6 -- Polynomial and Multh iriable Relationships, Aualysi: by the Method of Least Squaresi

AMCP 706-111 (SECTION 2)- ANALYSIS OF ENUMERATIVE ANDCLASSIFICATORY DATA

7 - Characterizing the Qualitative Performanee of a Material, Product, or Process8 - Comparing Materials or Products with Resp-et to a Two-Fold Classification of Performance

SCoiuparing Two Percentages)

9 - Comparing Materials or Products with Respect to Several Categories of Performance (Chi-SquareTosts)

V -Sensitivity Testing

AMCP 706-112 (SECTION 3)-THE PLANNING AND APALYSIS OFCOMPARATIVE EXPERIMENTS

11 - General Considerations in Planning Expriments12 - Facoria! Experiments13 - Itandomized Blocks, Latin Squares, and Other Special-Purpose Designs14 - Experiments to lDetermin, ()pt imunm Conditions or LevelsII

AMCP 706-113 (SECTION 4) - SPECIAL TOPICS

15 - Some "Short-Cut'" Teots for Small Samples from Normall Populations

16 - Some Test, Which Are Independent of the Foem of the Distribution17- The Treatment of Outliers1 - The Plave of Control Charts in Experiment. I Work19 - Statistical Techniques for Analyzing Extreme-Value Data20 --- Tile Use of Trai sfornkations ,

21 -- The Relation Bletween Confidence Intervals and Tests of Significance22- Notes on Statistical (omputat ions23-Expression of the Uncertainties of Final Results

Index

-F AMCP 706-114 (SECTION 5) -TABLES

Tables A-I thrmugh A-37

xi

A PrD 706l-11l

CHAPTER 1

SOME BASIC STATISTICAL CONCEPTS ANDPRELIMINARY CONSIDERATIONS

1-1 INTRODUCTION

Statistics deals with the collection, anal- Inductive statistical methods are used whenysis, interpretation, and presentation of we wish to generalize from a small body of

numerical data. Statistical mnethods may be data to a larger system of similar data. Thedivided into two classes--escriptive and in- generalizations usually. are in the form ofductive. Descriptive statistical methods arethose which are used to summarize or de- estimates or predictions. in this handbookscribe data. They are the kind we see used we are mainly concerned with inductive sta-everyday in the newspapers and magazines. tistical methods.

1-2 POPULATIONS, SAMPLES, AND DISTRIBUTIONS

The concepts of a population and a sample (b) A production lot of fuzes.are basic to inductive statistical methods. (c) The rounds of ammunition producedEqually important is the concept of a disri- by a particular production process.bution.•sltn (d) Fridays the 13th.

Any finite or infinite collection of individ- (e) Repeated weighings of the powderual things--objects or events--constitutes a charge of a particular round of ammunition.population. A population (also known as auniverse) is thought of not as just a heap of (f) Firings of rounds from a given pro-Wlfirirags eciflied by enumerat"g Vtham _p duction lot.after another, but rather as an aggregate In examples (a), (b), and (c), the "indi-determined by some property that distin- viduals" comprising the population are ma-guishes between things that do and things terial objects (corporals, fuzes, rounds); inthat do not belong. Thus, the term popula- (i) they are periods of time of a very re-tion carries with it the connotation of corn- stricted type; and in (e) and (f) they arepleteness. In contrast, a sample, defined as physical operations. Populations (a) anda portion of a population, has the connota- (b) are clearly finite, and their constituentstion of incompleteness, are determined by the official records of the

Examples of populations are: Marine Corps and the appropriate produc-Spe otion records, respectively. Populations (c),

(a) The corporals in the Marines on July (d), and (e) are conceptually infinite. Off-1, 1956. hand, the population example (f) would

1-1________________

A iCP 706 AI0 BASIC CONCEPTS

seein to be finitc, becauze firing is a deistruc- with these examples, note that, as a generaltive operation; but, in order to allow for v~ri- principle, the distribution of a characteristicstion in quality aming "firings" performed or a group of characteristics in a populationin accordance wir the sMie general proce- is not completely defined until the method or

diae i .i soeties seulby nalgy vit methods of measurement or enumeration in-repetitive wetghingai, to regard an actual vle r ul eildfiring as a sample of size one ft 3m a conl- The distribution of some particular prop-ceptually infinite po'ulation of "possible" erty of the individuals in a population is afirings, any one C? which rnýight have been collective property of the population; anidassociated with Asje rticular round con- so, also, are the average and other charac-ceived. In thio c 4zr ian, note that, in exam- teristics of the distribution. The mcthois ofples (e) and (f) lti'c phopulations involved inductive statistics caable us to learn aboutare not completely defir Ld until thle weighing such population characteristics from a studyand firing proceduree concerned huve been of samnples.fully specified. An example will illustrate an important

Attention to somTh- char acteristic of the claws of derived distributions. Suppose weindividuals of a ;(,pulation that is not the select 10 rounds of ammunition from a givensame for every i.idividual leadi irmnediately- lot and measure their muzzle velocities whento recognition of tile disiribution of this the rounds are fired in a given test weapon.characteristic in the population. 'Thus, the Let X be the av(-rage muzzle velocity of theheights of tile corporals in the Marines on 10 rounds. If the lot is large, there will beJuly 1, 19563, the burning times of a produc- many different sets of 10 rounds which could(tion lot of fuzes, and the outcomes of sa.cces- have been obtained from the lot. For eachsive weighings, of a powder charge ("oh- such sample of 10 ciunds, there will corre-served weights" of the charge) are examples spond an average muzzle velocity Xj. Theseof distributions. The presence or absence of averages, from all possible samples of 10,

at.I-4te o ..IuinOAn attributc is a char-acteriStic or aai indi- thnmeive.s form a dis~iuun io 8it;iividual in a population, such as "tatooed" or averages. This kind of distribution is called"not tatoced" for the privates in the Marines, the sampling distribution of X for samples o 'f

This kind of characteristic has U pnrticul '%rly size 10 from the popuistion concerned. Sim-simple type of distribution in the poipulat~on. ilarly, we may determine the range R of

muzzle velocities (i.e., the difference betweencarateristic fo oec iniida th eads to the largest and the ;mallent) for each of all

caracteariste ics fria each rite o inuiiul leasto possible samples of 10 rounds each. Thesea niarat, ivrito, tlvaiate, ranges R, (U = 1, 2, . . .) collectively deter-

exrampeso disriution in g~ive Ppreviouslyhe ine the sampling distribution of the tangeweeexamples of pouniatlei t dietve rvioutily of muzzle velocities in samples of size 10

S3imultaneous consideration of th~e rnuzv.le from the population concerned. The methodsof inductive statistics are based upon thevelocities and weights of powder cliaryes, of mah atclppeisofspindsr-

- rondsof mmuitin fo~na gvenpro~w- butions, of sample statistics such as X and F.tion process deternihtes a bivnmriate 1i1-tribution of these characteristiin in ýhe Let us summarize: A population in Sta-population. Simultaneous recognition -:1 the tistics corresponds to what in Lo~gic is termedfrequencies of each of a variety of different the "universe of discourse"-it'n what weW

tye o cidnson Friday the 1ith leaids are talking about. By the methods of in- (to a multivariate distribution. In conhiectiou ductive sta.aistics wvc tvat learn, from a study

1-2

S~I

_____ STATISTICAL INFERENCES AND SAMPLING AMCP 704-119!

of sanriples, only about population character- crasies. The population studied may be largeistics.--oni,' about collective properties of the or small, but there must be a population; andpopulLions represented by the individuals it should be well defined. The cthwa-','4risticin the samples-not about characteristics of interest must be a collective pioj.trty ofof specific individuals with unique idiosyn- the population.

1-3 STATISTICAL INFERENCEL AND SAMPLING

1-1.1 $TATISTICAL INFERENCES inferences and the methods of inductive sta-tistics. Suppose thaL four cards have been

If we were willing or able to examine an drawn from a deck of tarb and have beenentire population, our task would be merely found to be the Ace of Hearts, the Five ofthat of describing that population, using Diamonds, the Three of Clubs, and the Jackwhatever nu-mbers, figu,-es, or charts we DaodteTreo lbalteJcwaredtever Sinubers, ies ord chartsy wco- of Clubs. The s1)ecific methods discussed in

care teuse Sine i isordiariy ieon the following paragraphs will be illustratedvenient or impossible to observe every item frotis eam pwe.

frow this example.in the popmuat ,on, we. take a sample-a por-tion of thn,ý ,opulation. Our task is now to First of all, from the example, we cangeneralize from ot.- observations on this clearly conclude at once that the deck con-por ion (v•hich usr 3l1y is small) to the popu- tained at least one H.art, at least one Dia-l lation. Su generalizations ebout charac- mond, and at L. ast two Clubs. We also cantoristis moe population from a study of one conclude from the presence of the Five and

Sor more o.ples from the population are the Three that the deck is definitely not atermed 8tcistical inferences. pinochle deck. These are perhaps trivial in-

Statistic. inferences take two forms: ferences, but their validity is above questionc.-tivrates o,. the magnitudes of population and does not depend in any way on the

>:iars•iz-v •, and t•ss of hyp.the. re- .. . orawin U r-.

&r.iag opodation characteristics. BoUh areu4.*etl .r dctermining which among t,' or In order to be able to make inferences of_:;e)re cuurses of action to follow in practice a more substantial character, we must knowwhen the "correct" course is determined by the nature of the sampling ope ation thatsome particular but unknown characteristic yielded the ssn ple of four cards actually ob-of the population. tained. Suppose, for example, that the sam-

pling procedure was as follows: The cards

Statistical inferences all involve reachisig were drawn in the order listed, each cardconclusions ale¢mt population characteristics being selected at random from all the cards I(or at least a•bng as if one had reached suchi present in the deck when the card was drawn.

conclusions) from a study of samples which This delines a hypothetical population ofare known or ;ssumed to be portions of the drawin ;s. By using an apprepriate tech-population coi'cerned. Statistical inferences nique of inductive statistics--essentially, aare basically predictions of what would be "catalog" of all possible samples of four,found to be the case if the parent populations showing for each sample the conclusion tocould be and were fully analyzed with respect be adopted whenever that sample occurs--to th~e relevant characteristic or eharactr- we can make statistical inferences aboutisties. properties of this population of drawings.

3)A simple example will serve to bring out The statistical inferences made will be rig.a number of essential features of statistical orous if, and only if, the inductive technique

1-3

AMCP 706-110 BASIC CONCEPTS

used is appropriate to the sampling proce- Hence, in the present ,*as, inferences aboutdure actually employed, the parameters of thi. population. of draw-

ings may be interpreted as inferences aboutThus. by taking the observed proportion the composition of the deck. This empha-

of Clubs as an estimate of the proportion of sizes the importance of selecting and em-Clubs in the abstract population of drawings, ploying a sampling procedure such that thewe may assert: the pro-ortion of Clubs is relevant parameters of the populution of60%. Since random sampling of th- type drawings bear a known relation to the cor-asoumed assures that the proportion of Clubs responding par. meters of the real-life situ-in the population of drawings is the same ation. Otherwise, statistical inferences withas the proportion of Clubs in the deck, we respect to the population of drawings carriedmay assert with equal validity: the propor- over to the real-life population will be lack-tion of Clubs in the deck is 50%. If the deck ing in rigor, even though by luck they mayconcerned actually wzs a standard bridge sometimes be correct.deck, then in the present instance our esti-mate is wrong in spite of being the best 1-3.2 RANDOM SAMPLINGsingle estin ate available. In order to make valid nontrivial gcner-

We know from experience that with am1- alizations from samples about characteristicsples of four we cannot expect to "hit the nail of the populations from which they came,on the head" every time. If instead of at- the swnples must have been obtained by atempting to make a single-number estimate sampling scheme which insures two condi-

we had chosen to refer to a "catalog" of tionr:interval estimates (see, for example, Table (a) Relevant characteristics of the popu-A-22*), we would have concluded that the lations sampled must bear a known relation (proportion of Clubs is between 14% and to the corresponding characteristics of the86% inclusive, with an expectation of being population of all possible samples associatedcorrect 9 times out of 10. If the deck was with the sampling scheme.in fact a standard bridge deck, then our (b) Genera!izations may be drawn from

coniusm s crretan thsisac.but it--such. samplesa in acor'dance withf a givenvalidity depends on whether the sampling "book of rules" whose validity rests on theprocedure employed in drawing the four mathematical theory of probability.cards corresponds to the sampling procedureassumed in the preparation of the "catalog" If a sampling scheme is to meet these twoof answers. requirements, it is necessary that the selec-

tion of the individuals to be included in aIt is important to notice, moreover, that oample involve some type of random seiec-

stricUy we have a right to make statistical tion, that is, each possible sample must haveinferences only with respect to the hypo- a fixed and determinate probability of selec-thetical population of drawings defined by tion. (For a very readable expository dis-the sampling operation concerned. In the eussion of the general principles of sampling,present instance, as we shall see, the sam- with examples of some of the more commonpiing operation waa so chosen that the pa- procedures, see the article by Cochran, Mos-rameters (i.e-, the proportions of Hearts, teller, and Tukey"). For fuller details see,Clubs, and Diamonds) of the hypothetical for example, Cochran's book40.population of drawings coincide with the The most widely useful type of randomcorresponding parameters of the deck. selection is simple (or unrestricted) randon

sampling. This type of sampling is definedby the requirement that each individual in

Tbo A-.TbW rofor~wW in tHi handbook an the population has an equal chance of beingerAtalau to Sl" 11, AMCP 706-114. the first member of the sample; i..-er the

1-4

_ _ -- -----

STATISTICAL INFERENCES AND SAMPLING AKCP 706-110

Anrt member in selected, each of the remain- Finally-, it needs to bc. noticed that & Par-

ing individuals in the population has ani ticular sainple often qualifies as "~a sample"o

equal chance of being the second member from anyv one of several populations. For eir-

of the manple; and so forth, For a sampling ample, a sample of %t rounds from a single

pling, it is not sufficient 'that "each individual the production lot of which the rounda in

in the population have sa equal chance of that carton are a portion, and from the pro-

appearing in the sample," as is souietimes duction process concerned. By drawing these

said, but it is sufficient that "each possible rounds from the carton in accordance with

sample have an equal chance of being se- a simple random sampling schcme, we car.

lected." Throughout this handbook, we shall insure that they are a (simple) random sam-

Uassume that all samples are random samples ple from the carton, not from the produc-intesense of having been obtained by sun.- tion lot or the producti n process. Only

ple Ijan-lom Samnpling. if the p'rj6ct-a pro-.-!- ir in P. "statce rd

It caninot be oven-Ttiphasized that the ran statistical control" may our sample also be

&nnnss f asamle a ihernt n te ~ considered to be a simple randomn sample

pling scheme employceI Lo obtain the sample from the production lot atid the production

and not anl intrinsic pi ',iperty of the sample process. An a similar fashion, a sample of

itself. Experience teach a that it is not, i.a reneAV-d weighings can validly be consid-

to assume that a sampilz selected hap) ia- ered to be a random sample from the con-

ardly, without any conikci -,us plan, cal be ceptual.y infinite population of repeatedregrde asif t hd een&i~ine bysimic weighings by the same procedure only if

randm saplin. No dcs itseem to b di weighing procedure is in a state of sta-

possible to consciously diew a sample 0 tisti ýai control (see Chapter 18, in Section 4,

rantdom. As stated by Coz!Lran, MostelleL,

adTukey") I i therefore important in practice. toWe nsit o soe smblnc ofmecanial dic, kowfrom which of several possible "par-

befre e toata smpl fom n eistnt opua- ent" populationsa:;mlwsobindy4+ ,.,~,simpie random smln.Ti ouaini

one just "grabs a handful," the individuals in the termed thc U

* hadful almost always resemble one another (on the quite different from the population of inter-averege) more thani do the mnembers of a simple est, termed the target population, to whichrandom sample. Even if the "grabs' are randomly we would like our conclusions to be applica-spread around so that every individual has an equal

chance of ente~ring the sample, they: are difriculties. ble. In practice, they are rarely identica,

Since the individuals of grab samples resemble one though the difference is often smaU. A siam-

another mare than do individuals of random sman- ple from the target population of rounds ofplea, it follows (by a simple mathematical argu~- ammunition produced by a particular pro-mnent) th~at the means of grab sampldes resemble duction process will actually be a sampleone another lessn than the means of tandom -Amples

of the gmne sim, troi'n a grab sample, therefore, from _n_ O_ flfr Irdcif lo (smpewe tend to undceastimate the variability in the population), and the difference between sam-

population, although we should have to overentimu&te pled and target populations will be smallerit in order to obtain valid estimates of variability if the sampled population comprises a largerof grab sample meansa by substituti.&g such an CSti number of production lots. The further thernate into the formula for the variabil ity of means smldpplto srmvdfo h

of sir.ple random san~plea. Thus, using simple ran- smldpplto srmvdfo hdom sample formulas for grab sample inca~ns intro- target population, the more the burden of

duces a double bias, both parts of which lead to an validity of conclusions is shifted from theanwarrantted appeArance of higher stability. shouilders of the statistician to those of the

Instructions for formally drawing a sample subject matter expert, who must place

Jat random from a particular population are greater and greater (and perheno -nwar-

given in Paragraph 1-4. ranted) reliance on "other considerations."

1-r


1-4 SELECTION OF A RANDOM SAMPLE

As has been brought out previously, the ing the numbers. Any direction may bemethod of choosing a sample is an all-im., used, provided the rule is fixed in advanceportant factor in determining what use can and is independent of the numbers o4curring.be made of it. In order for the techniques Read two-digit number& from the table, anddescribed In this handbook to b> valid as select for the sample those individuals whosebases for making statements from samples numbers occur until 10 individuals have beenabout populations, we must have unrestricted selected. For example, in Table A-36, startrandom samples from these populations. In with the seco,•d pate of thlt Tabhl (p. T-83),practice, it is not always easy to obtain a column 20, line 6, and read down. The 10pr aticenmphi is n , a given population. items picked for the sample would thus b1Unconscious selections and biases tend to numbers 88, 44, 13, 73, 89, 41, 85, 07, 14,enter. For this reason, it is advisable to use and 47.a table of random numbers as an aid in se-lecting the sample. Two tables of random The method described is uplicable for

numbert which are recommended are by obtaining simple random samples from any

L. H. C. Tippett(0 and The Rand Corpora- sampled population consisting of a finite set

tion(". These tables contain detailed instruc- of individuals. In the case of an infinite

tions for their use. An excerpt from one of sampled population, these procedures do not

these tables•'• is given in Table A-36. This apply. Thus, we might think of the sampled

sample is included for illustration only; a population for the target population of

larger tabsk should be used in any actual weighings as comprising all weighings whichproblew. Repeated use of the same portion might conceptually have been made during ( )of a table of random numbers will not the time while weighing was done. We can-satisfy the requirements of randomness. not by mechanical randomization draw a

S~random sample from this population, and soAn illustration of the method of use of

+o,-bes1 nf r ~ ia_,,m.ber follows. Suppose must recognize that we have a random sam-

the population consists of 87 item s, and w e -l ..... -r.. T...5

wish to select a random sample of 10. ' Assign will be warranted if previous data indicate

to each individual a separate two-digit num- that the weighing procedure is in a state of

ber behveen 00 and 86. In a table of ran- statistical control; unwarranted if the con-dom numbers, pick an arbitrary starting trary is indicated; and a leap in the dark ifplace and decide upon the direction of read- no previous data are available.

1-5 SOME PROPERTIES OF DISTRIBUTIONS

Althaugh it is unusual to exasnine popula- gram. Suppose we have a large number oftions in their entirety, the examrination of a observed items and a numerical measure-large sample or of many small samples from ment for each item, such as, for exarmple, aa population can give us much information Rockwell hardness reading for each of 5,0l0about the general nature of the population's specimens. We first make a table shcwingcharacteristics. the numerical measurement and the num-

One d&vice for revealing the general na- ber of times (i.e., frequeacy) this meaaurke-ture of a population distribution is a histo- ment was recorded.

~-.-----..-- .-- ---

f

SOME PROPERTIES OF DISTRIBUTIONS AMCP 706-110

RockwellHardnessNumber Frequency

6b 156 1757 13568 503

59 1,11060 1,47061 1,12062 49063 125

F126 Ua..~.

65 ROCWZLL HARNESS NUMBER

Data taken, by perni,,kmn, from Samupling lipeetio" b*Variab by A. H. bowker and H. P. Gotmd, Copyright, 1962.

McGraw-Hill Hook Company, Inc. Figure 1-1. Histogram representih the dis-

tribution oj 5,000 RockiweU hardnessreadings.

Rleproduced by permiaalon from Samelinea Inspectisea I Voin-able by A H. Wlicker and H. P. Goods. Copyrialti 1kI*

M.Craw-Hill Book Compan.-r. Ine.

From this frequency table we can make thehistogram as shown in Figure 1-1. Theheight of the rectangle for any hardnessrange ib determined by the number of itemsin that hardness range. The rectangle iscentered at the tabulated hardness value. If

- ,P .- - 'Iwe take trhe sumn of all tLe i-ectAnziilam

to be one square unit, then tie area of anindividual rectangle is equal to the propor-tiun of items in the sample that have hard-ness values in the corresponding range. AWhen the sample is large, a&s in the present 'instance, the histogram may be taken to Iexemplify the general nature of the corre- =esponding distribution in the population.

If it were possible to measure hardness in t t t•tiner intervals, we would be able to draw a ROCKWItLL HARCESS NUMSERlarger number of rectangles, smaller inwidth than before. For a sufficiently large tsample and a sufficiently fine '"mesh," we Figure 1-f. Norml cure fitted to the dis-would be justified in blending the tops of the tribueion of 5,000 Riocfktwell hardnes -

rectangles into a continuous curve, such as readings.that shown in Figure 1-2, which we couldexpect to more nearly represent the under- ,prdumd tv nusao, from Samit amIs.nby Vari-

able. by A. H. Bowk.r and H. P. Good. Copyright. 19E.lyingpopulation distribution. MXGraw-Hill Book Co.,pany. Inc.

1-7


If we were to carry out this sort of schemeon a large number of populations, we would At i a 0. - 4

find , hat many different curves would ari, , -" = -

as 1l.uatrated in Figrre 1-,3. Possibly, the - 0 C

ina.urity of them would resemble the clawsof symmetrical bell-shaped curves called"dirmal" or "Gaussian" distributions, an ex-ample of which is shown in the center of ___ ., __

Figure 1-3. A normal distribution is unl- -4 0modal, i.L., has only a single highest point mIor mode, as also are the two asymmetrical Fipure 1-. Three diffcrtt norrai.arves in the lower left and upper right of distributions. '

Figure 1-4.

A '%( rmal" distributio-1 is completely de- 1termined by two parameters: m, the arith- in is also the mode and the value which di-inetic mean (or simply "the mean") of the viles the area un,.¢r the curve in half, i.e.,distribution, and a,, the standard deviati n the median. It is useful to remember that(often termed the "population mean" and a- is the distance from m to either of the two"population standard deviation"). The vani- inflection points on the curve. (The inflec-

ance of the distribution is cr'. Since a nor- tion point is the point at which the curvemAhl curve ia both unimodal and symmetrical, changes from concave upward to concave

downward.) This is a special property ofthe normal distribution. More generally, themean of a distribution m is the "center ofgravity' of the distribution; o, is the "radiusof gyration" of the distribution about m, inthe language of mechanics; and o-2 is thesecond moment about m.

-,- -",, ,,, The parameter m is the location parnm-exfn"A P•InTVA WD42All~TII PO44ivl

MAWAU ,eter of a normal distribution, while o- is ameasure of its spread, scat-ter, or dispersion.Thus, a change in m merely slidea the curve

right or left -without changing its profile,- while a change in ar wi.ens or narrows the"ymc'. ccurve without changing the location of its

center. Thiee different normal curves are/• k shown ina Figure 1-4. .(All normal curves in

this section are drawn so that the area un-der the curve is equal to one, which is a

standard convention.) o-

•,•,, •,, .. m •-•,,Figurea 1-5 shows the percentage of ele-mentq of the population contained in various

intervals of a normal distribution. z is thedistance from the population mean in units of

Figure 1-.. Frequency distributions the standard deviation and is computed usingof "arious shapes. the formula z = (X-m) /r, where X repre- N

A4*ptod With po naaiou fru.o El,,vw% of Statiasl Smeisiu sents any value in the population. Using z (I)Jcaw L* & rSaps. Inc to enter Table A-i. we find P, the proportion .

1-8

SOME PROPEKTIES OF DISTRIBUTIONS AMCP 706-110

of elements in the population which havevalues of z smaller than any given z. Thus,as shown in Fig. 1-5. 34.13% of the popula-tion will have values of z between 0 and 1(or between 0 and -1) ; 13.59% of the popu-lation, between I and 2 (or between -1 and A-2); 2.14% betweer 2 and 3 (or between -2and -3) ; and .14% beyond 3 (or beyond -8).Figure 1-5 abowa these percentages of the 13.59%population i urious intervals of z. C

2.14"7For example, suppose we know that the

chamber pressures of a lot of ammunuition 14%may be rep-esented by a normal distribu-tion, with die a-erage chamber pressure m = -3 -2 -1 0 1 2 350,000 psi and standard deviation or = 5,000 N M

psi. Then z = '-0 nd we know (Pig.r",00()

1-5) that if we fired the lot of ammunitionin the prescribed maniner we would expect609% of the rounds to have a chamber pres-sure above 50,000 psi, 15.9% to have pres-sures above 55,000 psi, and 2.3%o to have Figure 1-5. Percentage of the populatin, inpressures above 60,000 psi, etc. various intervals of a normwl distribution.

1-9

AMCP 70o-1 10 . BASIC CONCEPTS

tS1-6 ESTIMATION OF m aid o

In areas where a lot of experimental work Nearly evury sample will contain differ-hms been done, it of Len happens that we know ent ;,.dividualh, and thus the estimates Xin or tr, or both, fairly accuratelý. However, and s' of m and u-2 will differ from sample

'in the majority of cae it will be our task to sample. However, these estimates arto ustimate them by means of i. sample. 6up- such that "on the average." they tend tj bepose we have n observations, X,, X,, . X. equal to rn and a-', respectively, and in thistaken at random from a normal population. sense are unbia~sed. If, for example, we haveFrom a sample, what are the besi. estimates a large number of random samples of sizeof m and a-? Actually, it is usual to corn- %, the average of their respective estimatespujre thc best unbiased estimates of m and a-, of cra v'ill tend to be near cr2. Furthermore,and then take the square root of the estimate the amount of fluctuation of the respectiveof Wr. Ps the astimate of o. These recom- u*'s about W (or of the X's about m, if wemended estimates of i and a- are:* are estimating m) will be smaller in a cer-

tain well-deflned sense than the flu.luaiionwould be for any estimates other thmi the

-.. recommended ones. For these reasons, Xand a' are called the "best unbiased" esti-mates of m and a-', respectively.*

S(X,, XAs might be expected, the larger the sam-__-__ ple size i, the more faith we can put in the

•-1 estimates •"and as. This is illustr.,ted in • J

Figures 1-6 and 1-7. Figure 1-6 shows theX and 81 are the aamaple meav and sample distribution of X (sample mean) for samples

-....e. nt.... .... of ,a. sczeo from 'he a•a- i normnaldis-

called "the sample standard de-iation," but tribution. The curvo for n = 1 is the disti-this is not strictly correct and we shall avoid bution for individuals in the population. All-the expression and simply refer to s.) Fo- of the curves are centered at m, the popula-ccmputational puiposes, the following for..mula for as is more convenient:

"On the other hand,: a Is not an unbiased esti-

mator of v. Thus, in samples of size n from a nor-m-al distribution, the situation is:

S1 4 - s is an unbiasedSampie size, n *niim Q uf:

* The Greek symbdL Z is often used as shorthand 0.797 0

to* "the sum of." For example, 4 0.021

X., = X, + X, + X. + X. 6 0.952

7 0.9598 0.965

S(•, +Y, yo v.+Y. (X, + Y, ) + +X Y,) 9 0.,+Y0S9 0.969•-,10 0.978

20 0.87

X, Y, -XY-" XY + Xly , 40 0.9+460 0.996120 0.998

c+ + 0 1.0-0

1-kCONFIDNCEW ,vr~OVAI C A %K• fOn.._l I J

tion mean, but th~e scatter becomes less-asn

gets larger. Figure 1-7 shows the distribu-tion of m (sarb ple variance) for samples of

various sizes from the same normaJ distri-bution. -.-----

4 3

163

o ~,2

Figute 1-6. Sampling distribut ion of I for Figure 1-7. Sampting di~stribution of sl forrandom somatnp of size n from a itormal sample size nfrom a normal population

polmhzioin wth mean m. with or 1

PRap~umed by voertatuion from Tho M@tAw& ofa S.' m islieom 1 sm..m

Son*. Inos. W . U"asw^lw

_ .1

1-7 CONRDENCE INTERVALS

Tnmureh "6 eatim-att-n d nsriin of ovary Ply the proporion of b sampleBi of sOie forfrom samiple to sample, interval estimates which intervals computed by the prescribedof m and a- may sometimes be preferred to method may be expected to bracket M (or"single-value" e~stimates. Provided we have 0". Such inte-vals are known as confidencea random sample from a normal population, intervals, and always ar associated with awe can make 'Interval estimates of m or c" prescribed confidence coefficiert. As wewith a chosean degree of confidence. The level would expect larger samples tend to giveof confidence is not associated with r. par narrower coiuidence intervals for the sameticular interval, but ih associated with the level of confidence.method of calculating the iiterval. The in- Supae we amr given the lot of ammnuni-terval obtained f rom a particular sample tipn mentioned earlier (Par, 1-e ) and wisheither brackets the true piaraeter value to make a confidence interval estimate of"(in or a", whichever we are estimating) or the average chamber pressure of the roundsdoes not. Tae confidence coefficient y is tim- in the lot. The true average is w0,000 psi,

wih cosn egeeofcofdece Te evl oud xpc lrgr amle tnd-11gv

AWCP "ov"'4101 BASIC CONCEPTS

although this value is unknown to us. Let taken, and the resulting confidence intervalsus take a raidom sample of four rounds and computed from each. If we ceomplite 50% 1from this sample, using the given procedure, (90%) confidence intervals, then we expectcalculate the upper and lower limits for our 60% (90%) of the computed intervals toconfidence interval. Cousider all the posai- cover the true value, 50,000 psi. See Fig-ble camples of size 4 that could have been ure 1-8.

60,000

50,000 11 1111,11 1 11 1 1 .1 , .

40,000 1- . 1 , • I . A.

CASE A.rO % CONFIDENCE INTERVALS

"60,000

40VO00,

30000 I . , * , I , I * I , I1

0 10 2O SO 40 50 60 70 80 90 100

CASE B.90% CONFIDENCE INTERVALS

Figure 1-8. Computed ronfidence intervels for 100 8ample' of size 4 drawn at random froma normal population with m = 50,007 psi, a = 5,000 psi. 'ase A shows 50% confidence

intervals; Case I? ,,ows 90% confide&-c intermals, (A UptW4 with pernimaio from AST# Ma n4l an Qsalty Controi of Mlatetad. Copyright. 1951, American Bodety for Tuting

I 1-12

-_7 77

CONFIDENCE INTERVALS AMCP 706-110

In Case A of Figure 1-8, 51 of the 100 statistics X and s. both of which vary fromintervals actually include the true mean. For sample to sample. If, on the othe] hand,60% confidence interval estimates, we would the standard deviation of the iopaplatIonexpect in the long run that 50% of the inter- distribution a were known, and the con-vals would include the true mean. Fifty-one fidence intervals were computed from the

out of 100 is a reasonable deviation from the successive X's and o, (procedare given inPar. 2-1.6), then the resulting confidenceexpected 50% . In Case B, 90 out of 100 of i t r as w ud al b h a e wd h neintervals wivuod all be the same width, and

Sthe intervals contain the true mean. This is would vary in position only.precisely the expected number for 90 % inter-vals. Finally. as the sample size increases, con-

fidence intervals tend not only to varm liaNote also (Fig. 1-8) that the successive in both position and width, but alas to

confidence intervals vary both in position "pinch in" ever closer to the txue value ofand width. This is because they were corn- the population parameter concerned, as iflus-puted (see Par. 2-1.4) from the sample trad i. Figure 1-0. i

m i.o

m-Ior n4fIO

-r 2a -1 •4 n100 1000

Figure 1-9. Compttd 50% confde e intterv~a for thoefp ation men m from 100_ miges of 4, 40 samples of.0o, an~d 4 sample of 1oo.000

Ad*wthd parmimion froan Stotivel M*tAN.fom the Viewpoiid of u Co%,91 by W. A. Sbart (ed by W. 34war*Deoblv), Cocyrrtht, li$. Or.vAuat 3cbA*L U.S. DeplartMet of Asrtieta.m Wuamipsto. D. 0.


14 STAI!NTICAL TOLERANCE UMITS

Sometimes what is wanted is not an asti- ,pproximately 99.7% of the chamber pres-mate of the mean and vwiance of the popu- sures lie within teese limits (*ee Fig. 1-5).lation distribution but, instead, t-ro outer If we do not know m ard a-, then we mayvalues or limits which contain nearly all of endeavor to approximate *the limits withthe population values. For example, if ex- atati-itical tolerance limits of the formtremely low chamber pressures or extremely X - Ks and X + Ks, based on the samplehigh chamber pressures might cause serioue statistics X and 8, with K chosen so that weproblems, we may wish to kiAow approxi- may expect these limits to include at least Pmate limits to the range of chamber tc tea- percent of the chamber pressures in the lot,sures in a lot of ammunition. More sPe- at some prescribed level of confidence a.citlcally, we may wish to know w.hin whatlimits 99%, for example, of the chamber Three sets of such limits for P = 99.7%,pressures lie. If we knew the mean in and corresponding to sample sizes n = 4, 100,tandtrd devistion ar of chamber prumures and 1,000, axe shown by the bars in Figure

in the lot, and if we knew the dist ibution 1-10 It should be noted that for samples ofof chamber pressures to be normal (or very size 4, the bars are very variable both inxearly normal), then we could take m - flar location and width, but that for n = 100 andand rn + So- as our limits, and conclh:dv• that s 1,000, Jiey are of nearly constant width

7I4 1

I f Hal no.)I1b. I 11+11 II II

I JIul I ff I a Is II Ilk LILH'

I,. I ,- fill-,

0) no 0O 400 5

-SAMPLE NIIDER

Fi~l4S 1-0. Crn paed ~~ La -,? limits for 99.? % of the populatim from100

saarplea of sszt 4, 46 &'M14'30 -h'ae 100, and 4 samples o.f =se 1000.

&4PAtad witk pecmainid l fom, St•lg,*io J ifth0"d tf,- iv 9 . 1jun,, QmUtp Caorol by W. A. 8hewhart (ed•Itd by W. Edward CR)miim),• 'Wi0tt lM. Unuat4, . 1.S. Departmeat of .i-in*tUri. WWaingtat, D. G.

1 14

USING STATISTICS TO MAKE DECISIONS AMCP 706-11 U

and position--and their ý','d points approxi- should be noted. A woll.1-, ienr1.• --! I. An

mate very closely to m ZV and m + 8&. In interval within which we estimate a givenother words, statisti u&t Wlerance intervals populatioA parameter to liq (e.g., the popula-tend to a fixed 3ize d',i, ,'depends upon P) tion mean w, with respect to some character-as the sampkl siz,- Z , whereas con- istic). Statistical tolerance limits for a givenfidence intervals • Jvni towards zero population are limits within which we ex-width s.iz. en., e.cL •; s'ai 1 iz, as illus- pect a stated proportion of the populationtrated in Y 4tr.-, .-9. to lie with respect to some measurable cbar-

actelistic. Engineering tolransce limits areThe difference in the mea.i•go of the specified outer limits of acceptability with

terms conftdea intervals, statistical toler- respect to some characteristic usually pre-ance limaita, and ergineerinp toler•nace liniat scribed by a design engineer.

1-9 USING STATISTICS TO MAKE DECISIONS

1-9.1 APPROACH TO A DECISION PROBLEM on the basis of the results obtained from

Consider the following more-or-less typical these ten a' alls alone, whether to keep on

practical situation: Ten rounds of a new making the standard shells or to convert

type of shell are fired into a target, and the our equipment to making the new shell, how

depth of penetration is measured for t5c can we make a valid choice?round. The depths of penetration arc 11.0, A very worthwhile step toward a solution11.1, 10.5, 10.5, 11.2, 10.8, 9.8, 12.2, 11.0, in such situations is to compute, from theand 9.9 cmo The average penetration dept, data in hand, a confidence interval for theof the comparable standard shell is 10.0 cm. unknown value of the population parameterWe wish to know whether the new type shells of interest. The procedure (given in Par.penetrate far••her On the arage t ithe 02-1.4) j-0 the .. reK...ig tstandard type shells.] penetration data for the new typc of shell

If we compute thei arithmetic mean of the yields the interval from 10.18 to 11.22 cm.ten shells, we find it is 10.70 cm. Our fzrt as a 95% confidence interval for the popu-impulse might be to: state that on the aver- lation mean depth of penetration of shellsage the new shell 'will penetrate 0.7 cm. of the new type. Inasmuch a: this intervalfarther than the standard shell. This, in- lies entirely to the right of the wean fordeed, is our best single guess, but how sure the standard shell, 10.00 cm., we are jus-can we be that this actually is close to the tifled in concluding that the new shell is,trut•? OVIe .. ing tha • mhtI c tch our nn the averare, better than the standard,is the variability in the individual penetra- with only a 5% risk of being in error."tion depths of the new shells. They range Nevertheless, taking other considerationsfrom 9.8 cm. to 12.2 cm. The standard devi- into account (e.g., cost of the new type, costation as measured by s calculated from the of changing over, etc.), we may concludesample is 0.73 cm. Might not our sample of finally that the improvement--which may baeten shells have contained somie atypical ones as little as 0.18 cm., and probably not moreof the nev typc which have unusually high than 1.22 cm.-is not sufficient to warrant

- penetrati~ig power? Could it be that the new conversion to the new type. On the othershell is, on the average, no better than the hand, the evidence that the new type is a&-

_ standard one? If we were obliged to decide, most certainly better plus the prospect that

1-15


the improvement may be asa ieat as 1.22 cm. However, all statistical decision probleinsma serve to recommend furthor develop- are not anenable to solution via confidencemental activity in the direction "pioneered" intervals. For instance, the question at issueby the new type. may be whether or not two particular char-

acteristi a of shell performartte are mutuallyA aomewhat different approajh, which independent. In such a situation, fmy one

provides a direct answer to our quAtLion of a variety of tests of significance can be4TMould it be that the new shell is on the used to test the null hypothesis of "no de-

4*S.rage no better than the standar'?" but pendence." Some of these may have a rea-not to the question of whether to convert to soriably good chance of rejecting the nullthe new type, is to carry out a, so-called test hypothesis, and thus "discovering" the ex-of siaCiti'.lmc e (or test of a statistlcal hy- istence of a dependence when a dependencepothesio). In the case of the foregoirg ex- really exists-even though the exact natureample, the formal procedure for the corte- of the dependence, if any, is not understoodsponding test of signifIcance (Par. 3-2.2.1) aad a definitive measure of the extent ofturns out to be equivalent (as explained in the dependence in the population is lacking.AMCP 706-118, Chapter 21) to noting whetheror not the confidence interval computed does A precise test of significance will be poasi-or doea not include the population mean for ble if: (a) the sampling distribution of somethe standard shell (10.0 ca.). If, as in the sample statistic is known (at k Ast to a good

-present instance, the population mean for approximation) for the case of "no depend-the standard shell is not included, this is ence"; and (b) the effect of dependence ontaken to be a tu gative answer to our quee- this statistic is known (e.g., tends to maketion. In other words, this is taken to be it larger). For a confidence-interval ap-conclusive evidence (at the 5% level of vig- proach to-be-possible, two conditions areitifeance) against the null hypothesis that necessary: (a) there mutt be agreement on"4the new shell is on the average no better what constitutes the proper measure (pa-than the standard." Rejection of the null rameter) of dependence of the two charac-•zypot'ieas in this ease is equivalent to acteristics in the population; and, (b) thereeepting the indefinite alternative hypothesis must be a sample estimate of this depend-that "the nswv aL..011io-bt~,o h ente -aapi- -hm nupln Ua-U-2than the standard." If, on the other hand, is known, to a good approximation at least,

the population mean for the standard shell for all values of the parameter. Confidenceis included in the confidence interval, this intervals tend to provide a more completeis taken as an affirmative answer to our answer to statistical decision problems whenquestion-not in the positive sense of defi- they are available, but tests of significancenitely confirnting the null hypothesis ("is are of wider applicability.zo better"), but in the more-or-less neutralsense of the abssnce of conclusive evidence 1-9.2 CHOICE OF NULL AND ALTERNATIVEto.....ntar. .IyrOtHISE$.S.

As the foregoing example illustrates, an A statistical test always involves a nulladvaztage of the confidence-interval ap- f,.pothesis, which is considered to be theproach to a decision problem is that the con- hypothesis under test, as against a class of"Aldence interval gives an i..dication of how alternative hknothe.6es. The null hypothesislrg the difference, if any, is likely to be, acts as a kind of "origin"' or "base" (in theand thus provides some of the additional sense of "base line"), from which the alter-information usually needed to reach a final native hypotheses deviate in one way or an-decision on the action to be taken neA. For other to greater and lesser degrees. Thus,many purposes, thiit is & real advaitege of in the case of the ceassical problem of theconlidence intervals over tests of significance. tosuing of a coin, the imll or base h~pothesis

1-16

W UA*V e7AYUCTwtC A WMP I rr I I, a- .

sp*_iflfi that the probability of "heads" on the level of significance of the test. The riskany single trial equals 1/2. If, in a par- of making an error of •isi second kind,ticular situation, the occurrence of "heads" var."i-, &.3 oi;e u•oli expcct, with the naUni-were an advantage, then we might be par- tude o& the real differe"e•, and is SumUL&-ticularly inteiested in the one-sided claw of rized Ly tht Operatinji Chkiwcteristie (WC)alternative hypotheses that tAe probability Curve of the teat. See, for example, Figureof "heads" on any single trial equals P, 3-5. Also, the risk P of making an error ofwhere P is some (unknown) fraction ex- the second kind increases as the risk a ofceeding 1/2. If neither "heads" nor 'tails" making an error of the first Miad decreases.were intrinsically advantageous, but a bias Compare Figure "-5 with Figure B-6. Onlyin favor of either could be employed to ad- with "large" samples ccn we "have our cakevantage, then we could probably be inter- and cat it too".--and then there is the costeated in the more general two-smded class of the test to worry about.of alternative hypotheses specifying that theprobability of "heads" on any single toss 1-9.5 CHOICE OF THE SIGNIFICANCE LEVELequals P, where P is some fraction (lessthan, or greater than, but) not equal to 1/2. The significance level of a statistical test

is essentially an expression of our reluctanceThe important point is that the null hy- to give up or "reject" the null hypothesis.

pothesis serves as an origin or base. In the If we adopt a "stiff" significance level, 0.01coin-tossing instance, it also happens to be or even 0.001, say, this implies that we area favored, or traditional, hypothesis. T1is very unwilling to reject the null hypothesisis merely a characteristic of the example unjustly. A consequence of our ultracon-

- selected. Indeed, the null hypothesis is often hervatisn in this respect will usually be thatthe very antithesis of what we would really the probability of not rejecting the null hy-like to be the case. pothesis when it is really false will be large

unless the actual deviation from the null

1-9.3 TWO KINDS OF ERRORS hypothesis is lMge. This is clearly an en-tirely satisfactory state of affairs if we are

In basing decisions on the outcomes of quite satisfied with the status quo and arestatistical tests, we always run the risks of only interested in making a chazge if theinalklu -•-Le- o-e or the other of two types- cha:! rorerintx a very substantial im-of error. If we reject the null hypothesis provement. For example, we may he quitewhen it is true, e.g., announce a difference satisfied with the performance of the stand-which really does not exist, then we make an ard type of shell in all respects, and not beError of the First Kind. If we fail to reject willing to consider changing to the new typea null hypothesis when it is false, e.g., fail unless the mean depth of penetration of theto find an improvement in the new shell over new type were at least, say, 20% betterthe old when an improvement exists, then (12.0 cm.).we make what is called an Error of the Sec-ond Kind. Although we do not know in a On the other hand, the standard shell maygivea instance whether we have made an te unsatifactory in a n--irofr.-iiectserror of either kind, we can know the prob- and the question at issue may be whetherability of making either type of error. the new type shows promise of being able

to replace it, either "as is" or with furtherdevelopment. Here "rejection" of the null

I-4.4 SIGNIFICA.NCE LEVEL AND OPERATING hypothesis would not imply necessary aban-CHARACTERISTIC (00 CMRVE OF donment of the standard type and shiftingA STAIISTICAL TESTA-S A Eover to the new type, but merely that the

( The risk of making an error of the first new type shows "promise" and warrants-' kind, a., equals what is by tradition called further investigation. In such a situation,

1-17

AMCP 706-110 BASIC CONCEPTS (91one could afford a somewhat higher risk of outcome anything whatsoever about the oddsrejecting the null hypothesis falsely, and for or against some particular set of wondi-would take a = 0.05 or 0.10 (or even 0.20, tions being the truth.perhaps), In the interest of increaning the Indeed, it is astonishing how often errone-chances of detecting a small but promising ous statements of the type "since r exceedsimprovement with a smal-,cale experiment, the 1%6 level of significance, the odds are 99In such exploratory work, it Is often more to 1 that there is a co.-relation between theimportant to have a good chance o." detecting variables" occur in reseaith literature.. Ho;\a small but promising improvement than to ridiculous this type of reasoning can be isprotect ones If against crying "wolf, wolf" brought out by the following simple exam-occasionally-because the "wolf, woll"' will ple("): The American Experience Mortalityjbe found out in due course, but a promising Table gives .01008 as the probability of an vapproach to improvement could be lost for- individual aged 41 dying within the year.

ever. If we accept this table as being applicable toliving persons todas- (which is analogous to

In summary, the significance level a of accepting the publ shed tables of the signif-a statistical test should be chosen in the icance levels of tests which we apply to ourlight of the attending circumstances, includ- data), and if a man's age really is 41, thening coot!. We are sometimes limited in the the odds are 99 to 1 that he will live out thechoice of significance level by the availability year. On the other hand, if we accept theof necessary tables for some statistical tests. table and happen to hear that some promi-Two values of a, a = .05 and a = .01, have nent individual has just died, then we e*nnotbeen most frequently used in research and (and would not) conclude that the odds arcdevelopment work; and are given in tabula- 99 to 1 that his age was different frm 41.tions of test statistics. We have adopted Suppose, on the other hand, that in somethese "standard" levels of significance for official capacity it is our practice to checkthe pur s of this handbook, the accuracy of age statements of all persons

who say they are 41 and then die within the£ -'." pAa........ Cum',g tflanuu,*.

Many per•na who regularly employ ata- bility of the American Experience Mort.illty.tistical tests in the interpretation of research Table) will lead us in the long run to suspectand development data do not seem to realize unjustly the word of one person in 100 whosethat all probabilities associated with such age was 41, who told us so, and who thentests are calculated on the supposition that was unfortunate enough to die within thesome definite set of conditions prevails. year. The level of significance of the test isThus, a, the level of significance (or proba- in fact 0.01008 (1, in 100). On the otherbility of an errur of the first kind), is corn- hand, this practice will also lead us to dis-puted on the assumption that the null hy- cover mis-statements of age of all personspothesis is strictly true in all respects; and professing to be 41 who are really some other,, the risk of an error of the second kind, age and who happen to die within the year.is computed on the assumption that a par- The probabilities of odr discovering suchticular specific alternative to the null hypoth- mis-statements will depend on the actualesis is true and that the statistical test con- ages of the person• makii.g them. We shall,cerned is carried out at the a-level of signifi- however, let slip b3- as correct all statementscance. Consequently, whatever may be the "age 41" corresponding to individuals whoactual outcome of a statistical test, it is are sot 41 but who do not happen to diemathematically impossible to infer from the within the year.

- 1-18

USING STATISTICS TO MAKE DECISIONS AMCP 706-110

The moral of this is that all statistcal the graLt ri•sk invobled in drawing oonclu-tests can and should be viewed in ternis of sivos frovm . exc... in •yn- sm... a '---o

the consequences which may be expected to comes manifest to anyone who takeos theensue from their repeated use in suitable time to study the OC curves for the statia-circumstances. When viewed in tids light, tical tests in common use.

REFERENCES

1. W. G. Cochran, F. Mosteller, and J. W. " 3. L H. C. Tippett, Random Sampling Nsu-Tukey, "Principles of Sampling," Jour- bers, Tracts for Computers, No. 15,nal of the American Statistical As- Cambridge University Press, 1927.8ociation, Vol. 49, pp. 13-5, 1954.(Copies of this article can be obtoined 4. The Rand Corporation, A Million Ran-from the American Statistical Asci- dom Digits, The Free Press, Glencoe,

ation, 1757 K St., N.W., Washington 6, Ill., 1955.D. C. Price: 50 cents.) 5. C. Eisenhart, "The Interpretation of

2. W. G. Cochran, Sampling Techniques, Tests of Significance," Bulleti, of theJohn Wiley & Sons, Inc., New York, American Statistical Association, Vol.N. Y., 1953. 2, No. 3, pp. 79-80, April, 1941.

SOB E RECO( i)M ND EJ,4NTA9 _ '" A 1 .) m7 r"Fnrf~raL~t~rn~1VIN1YL3 EJUJV-L11A I I LX'&11 & jJJ'JV1 A IjnW

A. H. Bowker and G. J. Lieberman, Engi- M. J. Moroney, Facts from Figures, Penguinneering Statistics, Prentice-Hall, Inc., Books, Inc., Baltimore, Md., 1951.Englewood Cliffs, N. J., 1959. L. H. C. Tippett, The Methods of Statietics,

4th edition), John Wiley & Sons, Inc.,W. J. Dixon and F. J. Massey, Jr., Introduc- New York, N. Y., 1952.

tion to Statistical Analysis (2d edi- W. A. Wallis and H. V. Roberts, Statistics,tion), McGraw-Hill IDook Co., Inc-, A New Approach, The Free Press,

S'T ,. .. "% r '" ("ph~n _ fI. 1XT A_

()111-19

AMCP 706-110

I |

DISCUSSION OF TECHNIQUES

IN CHAPTER5 2 THROUGH 6

The techniques described in Chapterb 2 Table A-37 is a table of three-decimal-placethrough C, apply to the analysis of results of random normal deviates that axemplify sam-experiments expressed as me&isurcinents in pling from a normal distribution with zerosomp convenltional uniLs on a continuous mean (n- = 0) and unit standard deviationV They d& not apply to the analysis of (cr " 1). To construct numbers that will sim-data in the form of proportions, percentages, ulate measurements that are normally dis-or counts. -trihuted about a true value of, say, 0.12, with

It is assumed that the underlying popula- a standard deviation of, say, 0.02, multiplytion distributions are' normal or nearly the table entries by 0.02 and then add 0.12.normal. Where this assumption is not very im- The reader who wishes to get a feel for theportant, or where the actual population dis- statistical behavior of sample data, and totribution would show only slight departure try out and judge the usefulness of particu-"

from normality, am. indication is given of the lar statistical techniques, is urged to carryeffect upon the conclusions derived from the out a few "dry runs" with such simulateduse of the techniques. Where the normality measurements of known characteristics.assumption is critical, or where the actualpopulation distribution shows substantial de-parture from normality, or both, suitable All A-Tables referenced in thew Chaptersw-•l5are given, are contained in AMCP 706-114, Section 6.

2-i

AMCP 706-110

CHAPTER 2

CHARACTERIZING THE MEASURED PERFORMANCE Or-

A MATERIAL, PRODUCT, OR PROCESS

2-1 ESTIMATING AVERAGE PERFORMANCE FROM A SAMPLE t

2-1.1 GENERAL Example: Ten mica washers are taken atrandom from a large group, and their

In this Chapter we present two important thicknesses measured in inches:kinds of estimati.s of the average perform-ance of a material, product, or process from .123 .132a s&mple. These include the best single esti- .124 .123mate, and confidence interval estimates.* .126 .126

.129 .129Specific procedures are given for obtaining .120 .128confidence interval estimates when:

(a) we have a sample from a normal In general, what can we say about thepopulation whose variability is unknown; larger group on the basis of our sample?and, We show how to answer two questions:

(b) we have a sample from a normal (a) What is our best guess as to the aver-population whose variability is known. age thickness in the whole lot?

When the departures froet nor.a.ity arc 0 Caii W wiv c n iC a ..te.! iih MAY

not great, or when the sample si-•es arc expect, with certain confidence, to bracket Imoderately large, interval estimates made the true average--i.e., a con 6dence interval?as described in Paragraphs 2-1.4 and 2-1.5 These two questions are answered in th2will have confidence levels very little differ- paragraphs which follow, using the dataent from the chosen or nominal level, shown above. Another question, which is

The following data will serve to ill-ustrate sometimes confused with (b) above, isthe application of the pirocedures. treated in Paragraph 2-s. This is the ques-

tion of setting statistical tolerance limits, orcst4m;an,. oan $nyu-al whir, will includle,

Data Sample 2-1--Thlckness of Mica Washers with prescribed confidence, a specified pro-

A portion of the individual items in the pOpU-Form: Measurements X 1, X1, . .. , X, of n lation.

items selected independently at ran-

dom from a much larger group. 2-1.2 REST S11GLF ESVýMATE

The niost .%mmon and ordinarily the best"The readr who is not familiar with the mean- single estimate of the population rieahn m iI

lug and interpretation of confidence intervals shouldrefer to Chipter 1, and to Paragraph 2-1.3 of this simply the arithmetic mean of the measure-Chapter. ments.

2-1

ANALYSIS OF MEASUREMENT flATA AMCP 706-110

Pr1cedlu of interval estimates might likewise be called

Compute the arithmnAic mean X of the t, "99% confidence intzrvals." Similarly, ifmeasurementsB x Y,...,x our intervals included the true average

"95% of the time"--strictly, in 95% of the1 4; times or instances involved-we would be

Is M operating at a 95% confidence level, and ourintervals would be calk 1 95% confidence in-tervals. In general, if in the long run we

Compute the arithmetic mean X of 10 expect 100(1 - a) 0¾ of our intervals to con-Smepaurements (Data Sample 2,1): tain the true value, we are operating at the

.123 + .124 + .126 + ... + .128 100(1 - a) to confidenrc level.10 We may choose whatever confidence level

1.260 we wish. Confidence levels y commonly used10 are 99% and 95%, which correspond to

.1260 inch a = .01 and a - .05. If we wish to estimatethe mean of some characteristic of a largegroup (population) using the results of a

INTERVAL ESTIMATES random sample from that group, the proce-

When we take a sample from a lot or a dures of Paragraphs 2-1.4 and 2-1.5 willpopulation, the Lw1nple average will seldom allow us to make intenral estimates at anybe exactly the same as the lot or population chosen confidence level. It is assumed thataverage. We do hope that it is fairly close, the characteristic of iDterest has a normaland we would like to state an interval which distribution in the population. We may electwe are confident will bracket the lot mean. to make a two-sided interval estimate, ex-If we made such interval estimates in a pected to bracket the mean from both aboveparticular fasnion a larje number of times, and below; or we may make a ozie-sided in-

2ad found that these intervals actuall3 did terval estimate, limited on either the uppereon•ain. the true -near. in 9%/V of thc e, or the lower side, whic iS expeCWJe eor-

we might say that we were operating at a tain the mean and to furnish either an upper$99% confidence level. Our particular kind or a lower bound to its magnitude.

2-1.4 CONFIDENCE INT54VALS FOR THE POPULATION MEAN WHEN KNCWLEDGE

OF THE VARIABILITY CANNOT BE ASSUMED

2-1.4.1 Two-Sided Confidence IntervalTh".- prullre _ives an interval which is expected to bracket m, the true mean, 100 (]-a) %

of the time.

- Procedure ExamplehPr-oin: What is a two-sided 100 (1 - a) % Problem: What is a two-sided 95% confidence

confidence interval for the true mean m? interval for the mean thickness in the lot?(Data Sample 2-1)

(1) Choose tL- desired conflder-e level, 1 - a (1) Let 1 - a - .95a- •

(2) Compute: (2)1, the arithmepLii mean (see Paragraph .1260 inch

2-1.2), and4' (-1) (6

S 4 L-Y- -I-- a.X k2.- 0.00369 inch

n(

2-

i. -

-Al~

AMCP 706-110 CHARACTERIZING MEASURED PERFORMANCE

( )Procedure

(3) look up t ,- i for n- 1 degrees of (3) t- t., for 9 dqrees o freedomfriedom* in Torle A4. -2.262

(4) Compute: (4)

xV - . + t - Xu - .1260 +

- .1286 inch

XL = - Xz = .1260 2.262 (.00W59)

- .1234 inch

Conclude: The interval from XL to Xu is a Conclude: The inmerval from .1234 to .1286 inch100 (1 - a) % confidence interval for the ii a 95% confidence interval for the lot mean;population mean; i.e., we may assert with i.e., we may assert with 95% confidence that100 (1 - a) % confidence that XL < m < Xv. .1234 inch < lot mean < .1286 inch.

2-1.4.2 One-Sildeod Confide-ce IntervalThe preceAing computations can be used to make another kind of confidence interval state-

ment. We can say that 100 (a/2) % of the time the entire interval in Paragraph 2-1.4.1 willlie above the true mean (i.e., X).,, the lower limit of the interval will be larger than the truemean). The rest of the time--namely 100 (1 -a/2) % of the time-Xi will be less than thetrue mean. Hence the interval from XL, to + a is a 100(1 -a/2) % one-sided confidence in-terval for the true mean. In the example, Paragraph 2-1.4.1, 100(1 -o/2) % equals 97.5%.Thus, either of two open-ended intervals, "larger than .1234 inch," or "less than .128b inch"can be called a 97.5% one-sided confidence interval for the population mean.

We now give the step-by-step procedure for determining a one-sided confidence interv•,for the popuiaLlurtiuaia PLIVU--ren cULLlaiuci no t-nnreea levl.

• In A Dt~iona- of Statis'ical T7crn, 41) we find the following, under the phrase"degrees of freedom":

"This term is used in statistics in slightly different senses. It was introduced byFisher nn the analogy of the idea of degrees of freedom of a dynamical system, thatis to say the number of independent coordinate values which are necessary to deter-mine it. In thin sense the degrees of freedom of a set of observations (which exAVot.heJi aem aubject to tamplinc variation) ia the number of values which could beassigned arbitrarily within the specification of the system; for example, in a "m_4pleof constant size n grouped Into k intervals there are k - 1 degrees of freedom be-cause, if k - 1 frequencies are speciflled, the other is determined by the total size a;

A sample of is variate values is said to have n degrees of freedom, whether thevariates are dependent or not, and a statistic calculated from it is, by a natural exten-sion, also maid to have n degrees of freedom. But, if k functions of the sample valuesawe held constant, the number of degrees of freedom Is reduced by k. For example,

the statistic (xi - ij whers i in the sample mean, is said to have t - 1

degros of freedom....

In) nthisexaznple,' a (X -X)'/( - 1) and has "-t " dog•a of freedom

9.-3

ANALYSIS OF MEASUREMENT DATA AMCP 706-110

PIcsduv, Example

-ProW*%: What is a oneided 100 (1 - a) % Problem: What is a value whiclh we expect, withconfidence interval for the true mean? 99% confidence, to be exceeded by the lot

mean? (Alternatively, what is a value which weexpect, with 99% confidence, to exceed the lotmean?) (Data Sample 2-1)

(1) Choose the desired confidence level, 1 - a. (1) Let 1 - a = .99a 0.01.

(2) Compute: (2)- .1260 inch

8 - 0.00359 inch

(3) Look up t - t.- for n - I degrees of free- (3) t = t.ts for 9 degrees of freedomdomn in Table A4. - 2.821

(4) Compute: (4)

xil t 8 Xt-- .1260

(or compute -. 1228

X• - : -t- •_'/(or X•ff .1292)

CoedSu: We are 100 (1 - a) % confident that Condu&d: We are 99% confident that the lot (the lot mean m is greater than XL (or, we are mean is greater than .1228 inch (or, we are 99%

i- - 00 (1 - x) % confident that the lot mean in confident that the lot mean is less than .1292

is leas than Xu), i.e., we 'nay assert with inch), i.e., we may assert with 99% confidence_1 a) - _; __ confdenc- t'ju,-^- **h> . . .I• an th!ckness in lot > .1228 inch (or,

that m < XU). that mean thickness in lot < .1292 inch).

2-1.5 CONFIOENCE INTERVAL ESTIMATES W IEN WE H'AVE PREVIOUS KNOWLEDGEOF THE VARIABILITY

We have assumed in the previous paragraph (2-1.4) that we had no previous infor•nation-'-A+- .... n- Pqtimate of vari-

about the variability of performance of items and weye ,, .... . . .... -a -L...

ability obtained from the sample at hand. Suppose that in the case of the mica washers wehad taken samples many times previously from the same process and found that, althougheach lot had a different average, there was always essentially the same amount of variationwithin a lot. We would then, be able to take a,, the standard deviation of the lot, as known andequal to the va'ue iLdicated by this previous experience. This assumption should not be madecasually, but only when warranted after real investigation of the stajiity of the variationamong samples, usiaZ control chart tcehniques.

The procedure for computing these confidence intervals is simple. In the procedures ofParagraph 2-1.4, merely replace 8 by o- and t by z anC' the formulas remain the same. Valuesof z are given in Table A-2. Note that t, for an infinite number of degrees of freedom(Table A-4) is exactly equal to z,. The following procedure is for the two-sided confidence (interval.

2-4

AMCP 706-11O CHARACTERIZING MEASUkUL) PERFORMANCE

Procodum Iwmtl

Prob/em: Find a two-sided 100 (1 - a) % confi- Problem: What is a two-sided 95% confidenredence interval for the lot mean, using known a. interval for the lot mean? (Data 8Unple 2-1;

aud a is known to equal .0040 inch,)

(1) Choose the desired confidence level, 1 - ,. (1) Let 1 - a - .95a - .05

(2) Compute: (2)X1?- .1260 inch

(8) Look up z - z2,i in Table A-2. (3) z.t- 1.960

(4) Compute: (4)

XU - I + z X - .1260 + 1.960(.04

- .1285XL - z i- XL -. 1235

Condu&: The interval from XL to Xu is a Condude: The interval from .1235 to .1285 inch100 (1 - a) % confidence interval for the lot is a 95% confidence interval for the lot mean.mean.

Disewsion. When the value of cr, the standard deVation in the population, is known, Pro-cedure Z-1.5 should always be used in preference to Procedure 2-1.4, which is independentof our knowledge of a-. Wheni a.'a.lable, Procedur .... 6- ..a, M _now) will umu 'sa ' ',W' tconfidence interval for the population mean that is narrower than the confidence intervalthat would have Leen obtained by Procedure 2-1.4 (o- unknown). This is the case for ourillustrative exanmples based on Data Sample 2-1, but the difference is very slight because

o, and s were both very small--only 0.03 9 of the mean.

Whatever level of confidence is chosen, the t value required for t.e application of Proce-dure 2-1.4 (a, unknown) will always be larger than the corresponding z value required forProcedure 2-1.5 (a, known). This is evident from Table A-4. For very small samples, thediffcren ,4 l.n • millrghle- Nevertheless, it can happen, as a result of unusual samplingfluctuations, that the value of a obtained in a particular sample is so small in comparison to

a that, if Procedure 2-1.4 (ar unknown) were used, the resulting confidence intervalwould be narrower than the confid•cne interval given by Procedure 2- 1.5 (ca known). Thiswould have been the case, for instanc, if Data Sample 2-1 had yielded an a less than1.960(0.0040)/2.262 = 0.00347. With samples of size 10 (i.e., 9 de,'rees of freedom for 8),the probability of such an occurrence is about one in three. In such a case, however, onemust NOT adopt the confidence interval corresponding to Procedure 2-1.4 (ar unknowm) be-cause it is narrower. To choose between Procedure 2-1.4 (c unknown) and Procedure 2-1.5(ow known), when the value of o, IS known, by selecting the one which yields the narrower) • confidence interval in each instance, would result in a level of confidznce somewhat lowerthar. claimed.

2-5


2-2 ESTIMATING VARIABIUTY OF PERFORMANCE FROM A SAMPLE2-2.1 GENESAL a sufficient number of decimal places. If

We take the standard deviation of per- too few places are carried, the subtractionsformance in tho population as our measure of Involved may result in a loss of significantthe characteristic variability of performance. figures in &'. Excessive rounding may evenPresented here are various ways of estimat- lead to a negative value for s'. The formulaing the population standard deviation, in- recommended for computational purposes iscluding: to be preferred on this sccount because only

(a) single-value estimates; one subtraction is involved; and with a desk(b) confidence-interval estimatca based calculator one usually can retain all places in

on random samples from the population; the computation of ZX,' and (ZX,'.and, We take

(e) techniques for estimiating the popula- X" 'X)2~ (tion standard deviation when no appropriate a - -random samples are available. 1)The first two procedures are illustrated by as our estimate of oa, the population standardapplication to the following data. deviation.

Example: Using Data Sample 2-2, ZX• -•Date Sample 2-2---Snlng Time of 27987.54, ZX4 = 519.8, and thus ci = 107.593;

Rocket Powder and a = 10.87 seconds.*Form: it independent measurements X,, X,, 2-2.2.2 The Sample Range as an Estimate of

.. X, selected at random from a the Standard Deviotlonmuch larger group. The range of n observations is defined as

Example: Ten unit amounts of rocket pow- tieder selected at random from a large the difference between the highest and thelot were tested in a chamber and TABLE 2-1. TABLE OF FACTORS FOR CONVERT-

-* their burning times recorded as fol- R1lG THE RANGE OF A SAMPLE OF n TO AN- lows (seconds): ESTIMATE OF o, THE POPULATION STANDARD

DEViIATVN• $TIMATE OF a - RANGEAiL.60.7 69.8 Size of54.9 .4 Sample d 164.8 66.1 a [See Not*]44.8 48.142.2 35.5 2 1.128 .8865 1.414

8 1.693 .5907 1.7322-2.2 SINGLE ESTIMATES A 2.059 .4857 2.0002-2.2.1 a$ and s 5 2.326 .4299 2.236

The best estimate of Lr, the variance of a 6 2.534 .3946 2.449ii uaz .... P... U1a. .0 n, 7 2.704 .3698 2.646

R, 2.970 .8367 3.000(____ -., s'- - 10 3.078 .8249 8.1620-1 is - 1 12 8.258 .3069 S.464

For computational purposes, we usually find 16 8.532 .2831 4.000it more convenient to use the following for- Not.: d. is approximately equal to I/Wfor 8 xmcda: z 10. Thus, for small n a quick estimate of a can

( InXa 'lbe obtained by dividing the range by rud8 - In e tina repodb values of h should be rounded

is (n - 1) to two significant figures, but as a basis for furtherealculations it Is advisable to retain one or two addi-

The formulas are algebraically identical. tional flgures, For fuller explanation, see ChoptersWith any formula, it is important to carry 22 and 28, Swrou 4, AMCP 706-113.

2-6

4~


lowest of the n values. For small samples, AU the aam-ple ca,- getz lar-er tLhe .,ne_ isthe samnple range is a reasonably efficient not only troublesome to calculate, but is asubstitute for 8 as an estimator of the stand- very inefficient estimator of CF. Table 2-1ard deviation of % norm,., population-not as givos the factors which convert from ob-efficient as a, but easier to calculate. Using served ringe in a -ample of % observationsthe range is particularly valuable for a to an estimate of population standard devia-"quick look" at data from small samples. tion or.

2-2.3 CONFIDIENCE INTERVAL ESTIMATES*

2-2.3.1 Two-Sided Confidence Interval Estimates

We are interested in determining an interval which we may conlldefUy expect to bracketthe true value of the standard deviation of a normal population.

Procedure !bnampie

Problem: What is a two-sided 100 (1 - a) % Problem: What is a 95% confidence interval forconfidence interval for v? v, the standard deviation of burning time in the

lot of powder? (Dal a Sample 2-2)

(1) Choose the desired confidence level, 1 - a. (1) Let 1 - a - .95a - .05 -

) (2) Compute: (2)

"- =n -. (: z 2) a = 10.87 secondsnt (n - 1)

(3) Look up Bu and BL for n -1 degrees of (3) Fqr 9 degrees of freedom,freedom in Table A-20. B L - .6657

Bu -1.746

(4) Compute: (4)SL - BLS 8z - (10.37) (.6657)

= 6.90 secondsSU - BUS So - (1037) (1.746)

- 18.11 seconds

Condlude: The interval from SL to SU is a two- Conclude: The interval from 6.90 to 18.11 is asided 100 (1 - a) % confidence interval estimate two-sided 95% confidence interval for a; i.e., wefor (r; i.e., we may assert with 100 (1 - a) % may assert with 95% confidence that 6.90confidence thaL sz, < T < au. seconds < r < 18.11 seconds.

2-2.3.2 One-Sided Confidence Interval Estimates

In some applications we are not particularly interested in placing both an upper and alower bound on o, but only in knowing whether the variability is excessively large (or,exceptionally small). We would like to make a statement such as the following: We can btate

S ) ' The reader who is not familiar with the meaning and interpretation of confidenc' Intervals should refer•- to Chapter 1, and to Paragraph 2-1.8 of this chapter. The remarks of Paragraph 2-1.8 c,ncer•ing confidence

intervals for thz average carry over to confidence intervals for a meature of variability.

2-7

ANALYSIS OF MEASUREMENT DATA AMCP 706410

with 100 (1 - a) 9, confidence that the variability as measured by cr is less than some valuesb computed from the sample. Similarly, but not simultaneously, we may wish to state with100(1 - a) 9b confidence that or is greater than sume value s8. Either statement is a one-aided confidence interval estimate.

Pr•mcjure ExamplePrwNem: What is a value #6 such that we may Probkvm: What is a value s6 such that we havehave 100 (1 - a) % confidence that a is less 95% confidence that a is less than sý? (Datathan 80? Sample 2-2)

(1) Choome the desired confidence level, 1 - a. (1) Let 1 - a - .95a - .05

(2) Compute: (2)6 a - 10.37 seconds

(8) Look up AI,- for n - 1 degrees of freedom (3) For 9 degrees of freedom,irn Table A-21. A.m, 1.645

(4) Compute: (4)Sjr A,-1 .. s6 - (1.645) (10.87)

- 17.06 seconds

,(5) With 100 (1 - c.) % confidence we can as- (5) We are 95% confident that the variabilitys ert that s is less than aj. as measured by a is ess Mhan s - 17.06

second&.(

Should a lower bound to o- be desired, follow Procedure 2-2.3.2 with s8 and A,., replacedby s! and A., respectively. Then it can be asserted with 100 (1 - a) 'jo confidence that c,> .2.

2-2.4 ESTIMATING THE STANDARLD DEVIA- (a) Estimate two values a, and b, betw n m

TION WHEN NO SAMPLE DATA AIE which you expect 99.7% (almost all) of tWe

AVAILAbLE individuals to be. Then, estimate

It is often necessary to have some idea of as 6the magnitude of the variation of some char-acteristic as measured by o-, its standarddeviation in the population. In plan ui ex- (b) Estimate two values as and b. between-periments, for example, the sample size re- which you expect 95% of the individualb? tquired to meet certain requirements is a be. Then, estimate Jas a2 -b.1function of a-. In ahnost any situation, one 4can get at least a very rougi, estimate of o-.The minimum necessary information in- If the distribution concerned cannot be as-volves the form of the dibLribution and the sumed to be normal but can be a~sumed tospread of values. For example, if the values follow one of the top four fornm in Figureof the individual items can be assumed to 2-1, then the standard deviatio'. may be esti-follow a normal distribution, then either of mated as indicated in the figivre. This figurethe following rulee can be used to get wn alao illustrates the distribution and rules for (2estimate of or: (a) and (b) above.

2.8

-- in I_-I I

NL:

&MICp -, 04 10 CURCF~ZIGMASIJRFD PERFORMANCE

DISTRIBUTION STANDARD V 7-DEVIATION

£ b

4.2

4.2

b•a

4.9

" _•• b,6 - as

as b

Figure 2-Y. The standard deviation of somesimple distributions.

Ada~p wvith parm•lo frWO SyoIe ?•/#ro 5i, byW. Edwards Dqming. Copjrigbt. 1960. JohnM1U44y Zons Inv.

2-3 NUMIEiR OF MEASUkEMENTS REQUIRED TO ESTABLISHTHE MEAN WITH PRESCRIBED ACCURACY

2-3.1 GENERAL wIn planning experimlenlts, we miay nead to accuracy. Suppose we are willing to allow

know how many measurements or how large a margin of error d, and a risk a that our

"a sample to take in order to determine the estimate of m will be off by an amount d or

meau of some distribution with prescribed greater. Since the sampling distribution of

2-9

ANALYSIS OF MEASUREMENT DATA AMCP 706-110 --

X is "normal" to a good approximation for Sometimes we may have available to ussamples of four or more measurements from one or more samples from the population ofalmost every population distribution likely interest, from which we can derive an esti-to be met in practice, we can ascertain the mate e of oa based on r degrees of freedom.required sample size n. if we have an avail- Other times we may have one or more sam-able estimate s of o, or if we are willing to pies from some other population that has theassume that we know a-. If we have not same standard deviation as the populationmade an estimate or are unwilling to assume of interest, but possibly a different mean.a value for o-, then we must use a two-stage Again, we can derih e an estimate s of (rsample. The two-stage method will usuaily based on v degrees of freedom. In eitherresult in a smaller total sample size. In the case, we can utilize this preliminary estimatetwo-stage method, we must start by guessing of o- to determine the sample size n requireda value of o-, but the end results ,lo not de- to estimate the mean of the population ofpend upon how good or bad is the guess. interest with prescribed accuracy.

2-3.2 ESTIMATION OF THE MEAN OF A POPULATION USING A SINGLE SAMPLE

Procedure ExampleProblem: We wish to know the sample size re- Problem: We wish to know the average thick-quired to ascertain the mean m of a population. ness of the washers in a given lot. We are willingWe are willing to take a risk a that our estimate to take a risk that 5 times in 100 the error in ouris off by d or more. There is available an esti- estimate will be 0.002 inch or more, From amate s of the population standard deviation a, sample from another lot we have an estimate ofbased on a degrees of freedom. the population standard deviation of s - .00359 K)

with 9 degrees of freedom.

S (1) Choose d, the allowable margin of error, (1) Let d - 0.002 inchand a, the risk that our estimate of m will a .05be off by d or more.

T(2) L eoo up ii-ls for a, degrees of freedoai in (2) = 1-.,, for 9 degrees of fre-edenmTable A-4. -fi2.262.

(B) Compute. (8)

J2 ss =(2.262)3 (.00359) 16.d' (.002)'

- 17 (conventionally rounded up tothe next integer.)

Conclude: If we now compute the mean X of a Conclude: We may conclude that if we now corn-random sample of size % from the population we pute the mean X of a random sample of sizemay have 100 (1 - a) %0 confidence that the n = 17 from the lot of washers, we may haveinterval I -- d to .9 -+- d will include the popu- 95% confidence that the interval . - .002lation mean m, to .9 + .002 will include the lot mean.

If we know a,, or assume some value for ar, replace a by or and t,-., by z,.I in the aboveprocedure. Values of zI./,I are given in Table A-2.

2-3.3 ESTIMATION USING A SAMPLE WHICH IS TAKEN IN TWO STAG-$It In possible that we do not have a good estimate of o-, the standard deviation of the popu-

lation. When the coat of sampling is high, rather than take a larger sample thau is really O fnecessary, we might prefer to take the sample in two stages. The method (sometimes called

2-10

A CP 706-110 CHARACTERIZING MEASURED PERFORMANCE

Stein's method) goes roughly as follows: Make a gavs for the .va-r w, o•. , , a ....mine n. the size of the first sample. The first sample will provide an estinuate a of the popula-tion standard deviation. Use this value of a to determine how large the second sampleshould be.

Procedure ExamplI

Problem: We wish to knowr the sample size re- Problem: We have a larg 1ot of devices, andquired to ascertain the mean rn of a population, wish to determine the average ef some property.We ame willing to take a risk a that our estimate We are willing to take a risk of .05 of tho esti-is off by d or more units. mate being in error by 30 wita3.

(1) Choose d, the allowable magin of errox, (1) Let d - 80and a, the risk that our estimate of m will a - .05be off by d or more.

(2) Let a' be the best possible guess for the (2) From our knowledge of similar devi- * ourvalue of o, the standard deviation of the best estimate of a is 200 units.population (see Paragraph 2-2.4).

(3) Look up z,_/, in Table A-2. (3) z.*&, - 1.960

(4) Compute: (4)

_d J(30),i t' is the first estimate of the total sample 170.7size required.

(5) Choose n, the size: of the first sample. n% (5) Let n, = 50should be considerably less than n'. (If theguessed value of a is too largc, this will pro-tect us agairzt a first aample which isalready larger than we need.) A rough rulemnightL +.e t e nj >_ 111-1DL unes v' <~ buin which case let n, be somewhere between.5n' and .7n'.

(6) Make the necessary observations on the (6) From tests on 50 Jevices chosen at random,sample of na. Compute a,, the standard si - 160 units.deviat-on.

(7) Look up ti.i,/, for n, -- 1 degrecs of freedom (7) 1 - t.97 for 49 degrees of freedomin Table A-4. - 2.01.

k8) Compute()

d- (80),., 114.9'- 115

s is the total required sample size for the firstand second samples combined. We then %, - 115 -50require a second uample size ofa , - n - #I. - 66

We will require an additional 65 devices tobe tested.

2-11

ANALYSIS OF MEASUREMENT DATA AMCr 706-110it

If now we obtain the sc )jd sample of siz. n. and compute the mean X cf the total Sampleof size n = n, + n,, we may have 100 (1 - a) % confidence that thl interval X -- d toX + dwill include the population mean In.

2-4 NUMBER OF MEASUREMENTS REQUIRED TO ESTABLISHTHE VARIABIUTY WITH STATED PRECISION

We may wiah to know the size of sample required to estimate the standard deviation with

certain precision. If we can express tihis precision as a percentage of the true (unknown)standLrd deviation, we can use the curves in Figure 2-2.

S•~~~~ooo - "-- "

500

400_ _ "-300 -

___oo_________ __

200 --- •

S%. _

esim VAte th stahdard dtIato wihi% of itsl!

109Figure 2-.r Number of degrees ,o Vo. ( rtuIr. to

esimat th It-r deiaio wihi P of' itsl•tru vau wit cofdec cofiin Iy

Adpti it prisio fom Jored of th ,_,in Sz, . A . u~w. Vl -] -46--5) .ts. irlteetild"anl

Sze. Reamired for Estimating the Standard deevattin as a Perceit of Its Trit Vakr b J. A. GreenworA sod X. U. Sanclomim.

"T0 inner of 1graphl I adapvted with permiwlon from ,t.*a.fwm Ma&, a by E. L. Crow, F. A. Da•.I. and I. W. MaxIld,

N A V O RD _eport

-|$i9. 948. U . 8. N aval O ron ina St .. . . D Publi l

2-12 .

%4^" "it"•, wAACTFRIZII4G MEASUUP PERFORANC.

Problem: I we he.mpa

Probem; If we are to make a simple sries of Problem: How iarge a samnple w4u

measurements, how many measurements are re- to estimate the standard deviatio within W)%quired to estimate the standard deviation with- of its true value, with confidence coeffieientin P percent of its true value, with prescribed equal to 0.95?confidence?

(1) Specify P. the allowable percentage devia- (1) Let P - 20%tion of the estimated standard deviationfrom its true value.

(2) Choose -r, the confidence coefficient. (2) Let , - .95 4

(8) In Figure 2-2, find P on the horizontal (8) For y - .95, P 20%, the required de-scale, and use the curve for tie appropriate grees of freedom equals 46.v. Reed on the vertical scale the requireddegrees of freedom.

(4) For a simpl- series of measurements, the (4) is - 46 + 1required number is equal to one plus the - 47degrees of freedom.

_ ) 2-5 STATISTICAL TOLERANCE LIMITS

2-5.1 GENERAL

Sometimes we are more interested in the Thus for the data on thickness of micaapproximate range of values in a lot or popu- washers (Data Sample 2-1), we could givelation than we are in its average value. Sta- two thickiess values, stating with chosentistical tolerbnce limits furnish limits be- confidence that a proportion P (at least) oftween, above, or below which we confidently the washers in the lot have thicknesses be-expect to find a pribtd poGp0rtiono ,, in- twin these two limits. We call the confi-dividual items of the population. Thus, we dence coefficient y, and it refers to the pro-.night like to be able to give two values A portion of the time that our method willand B between which we can be fairly cer- result in correct statements. If a normal dis-tain that at leasL a proportion P of the popu- tributiorn can be assumed, use the procedureslation will lie, (two-sided limits), or a value of Paragraphs 2-5.2 ard 2-5.$; otherwise useA above which at least a proportion P will the procedures of Paragraph 2-5.4.lie, (ene-sided limit).

2-5.2 T.-9;Dr.0D TOLERACE L!MfITS MR A NORMAL DISTRIBUTION

When the mean m and standard deviation o, of a normally distributed quantity are known,symmetrical limits that include a prescribed proportion P of the distribution are readilyobtained by adding and subtracting z. a from the known mean mn, where z. is read fromTable A-2 with a = I (P+1). When m and a- are not known, we can use an interval of theform X _ Ka. Since both X and s will vary from sample to sample it is impossible todetermine K so that the limits X _ Ks will always include a specified proportion P of the) -underlying normal distribution. It is, however, possible to determine K so that in a iongseries of samples from the same or different normal distributions a defi•,ite proportion y ofthe intervals X ± Ks wil! include P or more of the underlying distribution (s).

2-13


Procedum Exemple

Problon: We would like to state two limits. Problem: We would like to state thicknms limitsbetwi.n %hich we are 100 y percent confident between whicih we are 95% confident that 90%that 100 P percent of the values lie. of the values lie (Data Sample 2-1).

(1) Choose P, the proportion, and y, the confi- (1, Let P - .90dcrce coefficient. ' - .95

(2) Compute from the sample: (2)_ .1260 inca

8 a = 0.00359 inch

(3) Look up K for chose- Pmid j in Table A-6. (3) K 2.339

(4) Compute: (4)r Xu -. 9 + K8Xv - .1230 + 2.839 (.00859)I = 0.136 inchXL f -Ks XL f .1260 - 2.83V (.00359)

= 0.116 inch

Conclu&: With 100 7 % 'onfidence we may pre- Contlude: With 95% confidence, we may say

dict that a proportion P of the individuals of the that 90% of the washers have thicknesses be-population have values between XL and Xu. tween 0.116 and 0.136 inch.

r.k

2-55. ONE-SIDED TOL. RANCE LIMITS FOR A NORMAL DISTRIBUTION

.. ometimes we are interested only in estimating a value above which, or below which, aproportion P (at least) will li.ý In this cage the one-sided upper tolerance limit will beX, =- A + Ks; and X, = A - K3 val' be the one-ýideu lowr -Iuit'. The appropriivatz2e ....for K are given in Table A-7 and are not the saone as those of Pfragraph 2-5.2.

Procedurs EtampleProblem: To find a single value above which w( Problem: To find a single value ab3ve which we

iy predict with confidence -y that a proportioI' may predict with 90% confidence that 99% ofP of .he population will lie. the population will lie. (Data Samnple 2-1).

(1) Chor-. P the proportion and y, the confi- (11) Let P = ,99Sdence coefficient. ' = jj0

.(2) Compute: (2'IX 1~ .1260 inch. s 8 0.00359 inch

(6) Look up KC iii Table A-7 for the appropriate (3) K (10, .90,.99) = 3.532i•, 'y, a". P.

(4) X,. - Ks (4) Xz= .1260 - 3.6,32 (.00359)

. xM>3inch.Thus we are 90% confident that 99% of theS • ~mica washers will have thickne-•ses above

:: ] .113 inch.

V }2-14

IA


Note: Factors for s1.me values of n~, y, and P not .zovered in Table A-7 may be fourad inSandia Corporation M~onograph SCR143 M. Alternatively, one ulay com~pute Ij using tLa fol-lowing formulas:

# -~- -h---- (where z can be found in Table A-2)

K _ 2, + VZP - dlba3

2-5.4 TOLERANCE LIMITS WHICH ARE INDE- of n 60, then we may have a confidence ofPFNDENT OF THE FORM OF THE at least y = .95 that 100P%) 759o of theDISTRIBUTION population will lie between the fifth largest

(8 = 5) and the fifth smallest (r = 5) of theThe methods given in raragraphs 2-5.2 sample values. That is, if we were to take

and 2-5.3 are buied on the assumption that riany randon, samples of 60, and take thethe observations come from a normal distri- fifth largest and fifth smallest of each, webution. If the distribution is not in fact should expect to -find that at least 957o of thenorir al, then the effect will be that V.ie true resulting intervals would contain 75% of theproportion P of the population between the population.tolerance liinits -will vary from the intended Taible A-32 may be useful for sample sizes( ) P by an hmount depending on the amount of o :ý10 hstbegvstecniecdeparture from rPormality. If the departure of wi~th which teabl givest thetconfidencefrom normality is more than slight we can wthe whplaich wies mabertwe the atgn a0% douse a procedure which assiurnes only that. the thales- pouatilies betee the largptlandistributik n has no discontinuities. The tol- sals auso h ape

................ la ..ts s...btaineei will be- substantially -4L OiedtTgrrikb~

wider than those aswviming normality. (Disttibution-Fre.)

2-5.4.1 Two-Sided Tolerance limits Table A-31 gives the largest value of m(Distribution-Free) such that we may assert with confidence at

le-A yt that 100P% of a population lies be-Table A-30 gives values (r, s) such that low the ?nih largest (or above the M1tb small-

we may assert with confidence at least y that est) of a random sample of 'n from that pop-1001Po of a population lies between the rtb1 ulation. For example, from Table A-31 withsinallest. and the sth larvest of a random sam- y = .95, P = .90, and 7t = .90, we may saypie of n from that population. For example, that we are 95% confident that 90% ol afroma Table A-'_0 with ~y=.95, P =.75, and population will lie below the fifth hrgsest-n 60, we may say that. if we have a s&,t-nple value of a sample of size -74 50.

REFERENCLSF ~1. M. G. Kevdall and W. R. Buck-land, A. Dictiwiony of Statistical 'fetr~,v

2. D 79, Oliver and Boyd, London, 11.934.2D.B. Owen, Table of' Factors for ()n~e-Sided Tolierance Limuits for a

Normwl Distribution, Sandia Corporation Monograph SCR713, April1958.

2-15b

Imo - N, w~

CHAPTER

COM ARNGMATRILSORPRO UCS IT *. SJK TPQ *W 7 1

AVE A9PTERFOMAC

341 GENERAL REMARKS ON STATISTMAC TEST5

One of the most frequent uses of statistics test will result in making one of two deci-Lis in testing for differences. fe we wish to sions, as in the pairs giveu. In each case theknow whethier a treatment applied to a stand- pa~ir of a0ternativc decisions is chosen beforeI ard round affects itt- muzzle velocity, we may the data are observed-thi8 is important!conduct an experiment and apply a statisti- Since we ordinarily obtain information oncal test to the exlperimentai rmetilts to se one or both of the products by means ofa'whether we would be justified in concluding smuple, we may sometimes make an errone-that there is a differt;ýix, etween the per- ous decision. However, the chance of making)~ ~ ~ ~ ~~~A fomneo rae' .ra ons the wrong decision can be reduced by in-

-In another case, tioý ~. ~. ctuting proc- creasing the number of observations. There~esses may be availab..V.ý ýrcss A is cheap:er tw wysiwhcwean aeawrnand therefore preferao, jnless process B is drecition wasi- wihw cnwk rndemonstrated to be superior in some respect.- (a Whesi n: we coc-d ttteesadfAgain, we apply a statistical test to the ex- f(a)c Where ioncld fac t there is aoe weisa-perinientai results to see whether proces& B fehent wherae an fact o ther Fist nonewesa

k~dzUenSuprforty.IlkNA11- I?-e .ai to Mind at difference thatOrdinarily, the statistical test applied to really exists, then we say that we make Ar

the results observed on a sample will print Error of the Second Kind.the way to decision between a pair of alter- In any particular case, we nevt r can ýe abso-natives. For some tests, the two alternative lutely sure that the correct decision 1 as beendecisions will lie formally ritated as follows: m e u ecnkowtepoaiiyo

(a) There is a di1~erence, between th~e mading buei cnknwther typeaoiletyrf(populatiun) averages of two material. adgetetp ferrprodutts, processes, etc. The probability of niaking'an Error of the

(hc No difierence has been demonstrated. Flist Kind is usually denoted by a; and the__1--- aPw IrimA u Rrrn. of the Sec-

In other cases, the forinail statemnent of the *Yn Kind' isd-itdb .Teaiiyondv Kindnaiv isison dil bee:b .Th bliyotwo ltenatie dcinons illbe:given statistical test to detect a diffferente

(a) The (populAtion) average of product (e.g., between averages) will in general deA ille greater than that of product B. pend on the size of the difference 8; thus,

(b) We have no reason to believe that the pls omaiguls soitdwt

tpan that of) avragc ofpoucB.s rae particular difference S. The. value of 8, P (5),thanthatof poduc B.associated with a particular difference 8 will

In this Chapter and others, we shall con- dcesas8irae.Fo ap Ariuasider a number. of stat, itical tests of diffev'- statistical test, the ability to detect a differ-ences. The application of each atatistical ence will be determined by tairee quantitiee:

3-1

AMCP 706-110 ANALYSIS OF MEASUREMENT DATA -

a, p (8) and it the sample s'". The comple- convenient function of 8. Figures 8-1mentitry quantity 1-P (3) is termed the through 3-8 show OC curves for a number1pwe -of the test to detect a difference 8 with of statistical tests when conducted at thea sa aple of size n, when the test is carried a = .05 or a = .01 significance levels.

,.out at the a-level of significance.• The decision procedure Is a very logical A Ccredpcstedsrmntr

: he de p e is a ver lo power of a particular statistical test. For,one. Suppose we wish to test whether two specified values of n and a, there is a uniquetypes of vacuum tubes have the same resist- OC curve. The curve is useful in two ways.imce in ohms, on the average. We take sam-

Vples of each type and measure their resist- Ifwhaesciednadwectueteip OC curve to read P (8) for various values ofance. If the sanmple mean of one type of a. If we are still at liberty to set the sample,tube differs sufficievtly from the sample size for our experiment, and have a particu-Maean of the other, we shall say that the two lar value of 8 in mind, we can see what valuekinds of tubes differ in their average resist- -of n is required by looking at the OC curvesance. Otherwise, we shall say we failed to for specified a. If, for the a chosen, the

Sa difference. How large must the differ- sample size required to achieve a reasonably>iince be in order that we may conel de that small a (8) is too large, and if it really isSthe two types differ, or that the obf.-rved dif- important to detect a difference of 8 when it- ference is "significant"?* This will depend exists, then a less conservative (i.e., larger)

aon several factors: the amount of vriability value of ot must be used. Various uses of the--in the tubes of each type; the number of OC curves shown in Figures 3-1 through 8-8"tubes of each type; and the risk we are are described in detail in the appropriateIwilling to take of stating that a difference paragraphs of this Chapter.exists when there really is none, i.e., therisk of making an Error of the First Kind. It is evident that for any j8(8), n will in-We might proceed as follows: we would be crease as 8 decreases. It requires larger sam-willing to state that the true averages differ, ples to recognize smaller differences. In some4-- a difference larger than the observed dif- cases, the experiment as originally thought

j ýerence could arise by chance less than five of will be seen to require prohibitively large• times in a hundrei wnhen the true averages sanmpe sizes. We %hen ru.L ounpron-ise b-

'are in fact equal. The probability of an tween the sharp discriminatory power we7Xrror of the First Kind is then a =- .05, or, think we need, the cost of the r nount of test-

as v -e commonly say, we have adopted a .05 ing required to achieve that power, and theAiginificance level. The use of a significance risk of claiming a difference when none4evel of .05 or .01 is common, and these levels exists. If the experiment has already been-are tabulated extensively for many tests. run, and the sample size was fixed from other

,.'-ýhere is nothing unique about these levels, considerations, the OC curve will show what-however, and a test user may, choose any chance the experiment had of detecting aSi,- w for a that he feels is appropriate. particular difference S.

As we havo mentioned, the ability to detect To use the OC curves ii, this Chapter, wea diff rence will in general depend on the must know the population standard deviationSize of the difference 8. Let us denote by �o, or at least be willing to choose some rangep (8) the probability of failing to detect a for a-. It is quite often possible to assignopecified difference 8. If we plot/3 (6) ver- sore upper bound to the variability, evenj•us th• difference 6, we have what we call without the use of past data (see Paragraphxn 01 orating C'taracteristic (OC) curve. 2-2.4). After the experiment has been run,Actual y, we usually plot P (8) versus some a possibly better estimate of o will ho avail-

able, and a hindsight look at the OC curve"0r mor . ceuratelj,, statistically significant A using this value vvill help to evaluave tthe

idifferene rmzy be atatistizall~y *igxiticaut amid yet b experiment.pr•zcticaUy unimpwotant. ex.... .

S_ _-8-2

COMPARING AVERAGE PE~RFORMANCE A MC 1"7000611O

We outline a number of diff'erent tests ini not cruciail, particuharty' jf the &A-mple sitethis Chapter. For each test, we give the pi.oý- is not very small.cedure to be followed for a specifted signifi- Alternate precedurm for most of the testocance level a and sample size n. For most of in this Chapter are given in AMO)? 706-118,the tests, we also give the 00 curve which Chapters 15 and 16. Chapter 16 given testaenables us to obtain the (approxim&te) value which require neither normality azsump-of p for any given difference. Tables are tions nor knowledge of the variability of theprovided for determining n, the saxnape size populations; but, this greater generality isrequired when a, 6, and 63(8) have been achieved at the price of somewhat reducedspecified. The testas given are exAct when: discriminating power when normality can be

(a) the observations for each item are assumed and the knowledge about the vari-taken randomly from a single population of ability of the population~s, needed for tliepossible observations; and, tests of this Chapter, is in hand. Chapter

15 gives shortcut tests for small samples(b) the quality characteristic messured is from normal populations which involve less

normally distributed. within this populatior.- computation than the tests of this ChapterOrdinarily, the assumption of normality is with negligible loss of efficiency.

3-2 COMPARING !HE AVERAGE OF A NEW PRODUCTWITH THAT OF A STANDARD

The average performance of a standard prod- cases, i.e., where the variability is estimated) uct is known to be vi,. We shall consider three from the sample, and where a is known from

different problems: previous experience.(.t) To determine whether the average of a Syjmbols, fo be uzed:

new product differs frenm the standard, Para- -avrgofnwmtilpdutrgraph 8-2.1.m avrgofnw atrapdutr

(b) To determine whether the average of a process (unknown).new product exceeds the standard, ParagranhMb average of standard material, product

3-2.2.or process (known).(e) To determine whether the average of a average o! sample of n inewnuremuents

new product is less than the standard, Para- on new product.graph 3-2.3. a standard deviation estimate computed

from n measurements on the new prod-For summary of the procedures appropriate uct (used where a is unknown).

for each ef these three problems, see Table 3-. c the known standard deviation of the

It is necessary to decide which of the three new product.problems is appropriate before taking the oh-DaaSml -WeSiofPwrservations. If thi:,; is not done and the chr ice of Dt ape3-egto odthe problem is influenced by the bertis, For a certain type of shell, speci(Ications state(for example, Paragraph 3-2.1 vs, 3-2.2), the that the am-ount of powder should average 0.73'significance lt~vel of the test, i.e., the probability pound. In order to determine whether theof an Error of the First Kind, and the operating average for a new stock meeta the specification,characteristics, of the test may differ consider- 20 shells are taken at random, and the amountably from their nomizial values. of powder contained in each is weighed

Ordina~rily the variability of a new product is The sample average I - .10 pound.not known. At other times previous experience The sample standard deviation estirm;,teC) may enable us to stgte a value of LT. We shall s -. 0504 pound. In illustrating the known-voutline the solutiona of the three problems case, we assume k known to beequal to 0.0(Paragraphs 3-.2.1, 8-2.2, and 3-2.3) for both pound.

8-3

_- AMCP 706-110 ANALYSIS OF MEASUREMENT DATA

SAILS 3-1. MIMASYV CR YUOIQWS P08 COMWA8S INM AVBRAW 0' A NOW PSO4ICV WTH TAt 00 A SIMS %ft(04M DIAIU AM WMKS SYAMAN W PALANWLIES $4.1, S-U., AND 11-IS)

mad a - 41)

a 4ilifg "L22. uiknowa; a - atimate 81a ff .. 3-1 wTbeA.5r 0tad2ttranm. W4 od (rw .mimp. and tabusr alwue. FWr a .01, "4t

tabualar vain..

8-2.2.2 - S.1w (I .> a soa, Er8- U.,T" lA-9.(

a s mq 81L sunnon;# etiat ( -ns >a asUs TableA-t For a 8,ddta 4

Abuan of F-4. tnkowda esamp e. a and-) . 8 4 tabutlar vy.ue. For A-.0,nd 9 to -

8-2.212 qknown mea -) >v Se Fsoa 3-7 Uue.TableA-9and 88-"

- ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~go oof th vrqoo teNvnPout cinfomh Sodrd(rUnnw)

(we Tal A)>a 8"F 67 n abe.9

nd50 ptound1tW~~~~~~~~~~~~~~C~..1 to haewwwl Otw(ct wnisvzý)i d Du h wdaCwm~awmA nmv nMn t" tI

Do otesne type dAvers Prthtoducth Dife dero thtte Stnavrag (o Ukown)?fpwdri

Proceur 1 thtteaeael7j 5ExEmplZ(1) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ h Chroe ava.g ofe thfcnelve ftets. 1 e 0

ne tabpe Ae4.~ct

COMPARING AVEKAIGE KRFORMANCC V- a- 'I iV

-j - "-Slcurveaof the preceding teut for •. .O5 and - .O1,rmp~tvely, andvariouavwluasotti. .: ,-. -

the true absolute difference between the averages (unknown, of coum)"Some value of o',- ..

(One mnay use an estimate from previous data; lacking auch an estimate1 see Paragraph2-2.4. Uf the 00 curvie is consulted after the experiment, we may tus the estimate ••

from the experiment.)

*d 0or

We then can read from the OC curve for a given significance level a and sample size it, a value,of 5(3). The 0(a) read from the curve is #(a Iw, a, %), i.e., #(a, given ,, a, %)-the probability of failingto detect this difference when the given test is carried out with a sample of size S, at the a-level ofsignificance, and the population standard deviation actually is or

If we use too large a value for a, the effect is to underestimate d, and consequently to over-estimate 0(8), the probability of not detecting a difference of a when it exists. Conversely, if wechoose too small a value of a, then we shall overestimate d and underestimate A(6). The true value -.

of fi(6) is determined, of course, by the sample eize % and the significance level a employed, aad the9 true value of Y.

$&Wtion qJ Sample Size n. if we choose

6 im - m 1, the absolute value of the average difference that we deire to 4t!lcta, the significance level of the testj, the probability of failing to detect a difference a

and cornpute -

then we may use Table A-8 to obtain a good approximation to the required sample size. If we takea - .01, then we must add 4 to the value obtained from the table. If we take a - .05, then we-must add 2 to the table value. (In order to compute d, we must choose a value for. c. See Paragraph

-22.4 if no other information is available.)

As an example, suppoae that we plan to take c - .05, and want to have 0 - .50 for a differenceof .024 pound; that is, we wish to conduct a test at a significance level of .05 that will have a 50-50eb'uce of detecting a difference of 0.024 pound. What sairple size should we require? Supposeprevious experience suggests that a lies between .04 and .06 pound.

Taking - .04, with -m - mol .024, gives d - 0.6. Using Table A-8, with a = .05,01 -,# - .50, we find the required samplesize asn - 11 + 2 - 13. Taking. - .06, yields d - .4.

From the same table, we find that the required sample sire is 25 + 2 - 27. To bi safe, we woulduse n 27. For o _ .06, with a significance level of .05, this would give the *wrz-sided . test atJepst a 5O% chance of detecting a difference of 0.024 pound.

If, when planning an investigation leading to a two-sided t-test, we overestimate a, the eonse-quences are two-fold: first, wc overestimate the sample size required, and thus unnecessarily increase-- .he cost of the test; but, by employing a sample size that is larger than necessary, the actual valueef f(a) will be somewhat less than we intended, which will be all to the good. On the other hand, if

11-5

AM P 06-110 ANALYSIS OF MAUE NTDATA

00

rigtke 0- C OC~urve for the two-sided 9-test (a -. 05).Adwt4 wth enumloafro Aus.I ¶j~*A~o8~al a~Vai.Vl. 17=o. 2,lune 194ti. tp .17-197. from milioe m.UUG[ "Op.rtung

11", 'umgigfo h WComac,. L~atlj Use 1=1.10Sgiftoace hIC.D1. Fers .iGrubbt. amidC L. wasvw

Nok NThee ; v apply to the following tests:

(b) Doteaverages of two products differ?

N/q VW+t2"I 1 WA + n1

Where u.r A -#, by assumption, and Rh and n,. are the respective sainple sizes fromproducts A and B. See Paragraph 8311

3-6

r^LADADIh~f% AIJCDAi'±C DiZDf''CebLAA LI^ A 1't 4 0lA 11 fl

0L9IL 09

0.6

0.1I - ý Cm\.JN

4=. -t &aJ

Adapte wihepre ido fro = mgnwn St-i by assum .tion, e and G.A J.d U* & a oprighthe 9,9 Presecie azrle siEe from

prdct A m"dB See Paragruph 3-2.1.1.

(b) o th aveagesof wo poducs dife 76_ _ _ _ _ _ _ ,

AXCP 706-110 ANALYSIS OF MEASUREMENT DAIA

we underestimate a, we shall underestimate the sample size actually required, and by t'aing toormal: a sample size, Oka) will be somewhat larger than we intended, and our chances of detectingreal differences when they exist will be correspondingly leeaened.

The following brief table, built around the preceding example, nerves to illustrate thbe pointeuumerically for a situation where a -. 05, and it is desired to have f(b) - .50 for a - m e -

M.04, and a in fact is equal to .04 though this is unknown.

Value of Resulting Comres•ndisgu Assurmed (.024)

.08 45 .02

.06 27 .15

.04 (true value) 18 .50

.03 9 .64

.02 5 .80

Thus, if w actually is .04, playing safe by taking .06 has more than doubled the sample sizeactually needed, buL we have gained a reduction in f from .50 to .1.

Finally, it should be noted that, inasmuch as the test criterion u -fi _-.,j --I dces not depend on a,

an erTor in estimating 4 when plarming a two-sided t-test will noL alter the level of significance of thetest, which will be precisely equal to the value of a desired, provided that t,-12 is taken equal to the100 (1 - a/2) percentile of the i distribution for s - 1 degrees od freedom, where Y, is the sampleSim -uctually t*Mploet 4~

3•i.1.2 Do** th, Average of the tiw Product i0e. %Irom the Standard (a Known)?

ITwo-sided Nuial TestiProcedum Fxampie

87 ~ ~ R9LL I1_ T..i - --Irý_

(2) Look up zx- in Table A-2. (2) z.,, 1.960

(8) Compute (3)1, the mean of the a, measurements. - .710 pound

(Data Sample 3-2)

(4) Compute (4) is known to be equal to .AC pound.1.96 (.06)us 1., 7 =

= .0m3

(5) U1 -• e >u decide that the avorage (5) 1X - rk I = 1.710 -- .735M =.025. We con-of the new type differs from that oi the elude that there is no reason to b.Alieve that

,standard; otherwise, that there is no reason the average amount of powder in the newto believe that they differ. stocl: dffers from 0.735 (the specified

standard amount).

(6) Note that the interval 1 + u is a 100 (6) Note that t.710 =1= .0263) is a 95% con-(1 - a) % confidence interval estimate of fidence interval estimate for the truethe true average m of the new type. average m of ;he new stock. -

COMPARING AVERAGE PFRFORPAANCE AMCP 706,110

Operaling Charaurtidaw oJ ta Test. k-igures 3-3 and 8-4 give tVe operaitig c twt tiucs Or'"'-

preceding test for a - .05 and a - .01, respectively For any given n and d - '-n the V'I4Ler

of() go- p(8o,a,I n), the pi obability of failing to detect a diTemrnce o absolute size6 - nus - tol,can be read off directly.

Setecliop, of Sam* Size %. If we specify a, the significance level, and 0, the probability oi risk we

are willing to take of not detecting a difference of absolute size. a - Im- mlj, then we can uoaTable A-8 to obtain ni, the reqt.ired sample size. As an exmiple, if o is known to bL 0.04 polmd,and we wish to have a 50-50 cha--ce of detecting a difference of 0.021 pound, then d = 0.6. FromTable A-8, we find that tlFc required sample size is 11.

When we know the correct value of a, we can achieve a desired value of P(b) witb fewer observ--

tions by using the norms' test at the desired level of significance a than by ising the correspondig•t-test. The saving is 2 or 4 observati•ns according as a = .05 or .01, respectively.

Overestimnatir, :;. underestimating or when planning a two-sided normal test has somewbiatdifferent coes, uences than when planning a two-sideci t-test. If we overe&Wimate a and choose

a' > a, we alsc overestimate the sample size required as in the case of the t-test. In addition, we

overestimate the correct test criterion u = ziýs/• for the samp'e size n actually adopted, with the

result that the effective significance level of the normal test is reduced to ., which is related to aby the equation

The actual probability of not detecting a difference of 5, 6'(5), is related to the intended risk 0(6)by the equation

will be less than 0(o) when hi' > 0 for all (large) S fo0 which P(S) < 0.50; j9(5) will be largerthan #(6) for all (small) 6 for which 0'(6) > 0.50. For the particular a for -which P(4) = 0.50, 0'(6)also will equal 0.50. Conversely, if we underestimate a, then wc not only und!emstt'macd the sample

size ruired but elso the test criterion for the sample size actually used, so that the actual risk:Ian Error of the irst Kind a' will be larger than a, and the risk of an Error of the Second Kind1'(6) will be irncreased for large 6, and decreased for small 5.

3-9

AMCP 706-i AO .INAL. i eSo. iV1LM.JrKL1iLr

The following calculations serve to illustrate these points imieric;,lly for situations bordering

on the conditions aýtumed in tni, preccding sample-size calculation:

Intended significance level a = 0.0.Intended risk of Error of ihe Second Kind s(s) = 0.50 for 6 = 0.024.

TWO-1,I0EED NORMAL TEST

Actual Aclual Risk ofValue of a Sample Size Signlficat.ce Error of Seocond Kind,Assumed Indicated Level, a' f' (0.024)

S43 (45)Y .00009 (.05) 0.50 (.02)*.06 25 (27) .003(.05) 0. 50 (15).04 (true value) 11 (13) .05 (.05) 0.50 (.50).03 7(9) .14(.05) 0.50 (.64).02 3(5) .33(.05) 0.50(.80)

Values in parentheses are for corresponding two-sided t-test.

To obtain a numerical illustration of the more general case where 0(6) Y 0.50, let us modify theforegoing example by taking P(s) = 0.20, say, as the intended risk of an Error of the Second Kindfor 6 = 0.024:

Intended significance level a = 0.05.Irtended risk oý Error of the Second Kind #(a) 0.20 for 6 =0.02%.

WQO-WIQED NORMAL TEST

Actual Ac.ual Risk ofVoluo of ' Samp*l Size Significance Error of Second Kind,AsIn) Indicated Level. a' / ' (0.024)

CIA $L (190) .00009(.05)* .046 (.004)*.05 50 (52) .003 (.05) .103 (.01).04 (true value) 22 (24) .05 (.05) .20 (.20).08 18(15) .14(.05) .26(.43).02 6(8) .33(.05) .34(.70)

"Valuea in parentheas are for ,wrrvandi-W two-,%dred t-tet.

31

' 3-10

COMPARINC. AVERACE PERFORMANCE AkICP 706-ilO

00

1 0

: 0.2 --

d

Figure S-S. OC cun'mes for the tuo-sided normal test (a = .05).Adapted wit, ,lraon from Anptals of MV..?Aeqtaieal Staistio, Vol, 17, No. 2, June 1lt46, p'p. 178-197, from artidj entitled "Operat•ng

Charactcristit.. .e Cotmion Statistlcal Tm.ts of Significanxe" by C. D. Ferris, F. E. Grubbh. and C. L. Weaver.

I iiNote: These curves apply to the following tests: I

(a) Does tie average rt uf a new product differ from a standard ino?

5= m - 6

d I - m0 1 See Paragraph 3-2.1.2.

(b) Do the averages of two products differ?

=IMA - ME:

d --- ; Aand 7 are kwiwn. See Paragraph 3-3.1.3.

/I/I

-+"

A DU S,."•'41 '7 06 1. 'a.V...........S O ME:ASUREME:Ni DATA

-I \LII

0.60

I.J

S3 4

d

Fio'w $-i. OC curves for tie two-sided nor.,l test (a .01).

,ptdap with p~rjz orn m~o f mi m X cw p SiLatimscm by A. H. Sowkm and Q_ J, 4~kk"lykn;~ 17.,yr~gtt. 199.5p pre I-L-~I n

Noje: These curves apply to the following tests

d - . See Paragraph 3-2.1.2.

tb) Do the avtrage8 of two products differ?

S-IM, - MDII6 Ain -mVIB

d -- ,+-• and ca are known. See Paragraph 3-3.1.3, -

"3-12

I

COMPARiNG AVERAGE PERFORMANCE AnrICI 706-i 10

3-2.2 TO DETERMINE WI.ETHER THE AVERAGE OF A NE"V PRCDUCT EXCF1-1- "'Tr- .T,000APLý

3-2.2.1 Does the Average of the New Produd Exceed the Stand&;t (y Unknown)y[One-sidod M-est;

Proceudur Exampla

(1) Choose o, tho significance level of the test. (1) Let a - .95

(2) Look up t,. for it - I degrees of freedom (2) t.., for 19 degrees of free-iont 1 729in Ti'able A-4.

(3) Compute (3)k and s I - .710 pound

a - .0504 pound(Data Sample 3.2)

(4) Compute (4)5 1.729 (.0504)

U U VTO

0.01D

(5) If (9 - rn) > + u, decide that the aver- (5) (9-- in0) '- (.310 - .735) ... 02,. Weage of the new type exceeds that of the conclude thai there is no reason to belicvestandard; otherwise, tha! there is no reasonp that the averago of the new product ex.-to believe that the average of the new type cceds •h, specified atandard.exceeds that of tiie standard.

(6) Note that the open i:tervl from (9 - u) (6) Note that the open interval from .691 toto + ic is a one-sided 100 (1 - a) % con- + co is a one-sided 95% coi lfidence inter-fidenep intpri al fnr t-hp t.mnp men nf thp v•q fnr t-e avprse of the new nrcui.itt

new product.

Operating Characteristics of the Test. Figures 3-5 and 3-6 give the operating characteristic (OC)e--"'Pa o~f the albove test for v = .05, and a = .01, reSpoctivelly, -- ' various ,,slues of n.

Chows-:= (m - mi,), the true difference between averages, (unknown, of course)

Some value of o. (We may use an estimate from previous data; lacking r'rV.t, " .Pn,'inate,ee, Paragraph 2-2.4. If OC curve is cor.sultrd after the experiment, we

may use the estimate from the exerinment.)

Com •ute

We thc,. -wn read from the OC curve for a given significauce level a and sample sizx 14, a valueof 0(s). The A(B) reAd from the curve is f(bI a, a, it), i.e., 0(S given o,, a, n)--the probability of failingto detect this difference when the given test is cam ried. out with a sample of size n, at the a-levelof significance, and the population standard deviation actually is a.

F wc use too large a value for v, the effect is to underestimate d and consequently to overestii iate

f(6), the probability of nat detecting a difference of a when it exists. Conve~sely, if we choose toosmall %. value of a, then we shall overestimate d and underestimate #(a). The true value of 0(b) isdetermined, of course, by the sample size n and significance level a employed, and the truevalue of a.

3-13

AMCP 70-110 ANALYSIS OF MEASUREMENT DATA

4H)

0.50

C,65"0'+ N\ t

OL1

0.-_-s 0 4-04- r 02 0.4 Ob1A ID L2 1.4 1.6 1I2D 222 2.4 P-6 . 30 &3Z

d " d*

Figure 3-5. OC curves for the, one-sided t-test (a .05).

Adapt.1 with pwnmianM from E,•4(mwcag Sta5•U by A. H. Rowker Lnd G. J. IMbwrmma, Copyright, 1959, Prentlce-liall, I¢. 1INote: These curves apply to the following tests:

(a) Duai the average m of a new product exceed a standa-d in0?

d - See Paragraph 3-2.2.1.

(b) Is the average in of a new product less than a standard ma?

V1, - -- m

d = See - m 5e Paragraph 3-2.3.1.0f

(c) Does tht-average of product A exceed thaL of product B?

" '•-- ,m, 1 •n--r

do- MhA-lB 1flAttyV/A + •IbB -- A + 14'

whee 5 - o' by assumption, a-nd r-a and "B are the respective samuple sizes from f. jproducts A and FB, See Paragraph 3-3.2.1.

3-14I

f L/" ADA DI!3.Ic! Al•ADA, "l ItDIC'•OkA A&1/( A U'1"-T 0 4.. 4 1iA%.fl~I 1 1'%%.7 P% Lr.F%..1L I Lr*JI.IV~Ef. IITAI,X IVV'UL All

-- 0 '------- --

-zJ

d ord900

0.50 -

W -0-50 - -----

o 0,~20

-0.9 C46 -0.4-01 0 02 OAO0.AO.8 LO 12 1.4 LeAIt.0 21t.4ZA2A 1 5 01Md or d

I 3gr -6. OC' curv~es for the onc-sided N-es.'( .01).

AdaPtod A-itb jermiiaon frotu E'ginwrine Skgoics by A. H. Dowker and G. J. Liebernja'41, Copyright. 1959. Prentica-1IAI1 luc.

!Vute: These curves apply to thc followin; te,,ts:

(a) Des the average n of a new product exceed a standard tno?

9"- ?no

d &- Se Paragraph 3-2.2.1.

iI(h) Is the average mn of a new product less than a standard inO?

d - -- See Pragr'aph %.--2.3.1.

(c) Does the ave, age of product A exceed that of product B?

5 MA-fL ___ I_ B

17 vtA T 1i-1 fnA + nB'

where JA = a.1 - o by assumption, and nA and nB are the respective sample sizes fromproducts A and B. See Paragraph 3-3.2.1. 3-'4


Selection of Sample Size n. If we choose

( i (m -,)

a, tile significa•ice level of the test0, the probability of failing to detect a positi~e differcuee of .ize (m - 7v4)

and compute

a -

then we may ust Table A-9 to obtain a good approximation to the required sample size. If wý: arvusing a - .01, tLen we must add 3 to the table value. If w( are usingA - .05, then we must add 2 tothe table value. (In order to compute d, we must choose a value for i: see Paragraph 2-2.4 whenno other information is available.)

If, when planning an investigation leading to a one-sided t-test, %.; overestimate a, the conse-quences are two-fold: first, we overestimate the sample size required, and thus unnecessar ily increasethe cost of the test; but, by employing a sample rize that is larger than necessary, the actual valueof #(a) will be somewhat less than we intended, which will be all to the good. On the other hand,if we underestimnate vr, A e shall underestimate- the sample size actually required, aild by using toosmall a sampie size, 8(b) will be somtwhat larger than we intended, and our chances of detectingreal differevc.t:. wh m they exist will be correspondingly lessened. (A iwurnerical example for the•w. sided 1-te•, i.: i\, cn in Paragriph 3-2.1.1. The o:ie-sided case is similar).

Finally, it should be noted, that inasmuhe, as the test criterion u - f, ip does not depend on a,ita

c:l,, error in estim,,ting o when planning a one-sided t-test does not alter the level of significance ofthe test, which will be precisely equal to tie value of a desired, prorided that i-, is taken equal tothe 100 (1 - n) percentile of the t distribution fUr n - 1 degrees of freedom, where n is the sample#ize actually employed.

3-2.2.2 Does the Average of the New Product Exceed the Standard (a Known)?

[One-sided Normal Test]

Procedure Example

(2) Look up z_, in Table A-2. (2) z., = 1.645

(3) Compute (3)I, the wimple mean X = 0.710 pound(Data Sample 3-2.)

(4) Compute (4) e is known to be equa' to .06 pound.

ar 1.645 (.06) I

= .022

(5) If (. t -o) > u, decide that the average (5) (X - in)- .710 -- .735 = -. 025, whichperformance of the new type exc.eds that is not larger than u. We conclude that thereof the stamdard; otherwiie, that there is n., is no reason to belihve that the average ofreason to believe that the average of the the new product exceeds that of thenew typt exceeds that of the standard. standard.

(6) Note that the open interval from (T - u) (6) Note that the open interval from .688 toto + w is a one--ided 100 (1 - a) % con- +- is a 9b% one-sided confidence inter-fidence interval for the true mean of the val for the true mean of the new product.new pioduct.

3-16

M

COMPARIWIG AVERAGE PERFORMANCE AMCP 706-Ii0

Opcrating Ctarar.ristsof 'the Yc'st. Figures 3-7 and 3-8 givc the operating rharacteristica of the

above test for ti .05 and a = .C1, respectively. For any gv:wr t? Wnd d - -n , the value of

0(s) - • 1c, 'r, i), the pr'bability of failing to detect a positive difference 6 - (ti - m.k), can beread oft directly.

Selection of Samnple Size n1. If we specify

- (tet - m0), the magnitL ;e of a positive difference of interest to usa, the significance livt:l of the tLct9, the probability of iailing to detect a positive difference of size 6

and . pute

d - I -- ??10

then we may use Table A-9 to obtain the requi:rd sanple size.

VWhen we know the correct value of or, we can achieve a desired value of 6(6) with fewer observa-tions by using the normal tesL at the des~red level of significance a than by using the corespondingt-test. The saving is 2 or 3 observations according as a = .05 or .01, respectively.

OverestimAting or underestimating a whan planning a one-sided normal test has somewhat Jifl.crer.. con...quences thar l.hen ivi- nnip Pa nnncwed p .t e If i we O m we a c h and"os

we also overestimate the samrple size required as in the case of the t-test. In addition, we overestimate

the correct test criterica u - z_, for the sample size n actually adopted, with the result tOat the

cfrctive significance level of tfe nornml test is redutr _ u, ,. .,k, L. elated to c- by the equation

•--• (•~ • I-•

The actual probability of not detecting a difference of A, f'(8), is related to the intended risk 00()by the eq-iatio 1

i

#'(A) will he lei.s than 0(b) %hen o' > a for all (large) a for which ý(6) -: 0.50; 0'(6) will Le larger thaa1(b) for all (small) 6 for which k,'(6) > 0.60. For the particular 6 for which #(&) ý 0.50. 0'(6) also willequal 0.50. Conversely, if we u7iderestimate u, then wc not only undrestimate the sample size Irequired but also the test criterion for the .zanple size actually used, so that the actual risk of anError of the First Kind a' will be larger than Yr, and the risk of an Error of the Second Kind p'(S)will be iincreased for large A, and decreased for smn~all A. (Numerical examples for the two-sided normaltest are given in Paragraph 3-2.1.2. The onc-sided case is similar.)

8-17

'MCP 706-110 ANALYSIS OF MEASUREMENT DAlA

1.00 - 001-

W

. -LO.0 50 ao (50 1.00 Wo t5 •o0

d

Fig:i.e 3-7. OC ct'rve- for the o7.e-sided normal test (a - .05).

AdAta•d 'ish armimion from i•n • E ieriW Sialisi by A. It. Bowker and U. J. •leto man, Copyrdiht, 1959, Prentice-.1al, Inc.

Note: Them curmes apply to the foliowing teszm;

(a) Does the average n; of a new product exceed a standard too?

d - 'or. See Paragraph 3-2.2.2.(7

(b) Is the average rn of a new product. les6 than a standard in?

I -rn - m

d = See Paragraph 3-2.3.2.

(C) Does the average of product A exceed that of pruduwit B?

6 InA Mb

- MA Mb ndI,A ,A and u ae known. See Paragraph 3-3.2.3.

3-18 [ -

COMPARING AVWR.GE 'Ft-ORMANCE A 10C -; 0u-i,

-T

I 8 J

d -0.-6-- -__\1--P-agah -. 22

rr

d - \t - -S. Iaarp 3-2.3.2

fl -

.. . .. .- -o -

9-1

0

0

Q1 .000 .430 0.,0 030 Wo 1.50 2.00 2.50 3.CO

d

Fgurte S-8. (C zurtle5 for Ow oiuz-sufecdtwoira1 teai (a =.0i).

Au.pmu !LL.j..~ S4r ý-" V Bv~ra~c-~a4'. .J~emCy-,h.I~ ~i-iI. VUHuse Xn

Note: Thew. curve" ap-py to th'w following test.9;

(it) Does. the average m of a r..!w product exceed a standard m?n~

d n Set, Paragraph 3-22.2. (".

(b) Is the avecagt in of a new product lvc& th.mn a staindard in.!

d _ 17 -r See Paragrvyh 3-2-3.2.

(c) Does the average of product A exceed that of pr-di~ct B?

3 mA -7,

d fl 1118 and am, knowu. 4. Seu Par-agraph 03-3.2.S.

U-1

AMCP 706-110 ANALYSIS OF MEASUREMENT DATA.1

3-2.3 TO DETERMINE 'WHETHER THE AVERAGE OF A NEW PRODUCT IS LESS THAN THESTANDARD

4-2.3.1 Is the Average of the New e'roduct Les-- than the Standard (a Unknown)?

lOne-sided t-tost]Procedure Example

(1) Choose a, the sign;ficance leel of the test. (1) Let a .05

(2) Look up t,- for n - 1 degrees of freedom (2) t.96 for 19 degrees of freedom = 1.729in Table A4.

(3) Compute (3) X - .71.0 poundX and a s = .0504 pound

(Data Sample 3-2)

(4) Compute (4)1.729 (.0504)

U U= I'~

=0.019

(5) If (mo - X) > u, decide that the average (5) .735 - .710 = .025. We conclude that. theof the new type is less than that of the average of the new type is less than that of jstandard; otherwise, that there is no reason th' standard.to believe that the average oi the new typeis less than the standa-d.

(6) Note that th-e open interval from - c to (6) Note that the open interval from - o to(.9 + u) iz a one-sided 100 (1 - a) % con- .729 is a one-sided 957(. confidence interval

fidence interval_ for the true mean of the for the true mean of the new type.ne• type.

Operezting Characteristics of the Test. Figures 3-5 and 3-6 give the operating characteristic (OC)curves ogf the above test for a = .05, and a = .01, respectively, for various values of n.

Choose:= (m, - m), the true difference between averages (unknown, of course)

Some value of a. (We may use an estimate from previous data; lacking, such an estimate,see ParagTaph 2-2A. If OC curve is consulted after the experiment, wemay use the estimate from the experiment.)

Cr

We then can read from the OC curve for a given significance level a and sample size rn, a valueof P'`). The 0(b) read from the curve is 0(6 j o, a, n), i.e., #(5 given a, a, n)-the probability of failingto detect this difference when the given test is carried out with a sample of size n, at the a-level ofuigaificance, and the population standard deviation actually is a.

If we use too large a value for o, the effect is to underestimate d and consequently to overestimateP(a), the probability of not detecting a differen -e of S when it exists. Conversely, if we choose toosmall a value of u, then we shall overestimate d and underestimate #(6). The true value of #(6) isdetermined, of course, by the sample size n and significance level a employed, and the true valueof •.

8-20

-t

COMPARING AVERAGE PERFORMANCE AM&CP 706-1.0

Seletion of Sample Size n. If we choose

a, the significance level of the testfi, the probability of failing to find at negative difference of size (Mi - Vn);

and compute

then we may use Table A-9 to obtain a good approximation to the required sample size. If we are

using a = .01, then we must add 3 to the table value. If we are using a - .05, then we must add2 to the table value. (In order to use the table, we must have a value for a. See Paragraph 2-2.4F if no other information is available.)

The effect of overestimating or underestimating a is the same as when a one-sided t-test is to beused to detect a positive difference of magnitude 6 = m - mo. See Paragraph 3-2.2.1.

3-2.3.2 Is the Average of the New Product Less Than That of the Standard (o Known)?

(One-sided Normal Test]

Proeedure Example

(1) Choose a, the significance level of the test. (1) Let a = .05

) (2) Look up z_ in Table A-2. (2) z.,& - 1.645

(3) Compute (3)X•, the sample mean 2• =f 0.710 pound

(Data Sample 3-2)

(4) Compute (4) T is known to be equal to .06 pound.

U = 1.645 (.06)

= 0.022

(5) If (me, - X) > u, decide that the average (5) (m, - X) (.735 - .710) = .025, whichof the new type is less than that of the is larger than _. We conclude that thestndard; otherwise, that there is no reason average of the new type is less than theto believe that the average of the new type standard.is less than that of the standawd.

(6) Note that the open interval from - • to (6) Note that the open interval from - ao to(I + u) is a one-sided 100 (1 - a) % con- .732 is a one-sided 95% confidence interval•_ýence interval for the true mean of the for the true mean of the new type.new type.

Operating Characteristhcs of the Test. Figures 8-7 and 3-8 give the operating characteristics of

the test for a = .05 and a = .01, respectively. For any given ta and d - V1 - the value of

p(s) = (S( a, a, n), the probability of failing to detect a negative difference of size (mo - m), canb e r e a d o ff d i r e c t l y . 8 -2 1

8-21

__________________-__

AMCP 706410 ANALYSI" OFM ASUREMENT DATA(

Sekleii<, of Sa4rnpl Sz,. n. If we specify

z - (,m.• - n), the magnitude of a negative difference of ihtcresl to usa, thy significanee level of the tpt,, the y'r.balii•.y of failing to dct-et a n.gative ditickrncu of iaizei,

mnd cempute

then we may use Table A-9 to obtain the required sample tize.The effect of overestimating or underestimating o is the same as when the one-sided normal test

is to be used to detect a posntve diference of magnitude B = vi - tn. See Paragraph 3-2.2.2.

3-3 COMPARING THE AVERAGES OF TWOMATERIALS, PRODUCTS, OR PROCESSES

We con.sder two problems:(a) We wish to test whether the averages of two materials, products, or processes differ, and we

are not particularly concerned which is lafger, Paragraph 3-3.1.(b) V"z wish to teat. whether the average of material, product, or process A exceeds that of

material, product, or PV,,.Z B, ', uauraph 3-3.2.

TABUL S3 ARY Of IU.O"QUES FOR tOMPARW THE AVERAGE PIlF0RMANE Of TWO PRODUCTS(tOR OTAIkt AND WO.tKED XAMPIUS, SEE PARAGRAPHS S4.1 AND 34.2)

we Wattoo- Knowtodge of Op- 1 _______wol -__

""11 o 9- of TNtI I -

m, differ% "-.1.1 ..... ;both I,-1>_,_whe_ IFor. - .05 and Use TableA-.For l(fA_1 + .- _3jfrotraW unknown • -'-. a - .01 ee Figs. a -. 06,addto

'u " . 1- and 3-2" and the t.bular value."Par. S8.1.1. For - - .01, add 2to the tabular value.

&4.3.2 o, P, .. ;bhothb jX.L I > w, where V is the value of 4,_,, foi- the effee-kkowu. 3_tive nmtrber of degreoe of fraedomS: : . R D + - I _e . _+ / - 2'

•3-S.l •,v,,, ot ] i , - , w For h-e .05 a Ute Table A-&

ui4 INA6reater &3.2.1 ..~=e;hoth (a 11 e) > U Whereb or , - .05 and Use Table A-9, Forthan tag unkwim a .01 ow Fg.% a - .05. add I to O

a tI..P-t - S-5+and 3-4 a. the 'L tblrVale R+a,."A 144 Par. "4.2. 1. For a .01, sAdd2i

totetWt-alua.'ate.

4.2*,d re4oth ( ". - 1.) >. whom 1-.or 1e , if.

8 -2f .

COMPARING AVERAGE PERFORMANCE AMCP 706-110

It again is import-bn to decide which problem is appropriate before making the observations.If this is not dine and the choice of the problem is ir~flucnced by the observations, the significancelevel of the test, i.e., the probability of an Error of the First Kind, and the operating characteric"'icsof the test may differ considerably front their nominal values. It is a&surred that the appropriateproblem ha- been selected and that rA and nB observation, are taken from products A and B,respectively.

Ordinarily, we will not know A or a,. In some cases, it may be safe to assume that 'A is approxi-rmateiy equal to F.,* Wi give the solation4 for tlh two problems (Paragraphs 3-3.1 and 8-3.2) forthree situadions with regard to knowledge of the variability, and rur the special case where theobservations are paied.

Case 1-The variability in performance of each of A and B is unknown but can be assumed to beabout the same.

SCase 2--The variability in performance of each of A and B is unknown, and it is not reasonableto asnume that they both" have the same variability.

Case 8-The vwiability in performance of each of A and B is known from previous experience.The standard deviations are C and .7, respectively.

Case 4- --The obser'.rationi are paired,

\

3-3.1 DO THE PRODUCTS A AND B DIFFER IN AVERAGE PERFORMANCE?

3-3.1.1 (Cfse !)--Variability of A and B is Unknown, But Can Be Assumed to be Equal.

Data Sampie 3-31.1.-Laieni t4aso of. Fusion Q, IceTwo methods were used in a study of the latent heat of fusion of ice. Both Method A (an electrical

method) and Method B (a method of mixtures) were conducted with the specimens cooled to-0.72*C. The data represent the change in total heat from -0.72°C to water at 00C, in caloriesaper gram of mass.

Method A Method B

80.02 - -

50 04 79.94 - I80.02 79.98 -

80.04 79.9780.03 79. 9780.03 80.0380.04 79.9579.97 79.97 -

80.0580.0380.0280.00

S ) 80.02

For a procedure to test whethr rA aid o difter, wee Chapter A.

8-23


[Two-slded ttsstl

Procedure Example(1) Chow a, the significance level of the test. (1) Let a - .05

(2) Look up t.,, for V - ("A + n± - 2 ) de- (2) nA - 13

grees of freedom in Table A4. nB - 89- 9eg,'ces of freedom

t.171 for 19 d.f. - 2.093

(3) Compute: -A and sA, iZn and sR,, for the nA (a) A - 80.02and nq measurements from A and B. -- .000574

9D - 79.98al - .000981t

(4) Compute (4) (,

S-4 -1) + (nB -- ) .065A)1+ 7 (.000981)nIA + nD 2 19"

- �.�.0725- .0269

(5) Compute (5)

U- Sp A u - 2.093 (.02G9)9)

- (.05630) (.4493)- .025

(6) If I -I 1i> u, decide that A and B -`n) IA - XB .04, which is larger than u.dil a r -4. t... -I b +,. ....... I.n " d-,• tC"ht A ar B iffnr with regWard

&nee" ALherwise, that there is no reason to to average periormance.

belXve A antd B differ with regard to theiraverage perlormance.

1(7) Let MA, mn be the true average perform- (7) The interval .04 : .025, i.e., the interval-•,>':nces of A and B (unknown of course), from .016 to .065, is a 95% confidence inter-

_I-t is worth noting that the intelval vl fo. t.he true differenec between th2-(S,, - 1z) =L u is a 100 (1 - a) % con- averages of the method&.

-fidence interval e.Atinatc of (mA - mA).

SOp ingW C& r~t'do~ic, of the Tst. Figures 8-1 and S-2 give the operating characteristic (OC) curvesof the above teat Or a -. ,05 and a - .01, respectively, for various values of n - nA + It - 1.

A- WI - I, the true absolute difference between the averages

Some value Of q (- VA - va), the common standard deviation.(We may use an tmate from previous data; lacking such aix estimate, uee Para-graph 2-2.4. I1 OC murve is consulted after the experiC 3, we may use the estimatefrom tht ex•crimerAt,'

mA ~ ~ " -__BCd1 V-44 + tip _ A I

8-24


We .hen can read a,% alue of #(a) froin the OC carve for a given significance level and effective =ampleSIZe It - ttit + ?lR - 1. The i60) read froin the curve is ý(S a~ , (WnA, 11A) i.e., 0(, YFlWI 9, a, WAi and 'i.)the probability of failing; to detect a real difference between the two population mecans of magnitude3 = II - 7n,,) when the te~st is cArried out. with samples of sizes3 lA and sma, respectively, at the

&-level of significance, and the two populatior stanlard de friations actually are b)oth equ"lt.

If we use t'.o large a value for a,, the effcct is to mak, - us underestimate d' and consequently to-overestimate b). Conversely, if we choose too small a value of a, then we shall overestimate d*and underestimate 0() The true value of p(a) is deter-mined, of cours, by the sample Sizes nA and )

"dA significance level a actually employed, and the true value of a VA- CB)-

Since the test criterion if does not depend on the value of a (- CrA -ai), an error in estimatingwill not alter the sigioificance level of the test, which will be precisely equal to the value of a desired,piavided that the Nalue of (Ij is taken equal to the 100 (1 - a/ 2) percentile of the t-distributionfor "lA + n,. - 2 degrees of freedomn, where 11A and nu, are the sample sizes actually employed, andit actually is true that CA 9B.

If Or Fdcr, then, whatever may be the ratio TA/Ordy the effective significance level a' will not differseriously from the intended value a, provided that flA - iD, except possibly when both are as smallas two. If, oiL the other hand, unequal sample sizes are used, and Vr rt O's. then. the effective levelof significance a' can dliffer considerably from the intended valae a, as shown in Figur 3-9 where

a.05.

e.1/. (LOGARITHMIC SAE

Fijiurs 8-9. Probabilityi of re'tit Of hkpoth~aj,WA - BWplt iJed aganht 8.

IN1, ro arioe.nd~1ad "The 90SWAfta* o Lhtwo m" wbwthe a kiwltiton V"Oftan VA-

equal" kwn B. L. W"e. 82

*,MCP 706-110 ANALYSIS OF MEASUREMENT DAT/. (* •Isction of Sample Size n. If we choose

4 - mA - mE,, the absolut- value of the svera!,e difference that we desire to detecta, the significance level of the testP, the probability of failing to detect a difference of absolute size 3,

and compute

d - ,rnA , where a , - C,

then we may use Table A-b to obtain a good approximation to the required sample siz-cI (- WA - niB). If we tak( a - .01, then we must add 2 to the value obtained from the table.If we take - .05, then we must add 1 to the table value.

In order to compute d, we must choose a value for t (- 4 'A - j). (Se,, Paragraph 2-?.4 if noother information is available.) If we overestimate a,, the consequences are two-fold: first, we over-eitimate the sample size n (-= n = n8 ) required, and thus unnecessarily increase the cost of thetest; but, by employing a sample size that is larger than necessary, the actual value of 0 8) will besom-what less than we intended, which will be all to the good. On the other hand, if we under-estimate a, we shall underestimate the sample size actually required, and by using too small a

ýssample size, P(a) will be somewhat larger than we intended, and our chances of detecting real.differences when they exist will be correspondingly lessened. These effects of overestimating orunderestimatinga (- -. = o-) will be nimilar in maýiaitude to those considered and illustrated inParagraph 3-2.1.1 for the case of comparing the me.. m of a new material, product, or process,Tfith a standard value mo.

As explained in the V discuSsion. of ... O. "'-"" ctmractei.... or. the et, nerror 1"-estimating 6 (-= T - a•) will have no effect on the significance level of the test, provided that thevwlue of tl/2 is taken equal to the 100 (1 - a/ 2 ) percentile of the t-distributiun for n.- + nv - 2degi -les of freedom, where nA and no are the sample sizes actually employed; and if arA rBa, the-effect will not be sericus provided that the samplc sizes are taken equal.

-44.1.2 (Case 2)-Variability of A and B Is Unknown, Cannot B. Assumed Equal.

Datc Sample 3-3.1,2-Comp.esslve Sreng&. cf Concraet

Two investigators using somewhat different techniques obtained specimen corvs to determine thezompreu re astmngth of the concrete in a poured slab. The iollvwing results in pl i were reported:

- ~A8128 1939

- 8219 16978244 80308078 242

202029091816 ( )2022

3-26 2810

COMPARiN•i AVERAGE PERFOKMAY1 : 'A,•,, ,v,-U,, •_.177

.roc .Jurv EIamp!-

(i) Choose a, the significance kvel of the test. (1) Let • - .05

(Actually, the procedure outlin,-d will give

a significance level of only approxiniate!ya).

(2) Compute: SA and e,, S i and A , foi- the rtA ( SAX - 8166.0

and nu measurements from A alrd B. 4 - 6328.67

Se"2240.4 4

9A

(3) Compute: (3) 6328.676328.67 : •

VA=, VA = 4

- 1582.17

and2,1--

nq 9- 24629.03 A '•:

the estimated variances of 1A and I2,9respectively. . 7;

-(4, Compute the "effective number of degrees (4),of freedom"

f V• + V " 1. ;Vý'•: )

VfA- V - 2" 500652.4 + 60658911.9 2

nA + 1 + B + 1t,3' 7027G605 2C31159564

11.233 - 2- 9.233

(5) Look up _,.,2 forf' degrees of freedom in (6) f' - 9

Table A4, where ' is the integui nearest to t =g,, -2.262 £f; denote this value bý I

(6) Compute (6)

U- t-.., VV A u - 2.262 V' ii.26- 2.262 (161.9)- 366.2

(7) If IX - In I> u, decide that A ar.d B (7) 19A - X51 - 925.6, which is larger than

differ with regard to their average performn- u. Conclude that A and B d'fk.r with

mnce; otherwise, docide that there is no regard to average performauca.

reason to believe A and B differ in averageperformance.

&e. footaot* on page 8-28.

8-27

"AAMCP 706-110 ANALYSIS OF MEASUREMENT DATA

Procedure* Examle(8) If mA, MB are the true average perform- (8) The interval 925.6 L 366.2, i.e., the inter-

anceJ of A and B (unknown -f course), val from 559.4 psi to 1291.8 psi is a 95"fthen it is worth noting that the interval confidence interval for the true diffc,ence(IVA - -i) t- U is approximately a between the aver %ges of the two methods.

100 (1 - a) % cc-rfidcnr:c interval estimateof mA -- ra.

To gain some understanding of the nature, properties, and limitations of t'is approximateprocedure, note first that Vj and VR are unbiased estimates of the true variances nI/n A and a 'tBof Lhe means 2A and XB, respectively. Consequently, VA + Vn is an unbiased estimate of the truevariance of the difference .4 - 5,,, provided only thac 5A and In are the means of indepcndentrandom sampleg of %A and nB observations from populations A and B, respectively. Note next thatthe effective number of degrees of freedom f, definci by the expression in step (4), also can beexpressed in the form

1 ~e' (1_-__)f + T+ T +2

wherefA-= n-I and f f-nB--I I

are the degrees of freedom associated with the variance cstimatts V and VD, respectively, and,

P _ VA. nnd I -- P. __

VA + V . V '- s V

are the fractions of the "stimated variance of the difference S.A - .5 that are associated withA and Xs, respectively. Frtnm this expression forj, it is evident that f can never be less than the

smnaller of fA (- TS - 1) andre (f- nB - 1), andJ cannot be larger than

(A + 2) + (In +2) - 2 - 1A + nB.

W/hen VA is so large in comparison to V8 that V8 is negligible, then c • 1 andf If- A, which isintuitively reasonable-the i, degrees of freedom upon which V, is based are not making a useful

the Ul itulation, O then ca~e -, 0thndI•UOZ1F|LfLPLIUII LU LIV• •iJIIIL• W.PI tUf V81iI4W UV UI1VI.VSC -AA -.. A- •1LI1MJ•1Y, WIM11 V a

dominates the stuation, then c =- 0 andIf .-_ f. In intermediate situations where neither VA nor V8can be neglected, both the fA and the fI degrees of freedom make useful contributions, and theeffective number of degrees of freedom f expresses the sum of their joint contributions. Thus, inour illustrative example, fA - 3 and fi -n 8, but f - 9+. Both samples make their maximum con-tributions, that is, f achieves its maximum of nA +- n, only when VA/Vs - ("A + 1)/(na + 1),i e., wheu */14 - .. (11A + 1)/".g fnA + 1)

The tast procedure given here to an approxlnmuton, I.e., the stated signiflcance level Is rnly approximately achieved.The approximation is good provided Ma and %, are not too small. A more accurate proeedur Is given in BiomadrkaTao~uJfor SUciani.0su) which (in the notation of the present procedure) provides 10% and 2% sCgnifltence Ilvels ofIt-(X -X - ("A - MB11/4YI/vT V, . for A. 6, si ýt 6, and 0 : V/ ,(V. + VR) 1, 8% fd 1% .e1j.

n 1o for 1`1A ;->8and its '-: 8 and the rams ranae of VAl(VA + V.3) are Olven by Trickett, Welch' andJames."2 (When using either of the tutles (I) or (2), It should 5e nittlced that our c 'orreaponds to their "2a".)

The appropricte modification when the valuc of the ratio of the variances 0 - e,/t)j is known, but not their respictive avalues, is indicated at the end of the Dixuialon that follows this procmdure.

J 8-28

COMPARING AVERAGE PERFORMANCE AMOP 706-110

When srpe.of equal size (flA - 'iq - n) are involved, the present approximate procedure forCase 2 (, and ,B both unknown ard presuniably unequal) in Paragraph 8-3.1.2, und the exactprocedure fcr Case 1 (CA and arB presumnably eqial, but thver commi-on value unknown) given inParagraph 3-3.1.1 ar, the isanie in all respxects eycept for the valuc of ij to be used. In the exatprocedure for Cawe 1, the valuc of . to be used whie. iA , tin - n is the 100 (1 - a/2 ) per-centile of the t-distribution for ~'-2 (n - 1) degreeus of freedoin, and is completely determined by-the choice of significance level a and the common sample size it. In contrast, the value of 1v,-12 tobe used in the approximiate procedurk, for Case 2 When tnA - tn~ - v is the 100 (1 - a/ 2) percentilefor the integral number of degreets of f reedom f' nearest to the effective degrees of freedom

f + 1) (0. + 2#BA' + 81

and thus depends not only on the choice of significaaice level a and commnon sample size n, but alsoon the ratio sel/sp of the sample estimiates, of a,' and a' Furthermore, sincef can vary fromn (n - 1)to 2n, and equal. 2ni only when sA B~, it is clear that the two procedures may lead to differentresults when CA ý- 011, Consequently, when sainples of equal size (?LA - "B- n) arc involted, theproeedure for Case I of Paragraph 3-.11 huld be us&/ even when it car-not be assunted tOW UA =- CB.

If in fact 0` A, then the effective significance level a' will be identically equal to the intendedaigniflc~nee level a, arid the test will have maximum sensitivity with respect to any real differencebetween the population means 711 and in,.- If, on the other hand, C~i~A, then1 the effectiVesignificance level a' will differ from the significance level a. intended, but only slightly, as shown bycurve (A) in Figure 3-9; and the test. will tend to have greater sensitivity with respect to any realdifference between tnA and mu than would be t1he caae if the procedure of the present section

)>

were used.

In contrast, whe-n the sanmip!es; are of unequal size (tiA tiB a, the procedure oj the present sectionslould alzwayjs be used unless it is known for certain that o64 V-GB Otherwise, the effective significance

level 'uiuy 'l"IeU-ois1UdUralx rmIII iiuin ~ iila a.. intended, even when 64.

by curve (B) in. ýigure 3-9.j

When the smaller sample conies from the more variable population, the effective number ofdegrees of f re Adom f to be used with the procedure of the present section is likely to be much emallertha~n "A + as, - 2, the Cegrees of freedom to be used with the procedure of Paragraph 3811Nevertheless, the sinall advantage of greater sensitivity to real differences between MA4 and m, thatthe procedure of Paragraph 8-3.1.1 provides when a'A - aB is rapidly offset, as the inequality ofOrA and a,, increases, by the much firmer control of the effective significance level by the procedure

--f *- - hbn~ f ic vprv amall (jnv . .6).

Finally, it should be remarked that the effective number of degrees of freedom appiopriate to theprocedure of the present section is given more accurately by

-- kt)A + VB)' - + ?'A)'

ti4 - e.1 n.- 4A

+-.+

where

tiA and B

are the true vmiriances of IVA and 5ZB, respectively, and e 0 4

3-29

II

A TA 013 '70A_1 I A A IAVC I C ft A A r-A ?I I fr& A rhi A

It eir:V is shown thatf* never is less than the snialler of flA - I and nB -- 1, and never cxceeds"A + ,ni - 2. If we know the values of 4A and e., then we could evaluate f*; but under thewecircumstances we ;hould use the procedure of Paiagraph 3-3.1.3, not the present approximatepiocedure. If we do not know the values of mA and ,, but do know their ratio 0, then the exactprocedure (Case 3) of Paragraph 3-3 1.3 cari..ot be applied, but f* can be evaluated. Under thcncircumstances, the approximate procedure of the peseynt !ectiort should be followed, with f replacedby ]f* Wchnu we do not know the values of aA ,nd a', nor even their ratio 0, then we must rely onthe best available sample estimate of f'; nainely, f defined in Step (4) of the presunL procedure.

3-3.1.3 ((La6e 3)-Variabllily In Performance of Each of A and B Is Known from Previous Expedence,and the Standard Deviation* are OA and aB, respectively.

Data Sample 3-3.1.3-Loitent Hea of Fusion of ice

The observational data are those of Data Sample 8-3.1.1 and, in addition, it now is assumed tobe known that Ou - 0.024 and ay - 0.033

[Two-sicled No-mal Tsdj

Procedur, Example(1) Choose a, the levci of significance of the (1) Let a = .05 (

test.

(2) Look up z,.,,, in Table A-2. (2) z. 7, =- 1.960

(3) Compute: S and XB, tle means of the nA (3) A - 80.02and njs mcasurements from A and 1. aA = 0.000576

f1A - 13XB - 79.98

0"" - 0.001089

(4) Compute (4) .000576 ýq.001089

u = zl'/-i -0- -r -- v - 1.960 ,V 4 A v N 13 8

- 1.960 (.01342- 0.026

(5) If I1A - XSBJ> u, decide that A and B (5) ft - .•,J - .04, which is larger than u.differ with regard to their average perform- Conclude that methods A and B diifer withance; otherwise, decide that there is no regard to their averag-. s.reason to believe that A and B differ inav -:age performance.

(6) Let tnj, m, be the tru) average perform- (6) The interval .04 :- .026 i.e., the intervalanica of A and B (unknown of course). from .014 to .066 is a 95% confidence inter-It is worth noting that the interval val for the true difference between the ((1 - X') ± U is i 100 (1 - a) % con- averages of the metho&,.fidence interval msti, ate of (mA - in,).

3-3O

COMPARING AVERAGL PERFORMANCE AMCP 706-110

Operating (7haractcristics of the Tvst. Figures 3-3 und 3-4 give the oporating char-acteristic (0C)curves of the above test for a - .05 and a - .01, respectively, for variouj valuei of n.

If 3A -n, , and (71A - i 11 ) is the true difference between the tv.o averaiges, then putting

d =

we can read f, the probability of failing to detect a difference of size =t (0A - nIB)..

If ?1A c=nu, we cui put d = 1?!4----i, and, asing n nAh, we can read fi, the probability of

failing to detect a difference of size =- (?A - "B,). '

Selection of Sample Size. We choose

a, the sinitieiance level of the test6, the probability of failing to detect a difference o. 4ize (m,. -

If we wish 11A - 1"B = n, we compute

d = jA - 1+1B

and we may use Table A-8 directly to obtain the required sample size n.

I1 Wfý Wish to haVC '71 ad liB such thiat = - 01, then we may compute

d = 11m1 - '""&'4 + cg's

and use Table A-8 to obtain n = fnA.

2-3.1.4 (Case 4)-Tho Observations are Palred.

,.,ll, W,, •,pCI1II11•Z is, or" can be, designed so that the observations are taken in pairs. The twounits of a pair are! chosen in advance so as to be as nearly alike as pz-sible in all respects other thanthe zharacteristic to be measured, and then onc menb*'r of each pair is assigned at random to treat-rnent A, and the other to treatment B. For inoaance, the experimenter may wish to compare theeffects of two different treatment.; or, a particular type of de-Ace, material; or process. The word"treatments" here is to be understood in a :road sense: the two 'treatments" may be differentoperators; different environmrntal conditions to which a material may be exposed, or merely twodifferent methodi of measuring one of its properties; two difierent laboratories in an inter'aborato:ytest of a particular procebs of measurement or iranufacture. Since the comparison of tle two treat-reents is made within pairs, two advantagcs result from such pairing. First, the effect' of extraneousvaxiation is reduced ani there is co sequent increase in the precision of the comparison, and in itssensitivity to real . ,Terences between the treatrinens with resiect to the measured char.cteristic.Second, the test may be carried out under a wide range of conditions 'epres,-ntaive of actual usewithout sacrifice of sensitivity and precision, tnereby assuring wider applicability of any concl"uionsreached.

5-31

AWCP 706-110 ANALYSIS OF MEASUREMENT DATA

Data Sample 3-3.1.4-CapacIty of SBteiiel

The data below are measurements of the capacity (in ampere hours) of paired b-tteries, one fromeach of two different manufacturers:

A B X = XA -X4

146 141 5141 143 -2135 139 -4142 139 3140 140 0143 141 21i8 138 0137 140 -3142 142 0136 138 -2

Procedure Example

(1) Choose a, tte significance level of the test. (1) Let a = .05

(k) Compute; 9. and sd for the n differences, (2) -L = -0.1X,o (Each Xd repreenvi an observation on sj = 2.807 4A minus the paired observation on B).

l(3) iozk up tl_,/t for n - 1 degreef of freedom (3) t.975 (9 d.f.) = 2.262in Table A-4.

(4) Compuite (4), U= _, / u -2.2622.0

is 2.62 3.162)= 2.008

(5) If I -Te > u, decide that the average,- dif- (5) d --- 0 ih. cere tan.Concludefer; otherwise, that there is no reason to that batteries of the two manufacturers dok•lieve they differ. rnot ,dif',•, ir ,,,-% ..... ae,

(6) Nomte: The interval Id + U is a (6) The interval -0.1 f. 2.0, i.e., the interval1'L00 (1 - a) % confidence interval estimate -2.1 to +1.9 is a 95% confidence intervalof the average difference (A minus B). estimate of the average difference in

capacity between the batteries of the twomanufacturers.

Olpvralinp Charteris.ice of Me Tat. Figures 3-1 and 3-2 give the operating characteristic (OC)curves of the above test for a = .05 and a - .01, respectively, for various values of n, the numberof palrs involved.

"3-32

COMPARING AVERAGE PERFORMANCE AMCP 706-10 t

Chovac:

- MA - 81, the true absolute difference between the averages (unknown,of course)

Some value of (- (rd), the true standard deviation of a signed difference X,.(We may use an estimate from previous data. If OC curve is consulted after theexperiment, we may use the estimate from the experiment.)

Compute

d

We then can read from the OC curve for a given significance level a and sample size n, a value ofof 0(b). The #(a) read from the curve is p(61 o, a, n), i.e., P0(, g•vt n , a, n)-the probability offailing to detect a difference of =- (mA - mia) when it exista, if the given test is carried out with npairs, at the a-level of significance, and the standard deviation of signed differences X, actually is a,.

If we use too large a value for a, the effect is to underestimate d, and con; equently to overestimate#(a), the probability of not detecting a difference of 6 when it exists. Conversely, if we choose toosmall a value of o*, then we shall overestimate d and underestimate P•(3). The true value of #(a) isdetermined, of course, by the sample size n and the significance level a employed, and the truevalue of a (= a,).

Selettion of Number of Pairs n required. If we choose5 = ma - ma!, the absolute value of the average difference that we desire to detect

a, the significance level of the test,6, the probability of failing to detect a difference of a

and compute

If M

where a is the standard deviation of the population of signed differences Xd for the type of pairsconcerned, then we may use Table A-8 to obtain a good approximation to the required number ofpairs n. If we take a = .01, then we must add 4 to the value obtained from the table. If we takea = .05, then we must add 2 to the table value. In order to compute d, we must choose a value for a,.

If, when planning the test, we overestimate a, the consequences are two-fold: first, we over-etntethe -umber of -a- euie, n-tu unn-ces-r-l iea.-etees fth et u,

employing a sample size that is larger than necessary, the actual value of U(b) will be somewhatless than we intended, which will be all to the good. On the other hand, if we underestimate a,we shall underestimate the number of pairs actually required, and by using too rmall a samplesize, ý(6) will be somewhat larger than we intended, ard our chances of detecting real differences,Yhen they exist will be correspondingly lessened.

Finally, it should be noted, that inasmuch as the test criterion u = t,/ = does not depend on c,

an error in estimating a when planning the test will net alter the level of significance, which will beprecisely equal to the value of a desired, provided that i•_./2 is taken equal to the 100 (1 - a/2)percentile of the t-distribution for n - 1 degrees of freedom, where n is the number of pairs actuallyemployed.

3-•3

AM fjr 0W A' 4l AIAI CI( -C I11cDr-AA"-PI I rAnIA.... . A( )

34.2 DOES THE AVERAGE OF PRODUCT A EXCEED THE AVERAGE OF PkODUCT B?

3-3 2.1 (Case 1)-Variability of A €Gnd B Is Unknown, but can be AssumL-d to be Equl.

Data Sample 3-3.2.1---Zurface Hardness of Steel Piites

A study was made of the effect of two grinding conditions on tthe surfL,-, hardness of steel platesused for intaglio printing. Condition A represents surfacc•3 "as ground" and Condition B representssurfac s after light polishing with emery paper. The observations are hardness indentation numbers.

Candition A Condition B

187 157157 152152 148164 158159 161164172

- -•: [~One,.,ided t-4,st]_j

Procedure Example-(1) Choose a, the signiiicnce level of the test. (1) Let a = .05

§ (2) Loek up Ji- fcr y -I& + n- 2 degrees (2) 11 7--

of freedom in Table A-4. nB = 5" = 10

:.9 for 10 d.f. - 1.812A(3) Compute: .9 and sl, Xs and 81, from the (3) 1A - 165

ani B, rspectively. I's 155.2

263= 26.

o : -- 9..544(5) Compute (4)

U =f t1- sp i---s (1.812) (9.[#44) 35

A (il , 17.294 ( 26.7)= 10.1.

.2(6) If (X4[ ._ Xiz) > is, decide that the average (6) (-A - X1,) = 9.8, which is not larger thanof A erceeds the average of B; otherwise, us. There is no reason to believ'e that the

* decide there is no reason to believe that the average hardners for Condition A exceedsav'erage of A exceeds the average, of B. the average hardness for Condition B.

('7)• Let ifA and m5 be the true averages of A (7) (GtA - :•a) - u = 9.8 - 1&.1 =-0.8."* '•rd b. Note that t,,e interval from The interval from -0.8 to • is a95% one- ,-

SI '•(- - X,) -- u to oois a 1 - ,•one- sided confidence interval estimate of the ( )" sided confidence in terval estimate of the true difference between averages.true difference ((A - m)).

3-,344


Operating Cha 'acteristics of the Test. Figures 3-5 and 3-6 give the operating characteristic (OC) eryes

of the above test for a - .05 and a .01, respectively, for various values of n -,-i- nA -%,

Choose: _4

6 - (mA MrB), the true difference between the averagesSome value of a (= CA - 'B), the common standard deviation

(We may use an estimate from previous data; lacking -

such an estimate, see Paragraph 2-2.4. If OC curve is -

consulted alter the experiment, we may use the estimatefrom the experiment).

Compute

d* - A -tf0A -Z W_+ n -1 'A +R

We then can read a value of 0(b) from the OC curve for a given significance level and effectivesample size n. The #(S) read from the curve is #(a3 a, a, WA, na) i.e., 0(0, given ff, a, "A, and nj,) theprobability of failing to detect a real difference between the two population means of magnitude3 - + (MnA - mi) when the test is carried out with samples Uf sizes "A aw.d nq, respectively, atthe a-level of significance, and the two population standard deviations actually are both equal to o.

If we use too la.,ge a value for a, the effect is to make us underestimate d*, and consequently tooverestimate fi(a). Conversely, if we choose too small a value of a, then we shall overeatimate d*and underestimate #(6). The true value of p(s) is determiried, of course, by the &ample slk&

) (anA and nY) and significance level a actually emplo~ed, and the true value of r (- GA - ao).

Since the test, criterion u does not depend on the value of CA ( -= 1a 7 B G), an error in estimatingwill not alter the Efgnifieance level of the test, which will be precisely equal to the value of a desired,provided Ukit the value of t-. is taken equal to the 100 (1 - a) percentile of the t-distribution for-"A + n, - 2 degrees of freedom, where nA and n-& are the sample sizes actually employed, and itactually is true that a,, aB.

If GA 9d a, then, whatever may be the ratio GA/GS, the effective significance level a" will notdiffer seriously from the intendtd value a, provided that =A 'ID, except possibly when both a-re assmall as two. If, on the other hand, unequal sample sizes are used, and GA o ay, then the effectivelevel of significance a' can differ considerably from the intended value a, as shown in Figure 3-9.

Selection of Sample Size n. If we choose

S- (MA - mB), the value of the average difference that we desire to detecta, the significance level of the test

a pn, the probability of failing to detect a difference of size S

d(MA - Mil)d wherea -CA=7Rr Ithen we may use Table A-9 to obtain a good approximation to the required sample sizen (- nA =i n). If we take a.- .01, then we must add 2 to the table value. If we take -a - .05,then we must add 1 to the table value..

In order to compute d, we must choose a valae for a (-= a'A - CB). (See Paragraph 2-2.4 if noother information is available.) If we overestimate e, the consequences are two-fold: first, weoverestimate the sample size n (= n. - is) required. and thus unnecessarily increase the cost ofthe test; but, by employing a scanple si:,e that is larger than necessary, the actual value of #(a) willbe somewhat less than we intended, which will be all to the good. On the other h•nd, if we under-

3-35

_7I


estimate e, we ishall underestimate t~ie sample size actually required.. and by using too small a

sample sLt,, ,ý) will be conlev. hat larger thaa we intended, va~d our chances of detecting real

differences when they exist will be correspondingzly lessened. These effectE of overestima'ing or

ýunderestimating a,(-i LIA - ufB) Will bt Bi Iilar iii magnitude to those conapidered and illustrated in

Paragraph 8-2.2.1 for the cawe of comipari.ng the meta, m of a new material, product, or procwn

w ith a standard value mo0 .

As explained in the preceding diviussion of tht Operating Characteristics of the Test, an error

in estimating cr( ' -~ o,) will have no elect c'.n the eignificance level of the test, provided that

the, value of $I- is taken equal to the 100 (1 - ce) percentiWe of the i-distribution for flA + ?&B - 2

-ýdegree of freedom, where ttA Wid iaB are the sawi.Ae sizcs actually employe-d; and if VUA 0 OD, th!

'effect will not be serious provided that. h~e ;1,-uple sizes are taken cqiial.

j 3-3.2.2 (Case 2)-Variability of A ane, 3 is Unkiiawn, Cannot Be Assumned Equal.

Consider the data of Data Sample 8-3.1.2. Suppose that (from a consideration of the methods,

,sd not after looking at the results) the qiieztion to Lue tasced was whether the average for Method A

g'xmod4 the average for Method B.

Pr*cedgurO* Example

(1) Choose a, the significance level o' the t~st. (1) Let a -. 05

-(2) Compute: 1A and 4!, lo and e, fror., the -(2) gA - 8166.0

§ 4221,661.3

(8) ompute: () V=__$A -'-

6328.67

and -1582.17

Si, 221,661.3

the estimated variances Of XA and Is,,A 6901 espectively.

_(4) ,_Compute the "effectivre nanber of degrees ___4~)~o reed.m

A" -r "- (26211.20)' -

'_r m I -2 .~500652.4 +60658911.9

W5A + 1 ftBN+ 1 11.233 -2

(5) Look up h-~ for f' degrees of freedom ir (5) ]"=9

Table A-4, whare f is the integer nearest to- 116 1.833f; denote this value by t~

U -tI.. VrA 11I- -/221.21.833 (161.90)

Wan footaot 31 ae~7. ~287

3.36A

7 . - -- ' -. ____

COMPARING ~ AVEAG PE -RA ~ AC 0-

of__ A xed h vrg fB tews, Cnld httl Ar&frMtoof B.

/ (8) IfJtTAadtBb h 14.~gdru, Lý_ f ()(A-I)-u -m 256 -29.7(A a -B.) Not thatci thatte anveage (7 from 62.8.Th 92intwihslervlfo 1. th an ts

aofnAe-sieed 1he avea1 onfiB;oterwiein Contevldesthmateo the tvruge forretov Aedervidesthmateo thertre idn cnt eifereve texeed the averages for teMethod B

that2. th(Case of VAreability thn aerfomage o aho n sKonf rvosEprec

V ~ ~ n th thnatd the vilon from 628.8 The ntrva frospe288itvely.

Thervbseevtimnalt of the true otdifference 3t9.e1n thfae haerages ofr sthel m athod. n

addition, it now is assumed thp:s", £hc ý-ariaL jity for the Zwo conditions was known from pr~elilusexperience to be UA =10.25 &A('~ ff~ -500-

10n.- Whd Normal Test)H.Procedure Example

()Choose a, the significance level .:4thie test. (1) Let a - .05

Look upE of. thebe -. (2 -1.4

(3) CoRIpute: 9Aand 1,,teni.c f h t 165ad na measuremen'tafrom Ar~( ~ r105 t

R'A

t,156.2

0'"A ItIE - 1.645 (4.472'- 7.4

of A exceeds teaverage of B; ot.'iarwise, Conclude that the averaga hardneas for

that the average of A exceeds the average for Condition B.

AadB Noethat th nevdfo ie ofdneinterval estimate of theI(A-X)-UIto coi ý-aoe redfee,*betwe-m averagea.

true differecee(MA -,MA)

*,e ~wnt4%adab h dsusinfth rpete ndlmtain o hs yeofpoe__i~a~fp " -.-.2.


OQmraing Characteristics of *ae Test. Figures 3-7 and 3-8 give the operating characwristic (OC)rgurves of the above test fur . - .05 and a - .01, respectively, for various values of n.

If "A - na - n and (mA - ri) is the true positive difference between the averages, then putting

d (VA--I

we can read p, the probability of failing to detect a difference of size (MA - MJI).

If iA - 01B, we can put

d -(MA -rMB)

v + c03,- nd again read P, the probability of failing to detect a difference of size (MA - )n.,).

-SC~ioR of S4mpi SiZe. We choose

a, the significance level of the 'estp, the probability of failing to detect a difference of size (MA - ma).

I we wisA itA - nj . n, we compute - -

said we may ue. Table A-9 directly to obtain the required sample size n.

Ij we wish to have 4 n and nr such that 1A - Mn., then we may compute

d- )

.- ,,,,,,ia. Tabni. A-A -o% obs--n i - .--,,Iw

3-4.2.4 (Cob- 4)-Th. Obsevatims are Paired.

Often, an experiment is, or can be, designed r* that the observations are taken in pairs. The two-inits of a pair are cld osn in advance so as to be a nearly alike as posible in &1l respects other than:the characteristic to be measured, and then one member of each pair is assigned at random to

-• stnent A, and the other to Treatment B3. For a discussion of the advantage of this approach,¾F at 'aragraph 8-8.1.4.

Daa• Sample 3-3.2.4--Molecular Weight of Dextrons

During World War HI bacterial polysacchaides (dextrons) were considered and investigated foruse as blood plaama extenders. Sixteen saples of hydrolyzed dextrons wcre supplied by variousmanufactuirer in order to assess two chemical methods for dete-raining the average molecular

2fýM od A _,id.@hod B X

~2,0)N$,400 £$300

-ý29'10L) Z7 , 5 Wxbý42,200 _"2 200

47,800 ,.46,80 M ,ow

40,000 3-7 100 2,W0048,400 37306,100

8... ,0 6,200 -400

85,8w00-48


Mathad A Method B ItX

8,.085,200 1,80044,200 83,000 .-V,20084,O00 82,200 2,10081,800 27,300 4,000 _88,400 86 100 2,800 -47,100 42,100 4,00042,100 88,400 ,042,200 89,900 2,300

Procedure Example

(1) Choose a, the significance level of ihe test. (1) Let a -. 05

(2) Compute the 9,1 and aj for the nit 'fter- (2' ;zd - 2875ences, X,. Each X, represents an obsTva- 8, - 2182.2 -

ticu on A minus the pairc'i observatioj on isn - 16 7 : * -•-•

B.(3) Look up t-. for n - 1 degiees of freedom1 (3) t.,9 for 15 d4.. 1.753

in Table A-4.

(4) Compute ,4)

U- U - 17.6

- 1.753 (545.6)- 956.4

_(6) If 1, > u, decide that the average of A (6) tg - 2875, which is larger than u. Con-exceeds that U B; otherwise, there is no elude that the average, for Method A -A

k ;-*, th6 f vc, o A -- A -... .

that of B.

(6) Note thst the open interval from 9, - u (6) 1 ui - (2875 - 956) - 1919. The in--:7to + o is a or.e-sided 100 (1 - a) % con- terval from 1919 to + c is a me-aided

-- " fidence interval for the trie difference 95% confidence interval for the true dif-(mx - mi). ferenue batweea the averages of the ,.wo

methods

Operating Characteristic, of the Test. Figures 3-5 and 3-6 gi-e the operating characteristic (OC)curves of the test for a .05 and a .01, respectiw'ly, for variou valu• of n, the number of pUar.

involved.

Choose: M

-6 (m -(MA ), the true difference between the a-ierages (unknown, of courge)

Some value of or ( a,), the true standard deviation of a signed diftemen .?4(We may use an estimate from previous datu. If OC curve is -. _consulted after the experimt we may use the eat from .

the experiment.) .---

"8-9


We can then read fr-om the O curve for a given significance level a and number of pairs n, a valueof 0(5). The 0(a) read from the curve is 0(61 a, a, n), i.e., 0(a, ritwn c, a, t) -tI e probability of failingto detect a difference (m, - ,n,) of magnitude 35 when the given test is carried out with n pairs,.t the a-level of aignificance, arl the population standard deviation of the differences Xd actuallyis C.

If we use too large a value for a, the eiTect is to underestimate d, and consequently to overestimate#(6), the probability of not detecting a difference (mA - roi) of size 4. 6 when it exists. Conversely,if we choose too small a value of or, then we shall overestimate d and underestimate F(a). The truevalue of f 6) is determined, of course, by the actual number of pairs n, the significance level aemployed, and the true valut of ; (- ad).

MSoctiten of Number of Pai, (n). If we choose

a = (MA - MB), the value of the (positive) average differenc- that we deaire to deitcta, the significance level of the test

Sdop p, the probability of failing to detect a difference of +a! .- and compute

d(mA MB)

"where o (- Id) is the standard deviation of the population of signed differences X4 of the typeSconcerned, then we may use Table A-9 to obtain a good approximation to the required number of

-pairs &. If we take a - .01, then we must add 3 to the table value. If we take a - .05, then weL add 2 to the table value. (rn order to compute d, we must choose a value for a.) I

-if oh.. ... linni . ne w' '. '.o wt -, "ý1'o- v, teanreuen eam two-fold: . .rst_, we over-

estimate the number of pairs requiiedl, and thus utinec----arily increase the cost of the test; but, by"'

employing a sample sie that is larger than necessary, the actual value of 0(a) will be somewhat lessSthan we intended, which will be all to the good. On the other hand, if we underestimate c, we shallt ounderbtimate the nubmber of pairs actually required, and by yusing too mall a sap!q size, #(a) will

be somewhat largevr than we intendea, and our chances of detecting real differences when they exist-Will be correspondingly leawued.

: doesnoon,S .. Finally, it should be noted, that inasmuch as the test criterion u - tj,_ dot, not depend on a,,rigF ... -n- 1-,, -•1, at iemifie-~ne• which1 will ba n error in estim at n or w h en p ia nu fi g , th tv ,a .. . .. . . . . ... n ot a 4,- ... ... lev el ... .. w h ch w ll

p )recisely equal to the value cf a desired, proide. that ti, is taken equal to the 00 - a) per-2:entile of the t-distribution for n - 1 degrees of freedom, where n is the number of pairs actwuau

3.4 COMPARING THE AVERAGES OF SEVERAL PRODUCTS

Do the averages of t products 1, 2, .. ., t differ? We shall assume that ni - n. .... nl -

- If the n's are in fact not all equal, but differ only slightly, then in .te following procedure we mayplo1awe # by the harmonic me.a of the n's, I

t= /(1/n 1 + i/nt 1-... - 1/r,)

and obtain a sati- 'aetory approximation.

$-40

X7 zt:

COMPARINkG AVERAGE PERFORMANCE AM CP 70t6-110

Tefloigdata re~eto breAking-trengthof cencnt brquettes (irpounds per sqtare inch),*

12 3

618 508 554 5M6 b86

53 28 579 650 528510) 534 5W8 G53 572544 63S8 544 540 W0

~x. 2670 2682 2818 2747 2634

534 r. 0 526.4 562.6 549.4 626. 81'XI 1427404 1440924 168Z141 1509839 1391354

IA S125780 1438624.8 158259,q.8 1509201.8 1387591.2 ~

zxI ';X 1624 2299.2 2547.2 687.2 8772.8 b8'406 574.8 636.8 109.8 I 943.2

Exterpted wlit pmwmkib~ froir Swaik.aJ Xxrm Pdrit 11, A sabo4a qf Vainaae &nd Aama-Wd T~ehAnie by' N'. L Jobanoa. Copyrfght. 1947Departmat of StatUstm UnIvgndtj Ce&k'. Lmidan.13

Prtcedur* Example

(1) Chuosc at, tLe significance level (the riak of (1) Let a - .01concluding that the averages differ, whenin fact all avexagva awe the samle).

(2) Compute: (2)

8 574.8 -Ma!-636.8

a,- 159.3 _

~-S43.2

(3) c~/t jSIf the n, are not all equal. the iollowiujg j540..-.

formula usually is to be preferred:

a- 23.82

(ft 1+n3+... +nj)-t

&477

A14CP 706-116 ANALYSIS OF MEASUPEMENT DATA

Q AProediure F uonplo

(4) LooIk up qi-. (t, r) in Table A-10 where (4)(, (a+f. +... +fs. + + 25 -5

.20tu~5

q.5 (6,20) -5.29

(5) Compute (5)ii. .5.29 (23.321

123.362.286

- 55.2

(6) li the absolute difference between any two (6) The greatest difference between samplesample means exceeds w, decide that the means is 562.6 - 526.8 -35.8, which isaverages differ; otherwise, decide that there l1w than w. We, therefore, have no reasonis no rason to believe the averages differ, to believe that the group averages differ.

N-1.: it is worth noting that we simult.neously can make confidence interval estimates for each of

the 2_L 2 paL-6 of differences between product averages, with a confidence of I - a that all of theestimates are correct. The confidence intervals e.e (Xi - Si =Lw hr • r ail en

of the, , wher•,•, produca l

REFERENCES

1. E. S. Pearson and H. 0. Hartley, Rionwtrika Tabies For Statiticians, Vol. I,(2d edition), pp. 27, 136-7, Cambridge University Press, 1958.

--2. W. H. Trickett, B. L. Welch, and G. S. James, "Further Critical Values for-the Two-_Meaw Problem," Biometrika, Vol. 43, pp. 203-205, 1956.

24

2/ : ., .

' (9

8-2

CHANI71t 4

COMPARING MATERIALS OR PRODUC'S WITH REWPECT TO

VARIABILITY OF PERFORMANCE

4-1 COMPARING A NEW MATFRIAL Ok PRODUCT WITH A STAISARDWITH RtEbPECT TO VARIABILITN OF -)kRFORMANCE

The variability of a standard material, product, or process, as measur -d by its standard devi~tie'n,is known to be co. We consider the following three problems.

(a) Does Lhe variability of tWe new product differ from that of Lhe standard? St.ý I'.ragraph

(b) Does the variability of the new product execeed that of the standard? Slee Paragraph 4-1.2.(c' , Is the variability of the n..w product less than that of thc S"Candard? iee Faragiaph 4-1.3.It is important to decide which of the three problerrs i.- appropriate beforeý takirZ the observa-

tions- If this is not dorc, and the choice of prot-eni is influeocA~ by the observati, ins, both theeignificance leve! of the test (i.e., tile probability of an Error of the First Kind) and the operaitingchzaractee-istics of the test may dit~er considerably from their nominal vai,,es.

(a)- thes- gbicveins for eanct, m w ioheorpoesac aeirndmyfo snl oulto *(a) tiestbsgerains a ore ean t, when:r rcs ret irndmyfomasnl pplto

of possible observations; and,(b) within the population, the quality cl., iacter'Astic measured is normally distributed.

4-1.1 DOES THC VARIABILITY OF THS I0hýW PRODUCT DIFFER FROM THAT OF THE STANDARD?The variability in the perfornianee: of a standard material, product, or process, -as measured by

its standard deviation, is known to be q,). We wish to determine whet her a given item) di-oer. :-Invariability of performance from the standard. We Nvish, from analysis ef the data, to make oneof the following decisions:

(a) The variability ;n performance of the iiew product diffurs fi-o-a that Of th~e -,tandard.t(b) There is no reason to believe the varial~ility of the new prod~uct is differert from that of the

standard.

Data Sample 4-1.1---Capacltr of batteriesThe standard deviation a,, of capacity for batteries of a standard type is known to be 1.66 ampere

hours. The following capacities (ampere hours) were recorded for 40 bitteries of a nevi type:146, 141, 135, 142, 140, 143, 138, 137, 142, 136.

We wish tn compare the ncw type of battery with the ste.-ndard type with regard to variability) of capacity. The ciuestion to be answered is: Does t..ý tnew type differ fromn the standard typewit)' ebpect to variability of capaicity (either a decrea,ý or an; increase is of interest)?

4-1k-


Procedum Example

(1) Choose a, the level of significance of the (1) Let a .05test.

(2) Look up Bu and B L both for n - I degrees (2) n - 1 -9of freedom in Table A-20. Bu for 9 d.f. - 1.746

EB for 9 di.,. - .6657

(8) Compute s, from the n observations ..)

S- *, X-(..... ... =......5:1960i6

Compute: .(4)

8L = EL , = (.6657) (3.464)=2.31

"av BuS sU = (1.746) (3.464)=6.05

(5) ,If . does not lie between SL and su, decide (5) Since c0 = 1.66 does not lie between thei ;:Ythat the variability in performance of the limits 2.31 to G.05, conclude that the vari- r )

m new prodtet differs from that of the stand- ability for the new typý does differ from the ,4 rd; otherwise, that there is no reason to variability for the standard type.

believe the new product, differs from the"7 standard with regard to variability.

tP I no~hnl~nth. teintertn h -1, (6V 'Ph,-. W.#,=..i frm-'sto Su a 100 (1 - a) % confidence interval hours is a 95% confidence interval es t imateestimate of c, tLe standard deviation of the for the standard deyjation of the new type.new product. (See Par. 2-2.3.1).

> t@perasing Characteristics of the Test. Operating-characteristic (OC) curves for this Nevman-,earson "unbiased Type A" test of the null hypothesis that , = o, relative to the alternative thatv 0 ,are not currently available except for two special cases considered in the original Neyman-P _Pearson memoir.') These special cases and more general considerations indicate that the OCc-ýcurvs for this test will not differ greatly, except for the smallest sample sizes, from the OC curvesion the corresponding traditional "equal-tail" test (see Figures 6.15 and 6.16 of Bowker and Lieber-

nman("). The OC curve for the present test for a given significance level and sample size n will lieabove the OC curve of the corresponding "equal-tail" test for a > re and below the O cuve for the"equal-tail" teat for r < au. In other words, the chances of failing to detect that a exceeds aro are

;omewhat greater with the present test than with the "equal-tail" test, and somewhat less of failingC detect that a is less than a.. The reader is remindod, however, that if there is special interest indetermining whether o > ac, or special interest in determining whether o < au, the problem andprocedure of this Paragraph is not at all appropriate, and Paraigraph 4-1.2 or 4-1.3 should be ( )consulted. - ....

4-.2

. . .- ----..

COMPARING VARIABILITY OF PERFORMANCEl AXCP 7064110

4-1.2 DOES THE VARIABILITY OF THE NEW PRODUCT EXCEED THAT OF THE STANDARD?The variability in perforrnLnee of a standard material, product, or process, as measured by its

standard deviation, is known to be go. We wish to determine wihether the variability in peýrform-r.nce of a new product exceeds that of the standard. We wish, from analysis of the data, to makeone of the followiný decisions:

(a) The variability in performance of the new product exceeds that of the standard.(b) There is no reason to believe the variability of the new product exceeds that of the standard.

In terms of Data Sample 4-1,1, let us suppose that-in advance of looking at the data!-theimportant question is: Does the variability of the new type exceed that ot the standard?

Procdure Exam*le

(I1) Choose a, the level of significance of the (1) Let a .05test.

(2) Look up A. for n - 1 degrees of freedom in (2) n 1 = §Table A-21. A.o. for .4 d.f. = .7293

(3) Compute ., from the n observations. (3) a = 3.464

(4) Compute 8 L = A, s (4) SL - .7293 (3.464)= 2.53

(5) If SL exceeds go, decide that the variability (5) Since 2.53 exceeds 1.66, conclude that theof the new product exceeds that of the variability of the new type exceeds that of) standard; otherwise, that there is no reason the standard type.to believe that the new product exceeds thestandard with regard to variability.

_A6) It is worth noting that the interval above (6) The interval from 2.53 to + oo is a 95%_-8L is a 100 (1 - a) % confidence interval -confidence interval estimate of the stand-:esimnat e of a, the sRandard deviation of the ard deviation of the new type.new product. (See Par. 2-2.3.4).

Operating Characteristics of the Test. Figure 4-1 provides operating-characteristic (OC) curves,.of the test for a - 0.05 and various values of n. Let or, denote the true standard deviation of the

inew product. Th~in tho Or. mrucm@ nf Mm im A- nu, thai nr 1%hihituf A A fl I AR~ M f,to conclude that a, exceeds go when 01 = -wo and the test is carried out at the a = 0.05 level ofsignificance using a value of s derived from a sample of size n. Similar OC curves for the case ofa = 0.01 are given in Figure 6.18 of Bowker and Lieberman.(') OC curves aie easily constructedfor other values of n - ond, if desired, other values of a - by utilizing the fact that if the test isconducted at the a level of significance using a value of s based on a sample of size n, then theprobability of failing to conclude that or exceeds g0 when a1 = ,ao' is exactly # for

X = N (a, 0, v) = N/i-4-. (n - !)/x) (f,- 1),

where xp (P) is the P-probability level of x' for v degrees of freedom, as given in Table A-8. Valuesof p (a, nfin) f (a, P, n) corresponding to a = 0.05 and a = 0.01, for fi = 0.005, 0.01, 0.025,) 0.05, 0.10, 0.25, 0.50, 0.75, 0.90, 0.95, 0.975, 0.99, and 0.995 are given in Tables 8.1 and 8.2 ofEisenhart(3) for n1 n -- 1 = 1(1)30(10)100, 120, •.

4-3


C'-I

LID _.0_4.0

ous41 prtn hrdeitc fteoesde -ett eeiewekrtesadr e0a

!i~~.I-. peatil caaceitc of th ne-wde productt textceneehehe the standard deviation ao ~tad( 0)

'TAdapted with pemiso from A suab o MathýmA~e Sta"Nes,4 Vol -17, N,% 3, Join. 1946, from wtaW etit ~led "Operadng Charactediatim fair the

'Selec.Aon of Samrple Size. If we chokxe

a, the significa wce level of the testand, 3, the probability of failirg to detec.' that n' eiceeds o~when Q'I

74hen for a-0.06 we may use thie 00' cWve§ of F-ure 4-1 to detrmijne the neciesary samnple

W~inple: Choose

a- 0.05=0.65

-'hen from Figure. 4-1 it is see.n that,, 3 0 is not quite sufficient, and n - 40 is more than sufficient.-Visual interpolation suggests n 3 5..,hltern..ively, one may compute the necessary sample size from the approximate formula (I

n-n (a~,) 1

4-4

0; .2 2

COMPARING VARIABILITY OF PEiCFOR"'UNCE AM-jr70-10"

(9 where z,. is the P-probability point of tiwshfu ~ vwarit, A z, ,i al 'lhjft erq A 1V& ii 4)RTable A-2 for various values of P. Thus, in the foregoing example we find

1. += 2 - (8.225)1 =1 + (,67.65)1

-34.8

which rounds to n - 35. Chand01) has found this formula generally quite satisfactory, and that"even for such a small value as a - 5" it "errs on the safe side in the sense that it. gives-(at least -

for a - #) a saimple size wbich will always be sufficient."

* ~ --hak: For n = 85, -

x(.06, .06 35) X!9 (34) 48.3

=1.50

Hence Pi = 0.06 for X -1.50.

4-1.3 IS THE VARIABIUTY OFt THE MEW PRODUCT LESS THAN THAT OF THE, STANDARD?The variability in performance of a standard material, product, or proces, es measured by it& j

standard deviation, is known to be cro. We wish to determine whether the variability ini perform-jance of the new product is less than that of the standard. We wish, from analysis of the data, tomake one of the following dkcision3:9 ~ ~(a) The variability in performance of the new product is lesa thanr that of the standard.I

(b) There, is no reason to believe the variability in performance of the iiew product is leas tharthat of the standard. -

Data Sample 4-t .3-Cutoff Bias of Tubes

manufacturer 1-1" reourded the caitoty bias &f a sarsapie often Cubes as fullowr -'''

12.1, 12.3, 11.8, 12.0, 12.4, 12.0, 12.1, 11.9, 12.2, 12.2.

The variability of cutoff bias for tubes of a standard type as measured by the, standard deviatiodis cGo = 0.208 volt.

Let us assume with respect to Data Sample 4-1.3 that the importaric question is: Is the viailty_of the new type with respect to cutoff bias less Owan that of the standard type?

Prec-4ure Example -

(1) Choose a, the level of significance of the ki Let , 1 .05test. -

(2) Uook up A,,for n - 1idegrees of freedom (2) 1 -i 9in Table A-21. A.%1 for 9d.f. = 1.645

(3) Compute a, from the n observations -8

S1826

4-5

-7T '&V*~

AMCP 706-IIG ANALYSIS OF MEASUREMENT DATA

..... oProcedu(e Example

"(4) Compute Bu - A-, 8 (00 - 1.645 (.1826)-:- 0.300

-- (5) If a, is less than vo, decide that the vari- (5) Since .300 is not less than .208, conclude-. ability in performance of the new product that there is no reason to believe that the

is less than that of the standard; otherwisc, new type is less variable than the standard.that there is no reason to believe the newproduct i3 less variable than the standard.

-(6) It is worth noting that the interval below su (6) TIhe interval below 0.300 is a 95% confi-is a 100 (1 - a) % confidence interval esti- dence interval estimate if the standardmate of o, the standard deviation of the devistion of- the new type.new product. (See Par. 2-2.3.2.)

Operating Characteristics of the Test. Figure 4-2 provides ope*ting-character.atic (OC) curvesof the test for a - 0.05 and various values of n. Let a, denote the true r.•adard deviation of thenew product. Then the OC curves of Figure 4-2 show the probability 0 = 13 (h 1.05, it) of failingto conclude tha' .a is less than ar when r -ao and the test is carried out at the a 0.05 levelWf significance using a value of s derived from a sample of size m. Similar OC eurves fcr the case-1 a 0.01 are given in Figure 6.20 of Bowker and Lieberman.N'• OC curves are easily constructed

- :I•-

"- Li]L

__ __ _____0

FPgros 4 l. OperjiWn characdr~istice of e onme-sided %-teWt to detennr•'e iAk Wh, standard devi'tiono' of a r"ew produN. is 18 than tke standard desialion ao of P, sr?ýnrd (a - .05).

ptd i io A � K( . 0snx 8tatwistia. Vol. 17. No.. 2-1une i110, from &rW1e .•_t,•Pd"opei Chw -.ftiztue for the

Aby C. V. ' F.4 F . Grub* and C. A. Wavw.

4-6

k-

p1- -l -.-- �-..-- � ' V

COMPARING VARIA�4ITY OF PER[ORMANCE AMCP 706410'71) -It

for other va4ues of 'i - and, i.� dcsred, other � 0' a - by utilizing the fact that if the tee�t is L�onducted at the a level of �ignificanc2 iwi'.i� � v�zlue of 8 based on a sample of size n, t�heui the -�'

probability o. 'ili�g to conclude thai� � u�. z '�n� � - � is exactly � for

- � - Vx�(ti - -

4 �br'� 4 (,) is the P-probability 1' vd �f x� fur w de�i e � of �reedom, as given in Table A4.

Selectioi� of Sample Size. If we choose

I:a, the Bignificance level of the test -�

and, �, the probability of failing to detect that Ci is less than �o when � -

then for a = 0.05 we m�y use the 00 curves of Figure 4-2 to determine the necessary sampla size ii.

Example: Chf.)oee

0.3

� 0.05K=) '� = 0.05

then from Figure 4-2 �t is seen that n = 10 is not quite aufficient, a�d it = 15 is more than sufficient.

Visual interpolation �uggesta n - 14.Altcrnatively, one may cor!1p1�tP the r�eeessary samvle size from the appro 1!4� forwui>a

urn

it = n(�r,.�)�) -

where z, is the P-probability point of the standard normal variable z, values of which are *ven inTable A-2 for various values of P. Thus, in the foregoing example we find �

1 + 1 (1.645 + (0.5) (i.645))2 __

1 (.i.4t��o 1+ (4±2 0.5 935)2 1 +�(24.35) 7.;

= 13.18

which rodadaton 13.

Cheek: For n = 18,

X (.05, .05, 13) X. 5.23 _� � �- 0.499 <0.50(7)

Hence, � - 0.05 for ?� = 0.50.

4-7

A?4CP 7G6-110 ANALYSIS OF MEASUREMENT DATA

4-2 COMPARING TWO MATERIALS OR PRODUCTS WITH RLSPECT TOVARIABILITY OF PERFORMANCE

We conbidt two problems:

(a) Does the variability of product A differ from that of prod-ict B? (We are not concernedwhich is larger). Sce Paragraph 4-2.1.

(b) Dows the variab'Ity of product A exceed that of product B? See Paragraph 4-2.2.

It is important to ducide whichi of thelm two problems is appropriate before taking the obse-rva-tions. Ii this is not done, and the choice of problem is influenced by the observations, both thesignificknee level of the test (i.e., the probability of an Error of the First Kind) and the operatingcharacteristics of the test may differ considerably from their nominal values. The tests given areexat when:

(a) the observations for an item, product, or process are taken randomly from a single populationof postible observations; and,

Ab) ithin the population, the quality ch aracteristic measured is normally distributed.

In the following, it is assumed the appropriate problem is selected and then nA, n.8 observationsame taken from items, processes, or products A and B, respectively.

4.2.1 DOES THE VARIABILITY OF PRODUCT A DiFFER FROM THAT OF PRODUCT B (3We wish to test %he'ber the variability of performance of two materials, products,, or procesres

dciffer, and we are not, particularly concerned which is larger. We wish, from analysis of the data,~-~omake one of tbe- followir-g decisions:

-- fWtk Th.. wn prpviefw diffar with n~rpprd tth thpir vurinhility

(b) Thereis no reason to believe the two products differ with regard to their variability.

Dale Sample 4-2.1-Div*-bombIng Methods.'The performance of each of-two different dive-bombin' methods is measured a dozen times with

N-Aethod A -Method B

~.:Z 26 414

A99 419627 .453

VV585 504459 459415 337

-460 598506 425

*450 436624 456650685

U~t us suppose that, in the case of Data Sample 4-2.1, the question to be answered is: Do the twomethods differ in variability (it being of interest if either is more variable than the other)?

71 A. 4-8

COMPARING VARIABI~LY OF PERFORMANCE AMCP 706-110

Pmodure1)Choose a, the level of siguificance of the (1) Let .-. 05

test.j

grees, of freedom, and Ft for (n, - 1, -I-U

VI 1) degrees of freedom, in Table A-5. -1,1) 8.48

(3) Compute 6A and el from the observations (3) eA- &45-

from A and B, rspectivly. 4 07'Y 8~(~Compute Y - ooA4 4)F -555/07

- 1.86

(5) .1?ý > F1....,, (nA - 1, nA - )Or (6) F.,7 (11, 11) - 3.48

F1.. , (a - , u, - ) F.~ (1, 1) =0.29

decide that the two productLe differ with Since F if not larger than 3.48, and is not--regard to their variability; otherwise, there smaller than 0.29, there is no reason tois no reasou to believe that they differ, believe that the two bombing methods

differ in variability.

(6) It is worth noting that the interval between (6) The interval between 0.89 (i.e., 0.29 XS1.36) and 4.78 (i.e., 8.48 X 1.36) is a 95%

co'dneitraAstmt ftertoo,-~a (A -lit - ) ~the-true vari-amce% OA/4

is a 100 (1 - a)%confidence interval esti-mnate of the ratio OA/0

Operating Charac*erisdi of daU Test. Operating-characteristic (00) curves for this tradttional4"equal-tail" test of the null hypothesis that O'A - 4B relative to the alternativc VA Pý -B ame given

-in Figures 7.1 a:.ý 7.2 of Bowker and Lieberman(2) for the case of equal sample sizes S1A - B=a,and signiflcancl, ie els a 0.VS and a=0.01, respectively. These curvee may be used to deter-i..ine ~he comino:ý !iample size ftA = it, = needed to achieve a preassigned risk # of failing todetect that vrA/0k X.) when the test is carried out at the a 0.05 or a - 0.01 level of significance.The reader is rtndhowever, that Vf there is speclaz irntarmt' iF& dtA-i-1si -A>;

the problem and of -,c4tthis Paragraph is not at all appropriate, and Paragraph 4-.2.2 should _

be consulted.

4-1.2 DOES THE VAU~RIAILIV1 OF PRODUCT A EXCEED THAT OF PRODUCT P?We wsh o tst hethx tl* narabilt~yin erfrmane o prduc A eceen tat f prduc B.f"'

We wish, to t esut ofheth i of~. aiblthe in perormakonce of pre ollctn A dee'lin ta o rdutBWe) wih, vasiabreslty of prodyuct A texdata th mke onf p ofucth B. lwn dcso)(a) Thrisn wotoblkc hathe variability of productt A exceed thet of productBof

product B.

4-9

V ---

Arlr',D '7• 1 A AW.AIVCICn (1 •M!•A4I ID FI.rJT flATA

*. .. .... .. ( )In teima of Data Sample 4-2.1, let us suppose that-in advance of looking at the datal-the

important quebtion is: Does the variability of Method A exceed that of Method B?

Prwcedu, Example

- (1 Choose a, the level of significance of the (1) LeW a - .05tesL

(2) 1ACk upY '_, fCr "A - 1, ta. - I degrees of (2) A 1 - 11freedom, in TabVe A-5. , - 1 - 11

F.* (11, 11) -I

(3) Compute OA, s4, the sample variances of (3) 4A- 45the observtions from A aid B, respec- a -4078tively.

)Compute F - sAI/ . (4) F- 1.86

(5) If F > F,_, decide that the variability of (5) Since 1.86 is not larger than 2.82, there is_ product A exceeds that of B; otherwise, no reason to believe thzt the variability of

there is no reaon to believe that the vari- Method A is greater than the varisbilityability of A is greater J=an that of B. of Method B.

(6) Note that the i.terval above (6)

, ; F j_. (" A I, % A - 1 ) s . ( 1 , ) " 0 .8 5

-is a 100 (1 - a) % confidence interval esti- The intetval above 0.48 (i.e., 0.35 X 1.36)• nate of 9•.4 is a 95% confidence interval estimate of the

WI&W.S kf 11"'Q W4 "V T Q& ACWAV-3ý9 ~A W It -

CCýi•g Chara'ctsxir of Ote Te•t. Figures 4-8, 4-4, and 4-F provide operating-characteristirS (C) curves of the test for x - 0.05 and various combinations of "A and ith. Let CA and a, denotethe true standard deviations of the proucts A and B, respectively. These OC curves show thep~robability 0 - 0 (XI.05, %) of failing to conclude that CA exceeds cr, when CA - Aer' with X > 1I•'*nd the test is carried out at the a - 0.05 level of significance using the values Of eA and ap derived

I from sampl~es o size KA and "B, respectively. Similar OC curves foi- the case of a -f 0.01 and

other valuem of 'A and % -- and, if desired, other values of a - by utilizing the fact that if the te-s iscmnducted at the a level of significance using values of SA and 8B based on samples of size nA and ng,rep•etively. then the probability of failing to conclude that wA exceeis am, when UA - XTB is exactly

S #for

N -p(A-," 1)- )/F,_. (A,, - 1, -8 - 1) 1, (ns - 1 -)

where F? (n,, %) is the P-probability level of F for nl and n, degrees of freedom, as fsiven in-Irable A-5. Values Of # (a, 0, fti, n2) - A' (09, 0, nA, fB) correspunding to a - 0.05 anid a - 0.01,

for 0 -. 0.005, 0.01, 0.025, 0.05, 0.10, 0.25, 0.50, 0.75, 0.90, 0.95, 0 975, 0.99, and 0.995 are givert inTables 8.8 and 8.4 of Eisenhart(g) for all comb~nations of valueb of ni - - 1 and %,2 = - 1derivable from the sequence 1(1)30(10)100, 120, -.

4-10

T-~i -P.

COMPARiWG VA\RIABILITY OF PERFORMANCE AM&CJ 70-~ 110

I V

AL

ffig4

-AA

J

ILI

t0sgiia.xlvl ftets

then fre 0-.05 wpertn -may acerisf the OC curve V Figue 4- to de~eitermineth the necessary comiiton___l sien

AMCP 706-110 ANALYSIS OF MEASUREMEN DUTA( )

m -- ____-

b

IA

M"0

UA of por'4dt A e•ceede the e~andard deviation GB of product B

SAdo~pte with perminiauim from Au~ua~o of MaLA,iwlfiol Sta4iak'., VoL 1"/, No. 2, June 1946.i, rvin a.IU~ fflldeatld "OJPeatiag Cbeura•'[ct'lslet~t 1,4"*,

FJxa m ple : Choose

S'-- - 1.5UrB

r "- 0.05a-- 0.05 N I l

then, from Figure 4-3 it is seen that ii - 50 is too smnalh and n -75 a bit too large. Visual inter- ( j)polation suggests X - 70.

4-12

-- . - --- --- - - - -. ..~ .- .

COMPARINGI VARIABILITY OF PERFORM'ANCE AMflCk 706-110

A

.4-141

402

405,0 k ~-

Figure 4-5. Operating characieristies of the ouc-sided F-test to determine whether the standard deviation

Commo Staiatfal CFAl018 of prodluct A exceedsf the standard deviation an of product B(41= .05; 11A = n8,

271A =

3 T4B, nA = ~2 fB).ý!th.. !r,~ ~, l.1..,, A-,l- -f L7.k.~4,~ V.4. 17, No 2- June 1946- from article entitled ChertngQaract~er~jticsfr h

Common.ta..stca..T.ts..-.igntificaince' yc.- P. Ferris, F. V~. Grubba: and CA. Weaver.fr e

Alternatively, fornvA =n.B n one may Compute the necessary Ksripie size from the approximate

formula

where zP is the P-probability point of the standard normal variable z, values of which are given in -

4-13 1~

AMCP 706-110 ANALYSIS OF MEASUREMENT DATA -

'Table A-2 for various values of 1'. Thus in the foregoing example we find

~ 2 + k-

-2+ (5) - 2+ (8.11)1-2w2+ 65.F- 68.

If instead of choosing nA - np we choose 311A -2flz or 2 nRA = RB, then for a 0.05 we may *use the' 00 curves of Figure 4-4 to determine the necessary combination of samPle sizes flA and nB.Similarly, Figu~re "- may be used if it is desired to have 2"A = 3flB or "A - 2n,,. Alternatively,one may evaluate the harmonic mean h Of A - 2 and nB - 2 from the approximate formula

1Z 1 .... ± Z , .~

-and h I.!g then determine the integer values of n,# and nB (sati~ying any additiortal requirements, e.g.,uiA,- 2nB) that most closely satisfy the equation

REFERENCES

_A. J. Neyman and E. S. Pearson, "Contribiu- -Selected Techniqu~es of Statistical Analyisis?Tiosto the Theory of Testing Statistical (Edited by Churchill Eisenhart, Millard

mfyothsews% I. Unbbin-sed Criitical Re- W. Hastay, and W. Allen Wallis.),gions of Type A and Type A,," Staistical McGraw-Hill Book Co., Inc., New York,1'Research Mfemoirs, Vol. 1, Department of N. Y., 1947.Ststistic~s, University College, Universityof London, 1936. 4. Uttam Chand, 'On the Derivation and

2. A. H. Bowker and G. J. Laieberman, Engi- Accuracy of Certain Formulas for Samplenwerinw Statisiiets, Prentice-Hall, Inc., Sizes and Operating Characteristics ofEnglewood Cliffs, N. T., 1959. Nonsequential Sampling Procedures,"

8.Churchill Eisenhart, "Planning and Inter- Journal of Research of the National Bureau,preting Experiments for Comparing Two -of Standards, Vol. 47, No._, pp. 491-501,

~StadardDeviations~"Catr8rlcmc 91

4-14

I 1

CHAPTER 5 - . -

CHARACTERIZING LINEAR RELATIONSHIPS

ETLEWEEN TWO VARIABLES -

5-1 INTRODUCTION

In many situations it is desirable to 'mow In this chapter, we deal cnsy with linear rela-something about the relationships between two tionships. Curvilinear relationships are dis-charactcristics of a material, product, or proc- cussed in Chapter 6 (see Paragraph 6-5). It isess. In some cases, it may be known from worth noting that many nonlinear relationshipstheoretical considerations that two properties may be expressed in linear form by a suitableare functionally related, and th', problem is to t-ansformation (change of variab.e). For exam-

--find out more about the structure of this relz- pie, if the relationship is of the form Y - siXb.tionship. In other cases, there is interest ininvestigating whether there exists a degree of then log Y = log a + b log X. Putting Yr

* association between two properties which could log Y, b0 = log a, b, = b, Xr = l,,g X, we bavebe used to advantage. For example, in specify-ing methods of test for a material, there may be the linear expression Y1 6b -I- b1X, in termstwo tests available, both of which reflect per- of the new (transformed) variables Yr and YT.

forman-e, but one of which is cheaper, simpler,*-or quicker to run. If a high degree of associa- A number of common linearizing transorma-tion exists between the two tests, we might wish tions are summarized in Table 6 1 and are di'-to run regularly only the simpler ivst. keuasd in Paragraph 5-4.4.

5-2 PLOTTING THE DATA

Where. only two characteristics are involved, a structural relationship, the plotted data willthe natural first step in handling the experi- show whether a hypothetical linear relationshipmental results is to plot the points on graph is borne out; if not, we must consider whetherpaper. Conventionally, the independent vari- there is any theoretical basis for fitting a curveable X is plotted on the horizontal scale, and the of higher degree. When looking for an empiri-dependent variable Y is plotted on the vertical cal association of two characteristics, a glance atscale. - the plot will reveal whether such association is

There is no substitute for a plot of the data to likely or whether there is orly a patternlews

give some ider, of the general spread and shape scatter of points.

of the results. A pictorial indication of the Tr, some cases, a plot wPil reveal unsuspectedprobable form and sharpness of the relation- difficulties in the experimental setup whichship, if any, is indispensable and sometimes may must be ironed out before fitting any kind ofsave needless computing. When investigating relationship. An example of this occurred in

5.-1


measuriag the time required for a drop of dye to ever, that the experimenter quite properly"travel betw~een marked distances along a water decided to find a better nicaws of recordingchannel. The channel was marked with dis- travel times before fitting any line at all.tance markers spaced at equal distances, and an I oo vosdfi ute r eeldb h-observe- recorded the time at which the dye Ifot ano obvou difclatienshi arereealedt by thnearpassed each marker. The device used for rt plote andlthe rYi-tion+hi appearartoyhshlinear,cording time consisted of two clocks hooked UP th'ahie daa ac+c1Xordinarilyshepoceuldrestso that when one was s~.opped, the other started: ftedo hdaacrintohepcdus'Clock 1 recorded the times for Distance Mark- piven in this Chapter. Fitting by eye usually isers 1. 8, 5, etc.; and Clock 2 ree~orded times for inadequate for the following reasons:

~the3ve-nuberd dstane mrkes. hen (a) No two people would fit exactly the Samethe elapsed tinies, were plotted, they looked line, and, therefore, the procedure is not oh-asoinewhat as shown in Figure 5-1. It is ob- jcievious that there was a systemnatic time differ- jcie-*nee between odd and even markers (presuma- (b) We always need some measure of howbly a lag in the circuit connecting L.je two well the line does fit the data, and of the uncer-iclocks). One could easly have fitted a straight tainties inherent in the fitted line as a repre-line to the odd-numbered distances and a dif- sentation of the true underlying relationship-ferent line to the even-numbered distances, with and these can be obtained only when a formal,approximately constant difference between the well-defined mathematicalprocedur-oftin

`two lines.__Xke off~qt. was soconaitent how- -is eMpleyed.

SPIM -- -----

4

3

S 5-2

F

LINEMAk RELATiONSHP .. TE- TW" VAR!A~tFS AMCP O6-lo to

5-3 TWO IMPORTANT SYSTEMS OF LINEAR RELATIONSHIPS

Before git ing the detailed procedure for fit- of mass x, is placed upon the pan repeatedly andting a straight line, we discuss different physical the position of the pointer is read in eachsituations which can be described by a linear instance, it usually is found that the readings Y,relationship between two variables. The meth- are not identical, due to variations in the per-ods of description and prediction may be differ- formance of the spring and to reading errors.ent, depending upon the underlying system.

In gnerl, e rcogizetwodiferet ad ~ Thus, correiponding to the mass x., there is aIn general, we recognize two different and im-

portant systems which we call Statistical and distrib.tion of pointer readings Y-; correspond-Functional. It is not possible to decide which is ing to msS xz, a distribution of pointer readingsthe appropriate system from looking at the Y,; and so forth-as indicated i;n Figure 5-2.data. The distinction must be made before It is customrry to assume that these distribu-fitting the line-indeed, before taking the tions are normal (or, at leasc symmetrical andmeasurements. all of the same form) and that the mean of the

distribution of Y,'s coincides with the true value 45-3.1 FUNCTIONAL RELATIONSHIPS + 0± 8 1x.

In the case of a Functional Relationship, If, instead of calibrating the spring balance in Lithere exists an exact mathematical formula (y as terms of a series of accurately known weights,

a function of x) relating the two variables, and we were to calibrate it in terms of anotherthe only reason that the observations do not fit spring balance by recording the correspondingthis equation exactly is because of disturbances pointer positions when a series of weights areor errors of measurement in the observed values placed first on the pan of one balance and then

2/ of one or both variable ). We discuss two casesof this typeh: on the pan of the other, the resulting readings(X and Y) would be related by a linear strune

FI-Errors of measurement affect only one tural relationship FII, as shown ini Figure 5-3,variable (Y). (See Fig. 5-2). inasmuch as both X and Y are affected by errors

FII-Both variables (X and Y) are subject to' of measurement. In this case, correspondingerrors of measurement. (See Fig. 5-3)_ to the repeated weighings of a single weight w,

Common situations that may be described by (whe tr--e ma-ss noeed not he known), there is aFunctional Relationships include calibration joint distribution of the pointer readingslines, comparisons of analytical procedures, and (X1 and Y1) on the two balances, represented byrelationships in which time is the X variable, the little transparent mountain centered over

For instance, we may regard Figure 5-2 as the true point (xi, VI) in Figure 5-3; similarly atportraying the calibration of a straight-faced points (x2, y,) and (r., ya), corresponding to re-spring balance in terms of a series of weights peated weighings of other weights wv, and to,,whose masses are accurately known. By respectively. Finally, it should be noticed thatHooke's Law, the extension of the spring, and th| FI mod-) is more general than the Flhence the position y of the scale pointer, shouldbe determined exactly by the mass x upon the model in that it does not require linearity ofpan through a linear functional relationship* response of each instrument to the independent

PIo P- x. In practice, however, if a wsight -variable w, but merely that the response curves

Note oi. Notation for Fuevtiona, Relationships: and Procedures and Examples for the FV ease, however,We have used z and V to denote the true or ac.irately we use X and Y bec•ause of the computational similarity

known values of the variables, and X and Y to denote to other cases discussed in this Chapter (i.e., the computa-t heir values measured with error. In the FI Relation- tions for the Statitical Relationships)..hip° the independent variable is alway without error, In the PH1 case both variables are subject to error, andan therefore ;n our discussions of the I case and in the :learly we use * nd Y everywhere for the observedparagraph heakdings we always use x. In the Worksheet, values.

S.. .. . . ..i.7 • .• :?

AMKCP 700- 110 ANALYSIS OF MEASUREMENT DATA

oi the two instruments be linearly related, that of F1 and Fil relationships. Detailed prob-is, that X - a + b -f(w) and Y - c + d - (w), lems and procedures with numerical ex~ampleswhene 1(w) may be linsear, quadratie, exponen- for FI relationships are given in Paragraphstial, logarithmic, or whatever. 6i4.1 and 5-4.2, and for F11 relationships in

Table 55-1 provides a concise characterization Paragraph 5-4.3.

yy

-Y

X4

Figure 5-2. Loiiwar functiona rclatiwn*As of Type F1~Ij(onli Y affected by~ measisrementt ortors).

-LINEAR RELATIONSHIPS BETWEEN TWO VARIABLES AXCP 706-.110

YtY3 A

JON SRBTOY21

OF X1AND Y

)~~igtsre5-8.Y resotnhi of, TyXFI

Figuno e-a.tLmaehematicala relationshof Type F1

X and Y; 'there is only a statistical associationbetween the two variableb as charateristics of ___

individual items from some particular popula- Strictly, we JOwuld writetion. If this statistical association is of b,:- th~ + A X,variate normal type as shown in Figure &-4, anthen the average value of the Y's associated with MX,. + 1%a particular value of X, say f x, is found to de- to conform to our notation of using mt to signify apeiid linearly on X, i.e., f x - o + #ý X; simi.. population mean. But thi. more exact notAtion tends

the verge vlueof te Xs asocited to conceal the paralieliarn of the curve-fitting processeslarly, teaeaevleothX'asoied in the F1 and SI situations. Cone~equeatly, MOMaerv

4) with a particular value of Y, say XZy, depends &eaane headInteseul e s jl v~linearly ~ ~ ~ ~ ~ ~ ~ ~ O onY(i.54y~. y=~+ ~ om.x and Xrin Name of oxt.y--and It should belinarl onY (ig.5-4 i~., 9y o' Y; rdruembered that these signify ponap toI ~maIe.

5.&

j~- - -~~~-$~7

_ . .. -. . .. : . , ••a

AMCP 706-110 ANALYSIS OF MEASUREMENT DATA_ . . .. ( <)

weight of one individual from his height, but wemight expect to be able to estimate the averageweight of all individuals of a giver. height.

The height-weight example is given as onewhich is universally familiar. Such examples

V .Ao, Ax also exist in the physical and engineering sci-ences, particularly in cases involving the inter-relation of two test methods. In many casestiere may be two tests that, strictly spealdng,measure two basically different properties of amaterial, product, or process, but these proper-

Sties are statistically related to each other in- -• -- some complicated way and both are related to

some performance characteristic of particular.. interest, one usually mo?.e directly than the

other. Their interrelationship may be oh-

scured by inherent variatiens among sample

units (due to varying density, for example).We would be very interested in knowingwhether the relationship between the two.'issufficient to enable us to predict. with reasonableaccuracy, from a value given by one test, theaverage value to be expected for the other-p:-'rticularly if one test is considerably simpler _or cheaper than the other.

Figure 5-$. A normal bivariatefrequene surJae. The choice of which variable to call X and'which variable to call Y is arbitrary-actually

diur-Iwi ,,Ui tihe association is found, ordinarily the variablepopulation, and the two characteristics X and Y asiai is found, ordinaril the variabe

a. smeaure oneac itm, hentypcaly . which is easier to measure is called X. Note Ia a m easu red on each ite rn, then ty p ically it is w el t a th si t e on y c e of i e r r l t o -found that errors of measurement are negligible well that this is the only case of linear relation-•in comparison with the variation of each char- ship in which it may be appropriate to fit twoacteristic over the individual items. This different lines, one for predicting Y from X andgeneral case is designated SI. A special c a different one for predicting X from Y, and the

only case in which the sample correlation co-(involving preselection or restriction of the enlicie in whic gfuh san elate o-

~efficient r is meaningful as an estimate of V eringe of one of the variables) is denoted by SI!. degree of association of X and Y in the popula-

N kwaionships. In this case, a random hion as measured by the population coefficient..sample of items is drawn from some definite of correlation p - V/•1•. The six sets of con-Spopulation (material, product, process, or tour ellipses shown in Figure 5-5 indicate thepeople), and two characteristics are measured mauiwer in which the location, shape, and orien-on each item. tation of the normal bivariate distribution

A classic example of this type is the relation- varies with changes of the population meansship between height and weight of men. Any (mx and my) and standard deviations (ax and ay)observant person knows that weight tends to of X and Y and their coefficient of correlation invary with height, but also that individuals of the population (Pr,).the smme height may vary widely in weight. It If p - =L1, all the points lie on a line andis obvious that the errors made in measuring Y - Po + PX and X -- + - Y coincide.height or weight are very small compared to If P - + 1, the slope is positive, and if p = -1,this inhe'ent variation between individilals. the slope is negative. If p - 0, then X and Y , FWe smrely would not expect to predict the exact are said to be uncorreluted.

S . ..- • . .. ' '• - =.. ._• :•L-y - • _:2• ,C ,, _ =.5 4.=

...... .......

LINEAR RELATIONSHIPS BETWEEN TWO VARIABLES AXCP 706-110 J2222

20~r~ I' •F a 202%,z

18- 2x• 1- -' P:20- 20--2.19191 -- - _ I

18 -- .4 msa6 isX - - -

-1m .2Z- myu2O14 15 16 J7 18 26 27 28 29 30 31 32 33 34

43 - 43-42 - 4 P.-.

41--- 41 ----

- ay 2 LI840 40 -

39 39 -- - ms3318 -m0 38--- ~ - mv0

14 15 16 17 18 a& 72829 3031 323334

)is 18

17- - , 17--- ---.-------- p.

,16 , 16 >1 1 1 _I1 Vx -1-

14.z16 , ,, -. A14. 15 16 17 I 26 27 28 29 30 31 32 33 34

Fiqu-e 5-5. Contour ellipses for normal bivariate distributions.- --- of L&. fivn parameters m... y'., . 'pv*

AdAptd with ,arml tn fr o A Infrst cl by Helen M. Welker and J0oeph Lev. Copy- U

"right. 913, býit, Rinshart ad Wrnstxn, I, NowYork, N. V.

S•I RelcIonshlps. The general case described those whose heights were between 5'4" and

above (SI) is .he most familiar example of a 5'8". We 'now are able to fit a line predicting

statistical relationship, buL we also need to con- weight from height, but are unable to determine

sid..r a common case of Statistical Relationship the correct line for predicting height from

(SII) that must be treated a bit differently, In weight. A correlation coefficient computed

Sl1, one of the two variables, although a ran- from such data iE not a measure of the true

dora variable in the population, is sampled only correlation among height and weight in the (un-

within a limited range (or at selected preas- restricted) population.sigmn values). In the height-weight example, The restriction of the range of X, when it is

suppose that th,. group of men included only conaikered as the independent variable, does

b-7

.,------ . ___ _____________*

7!

AMCP 706-110 ANALYSiS OF MEASUREMENT DATA

not spoil the estimates of ?x, when we fit the Y RoY ,

line Ix - be + bX. The restriction of the --range of the dependent variable (i.e., el Y in -

fitting the foregoing line, or of X in fitting the % --

line r b. b.' + bY), however, gives a seriouslydistorted estimate of the true relationship. XThis is evident from Figure 5-6, in which thecontour ellipses of the top diagram serve torepresent the bivariate distribution of X and Yin the unrestricted populAtion, and the "true"regression lines of f]' on X and ly on Y are yindicated. The central diagram portrays the j 0K X UNRESTRICTED)

ituation when consideration is restricted to my - 'X°X b.X-b

items in the population for which a < X < b. • xIt is clear that for any particular X in this in- -xterval, the distribution and hence the meanYx of the corresponding Y's is the same as inthe unrestricted case (top diagram). Conse-quently, a line of the form ?x - b0 + b,X fitted yto data involving either a random or selected set L-4_-VXoN X (UWSTCToD

of values of X between X - a and X - b, but - -- ON K (with %o selection or restrictions on the corre- Wy K (C.. .

sponding Y's, will furnish an unbiased estimate C --- - -

of the true regression line fx -. ,8o + P1X in the (..populatiou at large. In contrast, if considers-*ton is restricted to items for which c < Y < d, Figure 6-6. Diagram showing effect, of restrictions

--- inlcaetM in the bottom diagram. then it is of X or Y on the regression oj' Y on X.clear that the mean value, say 1*, of thervot•k.ted) Y's aasociated with any particular

value of X > tix will be less than the corie-spending mean value Y'x in the population as ci-nt r as a measure of the true coefficient ofa whole. Likewise, if X < inx, then the mean correlation p in the populations, when eiter XP1 of the corresponding (restricted) Y's 'ill or Y is restricted, see Eisenhartio and Ezekiel.(,)be greater than flr in the population as a whole.Consequently, a line of the form ?r - be + bjX As an engineering example of SII, consider afitted to data involving selection or restriction study of watches to investigate whether thereof Y's will not furnish an unblased estimate of was a relationship between the cost of a stopthe true regression line Fx = Po + PIX in the watch and its temperature coefficient. It waspopulation as a whole, and the distortion may suggestd that a correlation coeflic'.nt be com-he serious. In other words, introducing a re- puted. This was not possible because thestriction with regard to X does not bias infer- watches had not been selected at random fromences with regard to Y, when Y is considered as the total watch production, but a deliberateihe dependent variable, but restricting Y will effort had been made to obtain a fix-d numberdistort the dependence of Px on X so that the of iow-priced, medium-priced, and high-pricedrelationship observed will not be representative stop watches.of the true underlying relationship in the popu-lation as a whole. Obviously, there is an In any given caw-, consider carefully whetherequivalent statement in which the roles of X one is measuring samples as they come (andand Y are reversed. For further discussion and thereby accepting the values of both properLi•sillustration of this point, amd of the correspond- that come with the sample) which is an SI Rila- 0ing distortion of the smaple correlation coefli- tionahip, or whether on wects ownple which

LINE.AR RFI ATIONSHIPS BEIWEEN TWO VARIABLES AM(;P 706-110 1 '

_ _ _' . 'El

"• • "• .... '1o w o:

\- . . .

ro .0

A..


are known to have a limited range of values of X problems and procedures with num~erical e~xam-(which is an S11 Relations;hip). pies, are given for S1 relationships in Paragraph

¶To ..ie 6-1 give-s 4 brief sumnmary characteriza- 5-5.1 and for S1! relationships in ParagraphLion of SI8 and S11 Relationships. Detailed 6-5.2.

BASIC WVýK$HEET FOR ALL TYPES OF LINEAR RELATIONSHIPS

X denotes Y____________ 1 denotes

NYunber of points: n -

Step (1) IXY__ _____

(3) S.. t~4 Step (2)1

(4) ZXI -- (7) : . Y' (

(5) ( 2X)2 n -- (8) (Y) 2/n ___

(6) S. Se(4 Step () ()S1 Step (7) -Step (8)

Ste (5) (9 .

(10) b Step (3) Step (6) (14) p wihnueialeam__

(11) i(1) (n -2) sh = Step (9)- Step (14)

(12) b, 1 _ (16) st = Step (15)_- (n - 2)

(13) 60 - - =i. Step (11)- Step (12). = _______

tion li I n e: Estimated variance of the slope:

Sdnt4- b 1X dnt StS1tep (16) S p (6)

Estimated variance of intercept:

8b) =X I7 Y f l

(5) (X)'i (8) zY)'n + ._(6% ,sL =4{.•p.•• Se+() 9 8, } f - te (7 - Step (8

Note: The following are algebraically identical:

.- 2(X - .9)'; .- ((Y15 n -f)); S, 7t(X - X) (Y - 4).

Ordinarily, inr hand omputation, it is preferable to compt'te as shown in the steps above. Carryall Aecimal places obtainabl, - -i.e., if data are recorded to two decia. places, carry four places in_Steps (1) through (9) in order to avoid losing significant figures in aubtraction.

., - --- L - ..

--, fc~v"

LINEAR RELATIONSHIPS BETWEEN TWO VAidABLES AMCP 15-4 PROBLEMS AND PROCEDURES FOR FUNCTIONAL RELATIONSHIPS

5-4.1 Ft RELATIONSHIPS (General Cobs) Data Sample 5-4.1-Young's Modulus Vs.Temperature for Sapphire Rod*

Tlh.ere is an underlying rnatlieiatical (func- Observed values (Y) of Young's modulus (y)tional') relationship between the two variables, for sapphire rods measured at different tempera-

of te fom y~1o Te vriabe ~ t~uroýs (x) are gi ,,en in the following table. Thereof te frm y + Ox. Te vriabe xcan is as9sumed co be a linear functional relationshipbe measured relatively accurately. Measur~- bew nthtovaals ady.(rtements Y of the value of y corresponding to a purpose. of computation, the observed Y' valueegiven x follow a normnal distribution with inean were coded by subtracting 4000 from each. To

+ f+ jx and variance at.,, which is independent express the line in terms of the original units,of the value of x. Fr'rthermore, we shall, as- add 4000 to the computed intercept; the slopesume that the deviations or error8 of a saries of wiln*Aeafetd)Teosevddt robserved Y's, corresponding to the same or dif- potdi iue57ferent x's, all at-e mutually independr SeeParagraph 5-3.1 and Table 5-1. Cdd1

aY Young'sTemperature =Young's Modulus

The general case is discussed here, and the 0C Modulus minus 4000special case where it is known that po =0 (i.e., -___________________

a line known to pass through the origin) is dis- 30 4642 642cusdin Paragraph 5-4.2. The procedure dis- 1041 1

cusd206 4565 565 4cussed here also will be -.ralid if in fact 0, = 0 30 535131evern itbnih this f-aut 'is impt 'Known, beorehand. LflJ3 4476 476However, when it is known that ,3o 0, the pro- 500 4433 433cedures of Paragrazph 5-4.2 should be followed 600 4389 239becauise. they are simpler and somewhat moein044 347

efic.~t.800 4303 303900 4251 251

1000 4201 2011It will be noted that S11, Paragraph 5-5.2, is 3100 4140 140 .

hundled. compatationally in exautly the same 1200 4100 100maum'1 a4 El u ot h ndryn aa- 1300 4073 73

innie a "I, utbnJ11 WC x"ý .v.-q1400 4024 24tions and the interpretation of thfe end results 1500 3999 -1are different. * -1

tr

AKCP 706-110 ANALYSIS OF MEASUREMENT DATA

7777

4400-4300

42W

'4000

3900O 957i-

71(1

0 o 700 a0 900 m000 1100 200 • .

X STEMPEOHAThRE *C

u ~.Fgre 5-7. Young's modulus of kappkire rods as a functionm ..... --•-----

5-4.1.1 What Is the Res Line to bs Used for partic Mlar application of the general method ofEstimating y From, Given Values of x? least squares. From Dat-a Sample 6-4.1, the

of• b .. th ' -- '------ - ---- -

l -• ATrPTr.T. W• ... _tl^nenn 1% * tin nripatinen r"f tha fpltdi c inp the grrinral ienitqho io

line for prediction, outside the rane of data' ,.-t"from which the line was computed, may y = 4654,9846 - 0.44985482 x.- !ead to highly erroneous conclusions.

The equation in original units is obtained by

Frocedure adding 4000 to the computed intercept bo.Since the Y's were coded by subtracting a con-

Using Worksheet (See Worksheet 5-4.1), stant, the computed slope b, was not affecten.compute the line Y = bo + bx. 7 his is an In Figure 5-8, the line is drawn and confidence (estimate of the true equation I = Oo + #1z. limits for th2 line (computed as described inTh( method of fitting a line given here is a Paragraph 5-4.1.2.1) also are shown.

5-12

A2, -

.. 7 1 -

LINEAR RELATIONSHIPS BETWEEN TWO VARIABLES AMCP 706-110

WOVOU KSHEIEUT 5-4.1 TEM-PERATURrEXAMPLE OF Ft RELATIONSHIPYOUNGW$ MODULUS AS FUICTMON OF TEMPERATURE

X denotes Temperature, 'C Y denotes Young's Modulus - 4000

IX 12030 zY= 5068

751.875 = 316.75

Number of points: n = 16

(1) 'X1Y= 2,300,860.

(2) (ZX) (EY)/u = 3,810,502._

(3) S..,,59,2.

(4) ZX2 = 12,400,900 (7) xY' = 2,285,614

) (5) (Z.X)l/n = 9,045,056.25 (8) (2Y)/n = 1,605,9289.

(6) S=, - 3,355,843.75 (9) Sv = 680,325.

(10) b, -. 449,854,82 (14) • = 679 119--J314

- o1Ui- ~ ~ )~ j 5 f*

1-7JU.U03OU

(11) y (15) -- •; 0z.2)s

(12) bX - -338.2346 (16) sy - 86.074 1857

(13) bo - - b 1 654.9846 By 9.277617

b0 (in original units) 4654.9846

IFRnIAtion of the line: Estimated variance of the slope:

(in original units) .000 025 649 046Y= b, + bX-

4654,9846 - .449,854,82 X Estimated variance of intercept:

8b, .005 064sb= 4.458 638t = 4, + = 19,879 452

5-13

I;7

AECP 706-110 ANALYSIS OF MEASUREMENT DATA(

4MO

4200

:Iwo

4400

1~ 30 00 40 50 00 70 10 00 1000 1100 1200 8300 1444 150

Figure 5-8. Young' s modulus of sapphie rods as a fuiction

of temperature--showing computed regression lineSand confidence interval for the line.

Using the Regression Equation for Prediction. onlywith respect to a particular situation. The

.The fitted regression equation may be used for difference is that here we are concerned with

two kinds of predictions: relationships between two variables and there-

(a) To estimate the true value of y associated fore must always talk about the value of y, or Y,

_with a particular value of x, eg., given x - x' to for fixed x.

estimate the value of y' = $ + 01z'; or, The predicted y' or Y' value is obtained by(b) To predict a single new observed value Y substituting the chosen value (z') of X in the

-corresponding to a particular value of x, e.g., fitted equation. For a particular value of x,"given X ' X, t .ae-CA •ht viuue Of Za4 w 5Jgu, either type of prediction ((a) or (b)) gives themeasurement of iy'. same numerical answer for y' or Y'. The un-

hucertainty aspoeauecl with the prediction, how-f Which prediction should be made? In some ever, does depend on whether we are estimatingcases, it is sufficient 0o say that the true value of the true value of y', or predicting the value Y'V (for giver x) lies in a certain interval, and in of an individuaal measurement of y'. If theother cases we may need to know how large (or experiment could be repeated many times, eachhow small) an indivAdual observed Y value is time obtaining n pairs of (x, Y) values, considerlikely to be -wsociated with a particular value of the range of Y values which would be obtainedz. The question of what to predict is similar to for a given x. Surely the individual Y values inthe question of what to specify (e.g., whether to all the sets will spread over a larger range than fspecify average tensile strength or to specify will the collection consisting of the average Y's

minimum tensile strength) and can be answered (one from each set).

5-14iI

LINEAR RELATIONSHIPS BEIWEEN TWO VARIABLES AMCP 70&-110|CIC) -... K

To estimate the true value of y associated 3-4.1.2 What are the Confdnce intervtii ija-with the value x', use the equation kmates for: the Line as a Wh,,4ePoint an the Une,; a Future Valus of Y

V b -D + bix'. Corr""spnding to a Gven L Vaue Of A

The variance of V' - as a-x estimate of the iru Once we have fitted the line, we want to nakevalue y' = Po + Pi1x' is predictions from it, and we m ant to know how

r ' )good our predictions are. Often, these pre-Var 84..~ -dictions will be given in the form of an interval

V. -. .together with a confidence coefficient associatedThis variance is the variance of estimate of a with the interval-i.e., confidence interval esti-point on the fitted line. mates. Several kinds of confidence interval

estimate. may be made:(a) A confidence band for the line as a whole.(b) A confidence interval for a point on the

Young's modulus to temperature, we predict a line-i.e., a confidenee interval for y' (the truevalue for ti at x = 1200: value of V and the inean value of Y) correspond-

y" = 4654.9846 - .44985482 (1200) ing to a single value of x - z'.y, = 4115.16 If the fitted line is, say, a calibration line

Var y d = 86.074 .0625 + 1- 751.75)_] which will be used over and over again, we will1- 3,3+55,843.75 want to make the interval etimate. described

= 86.074 (.0625 + .0598) in (a). In other cases, the line as such may not=-86.074 (.1223) be Ao important. The line may have been

xy'=1.G3 . jtltl '..-tly to investigate or check the structure> . lationship, and the interest of theoberedvau o yor- :nter may be centered at one or two""To predict a single observed value of Y corre -"..,. ý'l the variables.

sponding to a given value (x') of x, use the sameequation Another kind of interval estimate sometiames

Y' = bo + bx'. is required:(c) A single observed value (Y') of Y corre-

U ,,Va v nce..ta ..& Y .a es.timate of a single sponding to a new value of x = .'.new (additional, future) measurement of y' is These three kinds of confidence interval state-

V Y . 1 + 1 (X"-- X)2_ 1 ments have somewhat different interpretations.

- Y '' L .. + SO J The confidence interval for (b) is interpreted as

The equation for our example is follows:Suppose that we repeated our experiment a

V 4654.9846 - .44985482 z. large number of titaes. ERuh titw, we, obtain n

pairs of values (xi, Yj), fit tihe line, and computeTo predict the value of a single determination of a confidence interval estimate for y' - # + Piz',

Young's mod..us at' x = M5, ..... hi the value of u corresponding to the particularequation and obtain: value x = x'. Such interval estimates of Y' are

Y = 4654.9846 - .44985482 (750) expected to be correc' (i.e., inclide the true4-4317.59 value of y') a proportion (i - x) c' the time.If we were to make an interval estimate of p(

VarY Y=sV 1 (X' - )1 corresponding t another value of x - xe, these

1 n S . interval estimates also would be expected to(750 - 751.87 include y' the same proportion (1 - a) of the

= 86.074 [1 +.0625 + 8355,84-•.75 time. However, taken together, these intervals5do not coustitute a joint confidence statement86.074 (1.0625) about y' and y" which vould be expected to

= 91.45 .be conct exactly a proportion (I a) of the5S~5-15

.AXCP 706-110 ANALYSIS OF MEASUREMENT DATA==. ... .. . .. . ( )I

ti-ne; nor is the effective level of confidence N-•/9 . This wider interval is the "price" we- a)', because the two statements are not pay for making joint statements about V for any

ir.3cpendent but are correlated in a manner number of or for all of the x values, rather than)ntimnately delendont on the values x' and z" for the y for a single x.'which the predictions are to be made. Another caution is in order. We cannot use.

The confidenee band for the whole line (a) the same computed line in (h) and (c) to make aimplies the same sort of repetition of the experi- large number of predictions, and claim thatAment except that our confidence statements are 100 (1 - a) % of the predictions will be correct.mot now limited to one x at a time, but we can The estimated line may be very close to the true

41alk about any number of x values simultane- line, in which case nearly all of the intervalSously-about the whole line. Our confidence predictions may be correct; or the line may be

atatement applies to the line as a whole, and considerably different from thetrue line, iv.therefore the confidence intervals for y corre- which case very few may be correct. In prc--aponding to all the chosen x values will simulta- tice, provided oar situation is in coutrol, we•meously be correct a proportion (1 - a) of the should always revise our estimate of the line to

* time. It will be noted that the intervals in (a) include additional infonnratin in the way ofare larger than the intervals in (b) by the ratio new points.

5-4,1.2.1 What is the (1 a) Confidence Band for the Line o•s a Whole? . jProcedure, Example

(1) Choose the desired confidence level, 1 - a (1) Let: 1 - a = .95

(2) Obwun SBy 1,ro1 ,Vo EEt. (2) -2= .77617from Worksheet 5-4.1

(3) Look up F_. for (2, n - 2) degrees of free- (3) F,9& (2, 14) = 3.74-domr in Table A-5.

(4) Choose a number of values of X (within the (4) Let: X = 30-"range of the data) at whici. to compute X =400

ý,Points for drawing the confidence band. x = 800

X -1200X ý 1500,

for example.

(5) At each selected value of X, compute: (5) See Table 5-2 for a convenient computa-LYi l + b1 (X-_) tional arrangement and the example cal-

and culations.

(6) A (J - a) eonfiden, band f9r the whole (6) See Table 5-2.-line is determined by -9-

Y. ±i W'.

5-16I_ _ _ __!

E-7

LINEAR RELATIONSHIPS BETWEEN TWO VARIABLES AMCP 706-IIQ

Ptmcedure L"umpte

(7) To draw the line and its confidence band, (7) See Figure 68plot Y. at two of t~he extreme selectedvalues of X. Connect the two points bya straight line. At i ach selected value ofX, also plot Y'. + W, arnd Y. - W1. Con-nect the upper series of points, and the

* 1ovver series of points, by smooth curves.If more points are needed for drawing the For example: W, (but not Y'.) hwas the samecurves for the band, note that, because of sym- value at X - 400 (i.e., 9 - 351-875)asametry, the calculation of TV, at n values of X X =1103.75 (i.e., I + 351.876). .-.

actually gives W1 at 2n values of X. .

TABLE 5-2. COMPUTATIONAL ARRANGEMENT FOR PROCEDURE 5-4.1.2.1

X X- ) Y. n S. V' s W, Y.+W 1 Y.-W 1

30 -721.875 4641.49 .21778 18.746" 4.8296 11.84 465333 4629.65.400 -851.875 4475.04 .09940 8.5558 2.9250 8.00 44831.04 4467.048W0 48.125 4295.10 .06319 549 2322 6.38 4801.48 4288.721209 4 1AM 251 4115-16 :12234 1050 3.50 .8 41.4 41.2

15 4.2 3989.-20 .2W8 19.7-351 4. 44=4 12.1 0012 .

.9 761.875 4 86.0741857 Y= ?+ b,(X- 9)

coded? f 316.75 1 =.0654

ginl uit) 41 ~ b5 - .44985482 2 .7 85 y

x 3 ,355,843.75 - - - -- J%/2P -2.786

5-17

"7T


8-4.1 ZR Qvo a 0l - a) Ceinflslngr !finorval Estimate for a Single Point on tho Line (Le., the MeanV.uqw, of Y Cwresponding to a -:Use Valueý of x =x')

Proceedure Example

(1) Choose the'desired confidenoe level, 1 - a (-j Let: 1 - a-.95a.05

(2) Obtain sy from Worksheet. (2) sy 9.277617- from Worksheet 5-4.1

(8) Look upt1 .for n - 2 degrees of freedom (3) 9.,7& (14) -2.145

-in Table A-4. -

1J4) Choose X', the value of A at which we (4) Let X' 1200 --

want to make an interval estimate of the -

mean v'alue of Y.

(5) Compute: (5) K)WIr + . ±. W3 2.145 (3.2451)

WA~~~ 41ai-6.96

Y. - + 11 (x1 - Y. =4115.16

(6) A (1 - a) confidence interval estimate for (6) A 95% confidcace interval estimate for the..,the mean value of Y corrm~ponding to mean valu.. of Y corresponding to X1 =1200

X X' is givenbý_y is4115.16 + 6.96

=4108.20 to 41'22.12.

Note: An interval esuira~tf ofthe intrempt of the line (0u) is obtAined ky settirig X" = Cin the~oy~rt~dure - -I

5-18

LIN.JEAR REIATIONSHIPS BETWEEN TWO VARIABLES AMCP 706410

) ".

-4.1.2.3 Give a (I - a) Confidence Interval Estimate for a Single (Future) Value (Y') of Y Ceorr.sponding to a Chosen Vaiue (x') of •A.

Piocadur.. Example - -

(1) Choose the desired confidence level, 1 -- a (1) Let: 1 .- ,95

(2) Obtain sy from Worksheet. (2) 8r - 9.277617from Worksneet 5-4.1

(3) Lock up t1-i, for -n - 2 degrees of freedom (3) t.9,1 (14) - 2.145 -

in Table A-4.

(4) Choose X', the value of X at which we (4) Let X' = 1200 _

want to make an interval estimate of a -.

single value of Y.

(5) Compute: (5)

1 + (x' .-_. ] w = 2A45 (9.8288)W, = tj-gsL + + S~,, J = 21.08

andY. + b, (X' - ) y.- 4115.16

(6) A (1 - 4) confidence interval estimate for (6) A 95% confidence interval estimate forY•' (the single value of Y corresponding to a single value of Y corresponding to .. .

"X') is X' - 1200 is -

4115.16 -L-21.08Y. : W. - 4094.08 to 4136.24 •

5... ..... 14 ,s ... ,-.. E.Imate for8 the Slope of the TruL Une y A - + i•X?

Procedure Examplo

(1) Choose the desired confidence level, 1 - a (1) Let: 1 - a .95S=.05

(2) Look up t41 2 for n -- 2 degrees of freedom (2) t.,7 (14) 2.145in Table A-4.

'

"(3) Obtain s6, from Worksheet. (3) 8 = .005064from Worksheet 5.4.1

(4) Computk- (4)

W, -rW, = 2.145r (.005064)-W .010862

(5) A (I - -) c:,z,-.e interval estimate for (5) b, = - .449865

Pi6 W, .010862 'A 95% confidence interval for 0, is the in-

b1 ± W. .terval -. 449855 L .010862, i.e., the inter-val from- -. 460717 to -. 438993.

5-19

• .;-4

AMCP 706-110 ANAIYSiS OF MEASUREMENT DATA (2)8-4.!.4 If We Observe *' New Valuea of Y (with Average V'), How Can We Use e FIttd Regretslon

'Uie to Obtain an Interval Estimat, of the Value of x that Produced These Volues of Y?

Example: Suppose that we obtain 10 new measurements of Young's modulus (withavenrge, F' 4500) and we wish to use the regi j',n line to make an interval estimateof the temperature (W) at which the measarem9e ts were made.

Procedure Example

(1) Choose the desired confidence level, 1 - a (1) Let: 1 - a - .95a .05

(2) Look up f1-11 for n - 2 degrees of freedom (2) t.97, (14) = 2.145in Table A-4.

(8) Obtain b, and se, from Worksheet. j3) From Worksheet 6-4.1,b .= -. 449855

(., - .0000256490

C ( - (C-pt)' ( C - .202370 - .000118C 6- .202252

- (5) A ( - a) confidence interval estimate for (5) A 95% confidence interval would be corn-the X corresponding to V' is computed puted as follows:from

+' -b(I I)X' 751.875 -4498 5 5 (4600 - 4816.76)2C .202252

=t• ,? ) + i.... 2.145 (9.277617)

W, - -- 75i.8+ -.07.252

7024 9.8 452v.iIiiI66.028

344.285 i:: 98.39452 V.28726I344.285 + 98.L9452 (.20706)

- 344.285 +- 20.374

The interval from X = 323.911 to X-364.659 is a 95% confidence interval for thevalue of temperature which produced the10 measurements 4 hose mean Young'smodulus was 4500.

5-20

LINEAR RELATIONSHIPS BETWEEN TWO VARIABLES AMCP 706-110 j j

5-4.1.5 UWing the Fitted Qi.-*s.liol Line, How Can We Choos a Value (x') of x Which We MayExp.tt with Confidence (1 - a) Will Prduce a Valu* of Y Not Less Than Some SpeciiedValue Q? 4 -

Example, What value (W') of temperature (z) can be expected to prociuce a value of 7-

Young's modulus not less than 4300?

Procedure Example _:

(1) Choose the desired confiden -e level, 1 -- a; (1) Let: I - a = .95and choose Q a = .05

Q -4300

(2) Look up ti, for . - 2 degrees of freedom (2) t.,5 (14) 1.761in Table A-4.

(3) Obtain bi and s8, from Worksheet. (3) From Worksheet 5-4.A,b, = -. 449855

= .0000o56490

(4) Compute (4)

C = bl, - ((_.), C = .202870 - .000080- .202290

(5) Compute (5) The value of X' is computed as follows:

X, - +• t bi S-. 4 98 55 (4300 - 4316.75)

-C + +•-i+(! AC 1.761 (9.277617) XS.202MxO

where the sign before the last term is + if ftjS2nn I I 1,7V-

b" is positive Or - if bj is negative. We -1 16have confidence (1 - c) that a value of

X - X' will correspond to (produce) avalue of Y not less than Q. (See discussion +of "confidence" in straight-line prediction - 8036466M. V'---i084 + .214933in Parag-aph 5-4.1s2).

- 751.875 + 37.249 j- 80.764662 V.2-15017

751.875 + 37.249 - 37.450

=751.674

5-21

- - -. - - -


&4,1.6 Is the A~sisnmlion of Unw Regression Justified?

This involves a test of the assumption that the mean Y values (1P.) for given x values do lie on astraight line (we assume 'hat for any given value of x, the corresponding individual Y values arenormally distributed with varianck Ay, which is ind'.pendent of the value of x). A simple test isavailable provided that we have more than one observation on Y at one or mome values of x.Assume that there are n pairs of values (zi, Yj), and that among these pairs there occur only kvalues of x (where k is less than n).

For example, see the data recorded in Table 5-3 which shows measurements of Young's modilus(coded) of sapphire rods as a function of temperature.

Each x iz. recorded in Colump 1, and the corresponding Y values (varying in number from I to 3in the example) are recorded opposite the appropriate x. The remaining colunms in the tableare convenient for the required computations.

TABLE 5-3. COMPUTATIONAL ARRANGEMENT FOR TEST OF LINEARITY

x V- Tum. - Young's 1ZY),

per- Modulus ZY (2Y)2 ZY' ni n niX" Zxy ,attire Mir is •00O

500 828 328 107584 107584 1 500 250000 164000 10584550 -1296 2 Air,6 87616 1 550 802500 1H2800 87616600266 266 70756 79756 1 600 360000 159600 70756603 260 244 504 254016 127136 2 1206 727218 303912 1.27008650 240 232 213 685 469225 156793 3 1950 1267500 445250 156408.3700 204 203 184 591 349281 116681 3 2100 1470000 413700 115427 1750 174 175 154 503 253009 84617 3 2250 1687500 q77250 84336.3800 152 146 124 422 178084 59796 3 2400 1920000 337600 59361.38W0 117 94 211 44521 22525 2i 1700 144500 179350 22260.5900 97 61 158 24964 1313P 2 1800 1620000 142200 1,2482950 e, 38 14-44 1441 1 t-50 902500 36100 1444

i000 so9 5 35 1225 925 2 2000 2000000 85000 612.5

TOTAL 4037 849003 24 18006 13952218 2756762 8462,96= =-22, T, T4 T, T,

5-22

_- - ar -V

LINEAR RELATIONSHIPS RTWEEN TWO) VARIABLES AXCP 706-113)

Procedure Example

(1) Choovea the significance level of the test. (1) Let: =.05

1 - -. 95

(2) Compute: (2)

-168.21

Ta18006 i

= ,the weighited average of-X.=750.25

(3) Compute (3)(T12ý6790,171.04

(T0 S, 8462%6- 679057.04.= 167238.96

(4) Compate -- 4

T,- T, T, 2756762 - 3028759.25

-2719217.25-44321&.5

-0-.6t368394

(5) Compute (5)

(6) Compute (

Ss Ts S, 849008 679057.04n= 169945.96

t7) IA-ok LDF 1..-for U. 2, - k) degrees,of (7) V 2freedom in ¶1 a~e A-5. kýi

F..~ for (10, 12) degrees of freedom 2.7kb

(8) Compute (8)

UF- S tL= F (1317.13 t2 -12)

=(.11715) (1,2)=0.14

(9) If F > F1 .,decide that the "array me via" (9) Since F is less than F,~, the hypothesis of1P. do not lie on astraight line. 1f F < F1-, line-trity is tiot disproved..the hypothesis of linearity is not disproved.

5-23


5-4.2 Fl RELATIONSHIPS WHEN THE INTERCEI T 4~, =(:c, -- x; 1)o,. Under th~ew~ circutivitaices,IS KNOWN TO BE EQUA L TO ZERO (LINLS qthe Y's will be n'ormally disti ibated With invansTHROUGH TM~ ORIGIN) OIXI, OIr 2 7 . - , 1r., respec tively, as before; and

In Paragraph 5-4.1, we ai~suzned: w~ith varjan1c~n i7 Xf ITS!)(ctivelly; bDut will(-I) that there is an underlying linear' fulic not, be independ'ent owing to the overlap among

tional relationship between x and y of the forni their respective errors.y= &. + Ox, with intercep~t &l and slope ~i 5-4.2.1 Line Through Origin, VWriance of Y's

both differeut from zero; Independent of x. The slopc of the(b) that onir data consist of observed values bAt-fittinx line of the- formi Y = ý11:r is given by

Y, I, . . . , V., of Y, Corresponding to accu-rately-known values xr, X2 , .. ,x,, of x; and,

()that. the Y's cani be regarded as being X

i'-.dcpeirdently and normally dis!-ributed with Xnmeans equal to their respective true values (i.e, ~mean of Y'. = fl *+- O~i i 1, 2, . . . , n) and and the estimated variance of b, isconstant variance C..-o

2 for aill x.

Furthermore, we gave: a procedure (Para- xgraph 5-4.1.2.2 with X' = 0) for determiningZconfidence limits for &, and hence for testing wherethe hypothesis that fl0 = 0, in; the absenze ofprior knowledge of the valae oi #,; and a proce- (Yj bi xi,)dure that is independent of the value of 0,j.-(Paragraph 5-4.1.3) for determining confidencelimits for 01, and hence for testing the hypoth- 4 XiYesis that 61 = 0. ~ ~-____

We now consider the analysis of data corre- Xsponding to an Fl structural relationship when-it is known that y = 0 when x = 0, so that theline must pass through the origin, i.e., when it is Consequently, we may effect a simplification of

know tht ~0. o bgin ith weassnle our Basic Worksheet-see Worksheet 5-4.2.1.as in (b) and (c) above, that our data consist of Using the values of b, and 8

b, so obtained,_observed values Y~, Y~, , Yý, of a dependent confidence limits for 0,, the slope of the true lineN~ariable y cor-responding to accurately-known through the origin, y = #,Y, can be obtained by--

_.alues xi, X2,. . xý of the indevendent variable fol lowing the priedr of Paragraph .5-4-1-.-xand that these Y''s can De regaroledt a,, being using ti-./ 2 for n - I degrees of freedom. Con-

Tinepedeniyand normally distributed with fidence limits for the line as a whole then areneaiis 0ixi, PIX2, O.,x., respectively, and obtained simply by plotting the lines yvariances a4., that may depend on x. We and y -. ptx, where #I' and 0,' are the upper and

cosdrexplicitly the cases of constant vani- lower confidence 'limits for 8, obtained in the____ anc (2.. = 0~), variance proportional to inanner just described. The limiting lines, in--

--x (Ay~ xau2), and standard deviation propor- t~his instance, also furnish confidence limits for--Aional to x (a. = ZG-). Finally, we consider the value y' of y corresponding to a particular-briefly the case of eumulative data where point on the line, s.ý.y ior x X!, so that an

form el + ?, + . .. +I ~ ei, that is, is the dence limits for a single future observed Y' corre-,zum of the error., ofl all preceding 17's plus a sponiding to x =x' are given byvfprivate enror" ej of it,3 own. Following t. '~+(' 2 sMandel,U') wt assumne that the errors (ei) areindependently and normally distributed with wheie 4. and s,, are from uý.r modified work-zero me~ans and with variances proportional to sheet and tiQ 2 corresponds to n I degrees of

* j the length of their generation intervals, i.e., free !oin.

5-24

LINEAR RELAT:ONSHIPS BEIWEEN TWO VARIABLES AMCP *106-110

WORKSHEET 5-4.2.1

WORKSMUET FOR Fl RELATIONSHIPS WHEN MviE INTERCEPT IS KNOWN .3 BE ZEROAtND THE VARIANCES OF THE i's IS IHD,"PENMNf OF x

X denotes Y denutes

___- -~~~x 1- _ _ _ _ _ _ _

Numbur of points: n =

"Step (1) zXY =________ ___________(2XY)' ___ _____

(2) 21X2 (5) (x x Y)

(3) ZjY2 - (6) (n - 1) s -= Step (3) - Step(5)

(4) b,= 1-X Step(1) + Step (2) (7) sI- = Step (6) + (n-1)Vx2 ----_

Equation of the Line: Estimated variance of the slope:

Y b,X Sg= X• Step (7) +Step (2)

5-4.2.2 Line Through Orgiat, Variance Propor- Using the values of b, and ab, so obtained,J •L.•l •,.I~a • •_Thes lore of confidence limits for Pi, the sl,)pe of the true line

the best-fittingz line of form Y =bix is given by through 'the origin, if - fti, can be otie •" =::• "followinxg the procedure of Paragraph 5-4.1.3

S..• y using t(_-/2 for n - 1 degrees of freedom. Con-b, fidence limits for the line as a whole then areS... = obtained simply by plotting the lines y = fiuand y , x#, where #1 and #" are the apper and

ihe ratio of the averages, and the estimated lower confidence limits for 61 obtained in theSvariance of b1 is manner just described. The limiting lines, in

stthis instance, also furt.,sh confidence limits forSz• on the line, say for x -- x'. Confidence limits•-,for a single future ob~v-ed Y corresponding to

where x = x', are given by

-i ,-l 1 ,qi V _ + W),811,

-X where s6, is computed at; shown above and t,.,/

,-1 corresponds to n - 1 degrees of freedom.

5-25

AMCP 706-110 ANALYSIS OF MEASUIPMEN1 DATA

5-4.2.., Line Through O3igin, Standard Devic- variabl' x. Thlus, )1 , • may denote: th,:ii Proporionai tax (rn., xo). The total w'igi1t lost; of a tire inder road t:',jt.

slope of the best-fitting iue of form Y • b1x is measured it. successive mil,;rges xr 2, . . .. ; orgiven by the weight g;rin of some niiaterkJl due to watcr

b E I ,) /.,absorption at successive time.s .r,, X, .... ; or the

T-1 total defleetion of a beam (or total comprcw-ion

of a spring) under owitimiafly increasing load,the atva~oa of tile ratios ,v' tleqsUred at load&x x, z 2 m.... ad so fort h. In

such cases, even though the underlying fuue-and the estimated variance of b, is cjonal relationship takes the form of a lihe

.sa - through the origin, y = ex, none of thc. pro-cedures that we have presented thuws far will t-.

where applcable, because of th,. cumulative elfoit (if/y~l2 err'ors of to'chaiqoe oil t 1Q Successive Y"; the

_1) (7 [: /t2)J deviation of 1,. from it.i true or expected vahle"(n )2 - , n y, will include the devition (YI - Y,-t) cf

Y,_-, from its true or expected value, plus anthat is, individual "private deviation or error'' c, of its

( 1,R). own. Hence, the total error of Yj wil! te theE sum (+ ±ec.+. . -[ e, + c) of the indi-

n (i t 1 vidual error contributions of Y1, Y2,. ,Y-,

for and its own additicnal deviaton.

firIf the test or experiment starts at x ,

Using the values of b1 and sb, so obtained, and the z's form an uninterrupted sequenceconfidhnce limits for 01, the slope of the true line 0 < x, < x. < . .. < x., and if we may regardthrough the origin, y ý 6z, can be obtained by the individual error co:.itributions el, ,2, . .. , asfollowing the procedure of Paragraph 5-4.1.3 independently and iormally distributed withusing t1 .t for n - 1 degrees of freedom. Con- zero means and variances proportiunal to 'hefidence limits for the line as a whole are then lengths of the x-intervals over which they ac-

'obtained aimply by plotting the lines y = #"x crue, i.e., if a'., = (xi - xi-1 ) a!, theni the bestamnd y = fx where 61' and 6L' are the upper and estimate of the slope of the uneriying linearlower confidence linits for R obtained in the functional reilatioim V = fiix is given bymanner just described. The limiting lines, in yý

:Ahis instamice, also fumnish cenfidence limits for -S-Ahe value V' of y corresponding to a particular*oint on the lneu; say for z = x'. Confidence and v•timated variance of b,

l3imits for a sirigie future observed Y correspond- - Y~-y

ig toz x x', are given by -1) Xi 7i-- X,

b X tI-.11 X,' N/X + s,, in which xo 0 and I = 0 by hypothesis.

where aR is • _npu•_td w sho•_ above and t_,sponds to n - degrees of freedoUsirg the values of b, and s,, so obtained,

f reeom.confidence limits for 01 , the sloIp 1f the true line

through the origin, y = 6,x, can ýw obtamned by

5 -4.2-.4 Line Through Origin, Errors of V's following the pnueedure of 1'arigraph 5-4.1.3Cumulativi (Cumulative Dala). In using tJ/2 for n. - 1 degrees of freedom. Con,

- meiy engineering tests and laboratory experi- fidence limits for the line as -. whole then aremenrts the observed values Yi, Y,,,. .. , ,..., cbtained simply by plottingr zho. 1ies y - 6ux

"cf a dependent variable y reprcent the cumula- and y =0--X, where p' and 03[ are the upper andtive magnitude of some effect at successive lower confidence limits for 0, obtained in the ,values .x, < x, <x < ... of the iidependent manner just described. These limit lines also

5-26

LIN4EAR REIATIUI~ !SHIPS BETWEEN TWO VARIABLE,` AJACLI706-110

provide conflauncei limits for a p:trticulrw point. Data Sonipia .5.4.3A--Po..,Vszn of (wooil thc( line, say the ;'lu' ' COi~r(cSpo~iding to Cobrima~rk r.~ MolA

x ''. Fx9 tile fitting of lint's of this ,sUrt to The following d.ata are u&,~! iesults of twocumulative data waieAr wort ryieral cur~idilions, c'.Aori :t-tiUlc methods for th, ý &brwinina ion 9f iand for other rcXited inatters, sce Mandel's hemlfical conuiwtrent. (The data liii-i beenarticle.(3) coded for a special purpGOs %V'iieli hiazz nothing

to do with this illustration). Tihe intcrt st here,5-4.3 FiI RULATIONSHM~S Of ..curst., is in thL relatiom-ihii between results

VUs~nguhshinj Featuries. There i.9 an1 under- given by the txvo mvehods, and it ija presuniedlyinig Imathlenlait ail (I'll ietional ) relationship be- tj aIt ther'Ie i!S a fun L.1 na j! ':iati nship with bothtween the two vai iables, of the form ni thuds subject Lo tii ro - of i-easuremient.

Y fuA -i __

Method, I Method 11Both X and Y are suibject. to errors of iesl- Sariipkl x Y'ituent. Read Paragraph 5-3.1 and Table 5-i. ______________

The futll treatmenvit of this case depends 011 the 1 3720 563GaSbullptioni we are willing to make aboat error 2 4328 6 i 95

distribibtions. For complet'e discussion of the 45 -2problem, see Actoni.") 4 4818 6C362

5 5545 75625-4.3.1 A Simple Met.1iod of Fittingj the Line In 6 7278 9184

the General Case. There is a (!nick 7 7880 10070and simple method of fitting a line of the form 8 10085 12519Y = bo + bX which is gerneraly applicable 9 11707 13980when both X arid Y' are subject to errors of ____________ ____

measurement. ~'his mothod is described inBartlett,(') and is illustrated in this paragraph. ()Th ltdinmutpstrog thSimilar methods had been used previously by wh hef~ er iemutpLshrghh;other authors.pon(,1)

(a) 1Vr the locatl.ion of the fitted straight line, 6668.4use as the pivot point the center of gravity oZ 8662.6Vall n observed points (X,, 1'j), that is, the po2in itt rus.Snetee i on

wn"h en cnodiial (s, F).inose

cqueiice, the fitted line will be of the form (b) To determine the slope, divide the pointsY - bo +±b1X with b, =V b~l, just as in it grip.Sic he a 9onsexactly~the least-squares inethod in Paragraph 5-4.1. o uagrusrebaid. -.

(b) For the slope, divide the n plotted points 1~~ 2190hixto three iion-overlapping groups when ccn- Y1 .599 5

sidered in the X direction. There nliould be an .- 9891equal number of points, k, in each of the two .~4234

exrm rus ihk itlos to as possible., ITake, as the slope of the line, -4- 99

6953991-4234

where1.0951Is average Y for 3rd grcup V- 1

1P= avervge Y for 1st group 8662.6- 6195(6M4Is, - average X for 3rd group 5657(6i8)

X,-average X for 1st group. 1360.0

6-27j

AMCP 706.110 ANALYSIS OF MEASUREMENT DATA

Ihe fitted liv~e joindy, from whll:h 100 (1 - W) % confidenlc

1360.0 +- 3.05'. x limits fcr the line as a whole can be dcrived.rlo stric" validity, they require that th(fe mes-i orreli-tnt elrors affe,'th'rg the observed X, be

Pro.cedures are given iiu Bar'tlett() for deter- sufflPently small in comparison with the spacingmining 100 (1 - t) confidence limits for of thtlhr true values X, thatL the allomfl ion of thethe true slope 3, and for detcrmining a obser vatozal pointy (X1 , ,%) to the three groups100 (1 - a) , confidence ellipse fur Ou and 63 is unaffected. These proccdu -s are forinally

140001

120001" y

- 10000

a

6000i

4000 6000 8000 10000 12000

METHOD I[

Figure 5-9. Rclatio. ship between two methods of determining ( ')a chtmical consf-itaent-an PlI relationship.

5-28

.-, 3- , -

LiNEAR RELATiONSHiPS BF1-FWEEN TW0 VAR-IlABLES) AriCPd: 7O -11

iuitethod in H1 situ Ations, bill involve nmore comf- ai own lljCr of preca~siglucd nom11inal o1- taLrgetpdex ealeulauiois. We do not consider themi values (x,, x,, . . ) of tho, in h'pciident variablofurther 1,ere. x, to which the exper Iiuenter equtictte the indi.--

jenidvint Variable ill his experimlent as besiL he5-4.3.2 An hinportant Exceptional Case. Untij canl, and tlhen observes, the corresponding yietds

Conliparat iveL.IN recenitly it was not reallized Jhat (Y,, .- .. of the depenlde'lt variable Y;there is a broad clw-s- (!f controlled experizuwnitl SeCwii, the exilerinientiM, in hi~s notebook,situations in which both X ,ind Y are 5sJbjfe't to reel., ds t le obse, e d y'ieILd (J', . Y,. . .) as corrv -er-o.-S otfetu~Žuit yet all of tile techniques sponiidng to, ard treats themn a.s if they were

atpr~iaeto the Fl Caaeo (Xr's acual producved b~y, the voiiiiiifli Or faiy'cr valuesknowNn, iueasurinieiit- errorns aiffect. thic )*:s oidly) (x, ,x., . . .) of the ilideIpendent variable -whiere-are4 strictly apllhie~iblc without cliange, as, strictly t hcy corrcý;pond to, nnd were pro-

A~s .il example, let. us consider the caýse of an duced by, the actual inpuit values kXiIy I.)anlalytical All :st who, in )rdkcr to obta,,in an which ordiziurily will differ somnewhat fromn theaccurate de' c- Iination of the conccn~tration of nom11inal Or target Values (X. , X, . ns aa pot assiuntl Sulphale sohititoii, dlecide,, to 1)10- reutof errors of technique. Flirt h1ruilore, thecced as follow.": Fromn a burette hie will draw efflct OTre auS (X1. . . . . . . ý. ) of the indep-.ndenllofT 51, 10, 15, and 20 nil samples of thie solution. variable actually realized in the experinieýnt areVolurne of solution is hlis; indepeuideiit variab',e not recorded at aill -nior even ieasured!:r, and his target values are xi = 5,2 30 It. ;s suriprising but nieverthieless; true that an

x3=15, and x, = 20, reswectively. The vol- underlying linear structural relationshbip of theurnes of solution that hie actually draws off form Y' = fl -+ Oix can be estiiated validlyXT,, X, X, and X, will, of course, differ from froni the results of such experiments, by fittingthe nominal or ta, rget values ,-s a result cf a line of the fori A' = b, + b,1r in a.Žcor'incee1'rors of technique, and ILe will not attemipt LJ with the procedures for Fl situations (r's .ownmeasure their volumies accuratle13. These four accurately, Yos only subject to error). Thissam11ples Of the p)otassium11 Aulphate Solution then fact was emphatically brought to the atttnitionwill be treated with exet ss barium chloride, and of the scierktific world by JosepTh Berkson in athe precipitated barium sulphate dried and paper(15 published in 1950, and for itv vAlidityweighed. UAt Y1, Y., L,~ and Y,~ denote the requires only the usual assumptions regarding

eoi'iepondin I i UaiS uIUI ~aiMkSilIPIM, the randominess and independence of the errorsyields actually will correspond, of course, to the of ineasurenient anld technique affecting bothactual inputs X 1, X,, X,, and X,, respectively; of the variables (i.e., causing tliq deiiations ofand will differ from the true yields associated the actual inputs X,, Xý, .. ... fromn their targetwith these inputs, say Ii,(XI), ý~(X,), Y3(X.), values z,, X2, . and the deviations of the ob-and Y4(X 4), respectively, a; it result of errors ot evdotusYls,..,fo hi rsweighing and analytical technique. The sull- tuieof,(), 1 X2 ,..). heonlsn?hate concentration of the original potassium also extends to the many-variable case con-sulphate solution then will be determined by sidered in Chapter 6, provided 4that the relation-

Pv!ilinit.incg the A1nnP b. o: the best fittinix -IL:-. :_ : i- tastraight line Y' b- 4 bix, releting the observed ~'

barium sulphate yields (Y',, Y,, Ys, and Y,) to Il = #0 + PIX + iiU + 9*V -I

(xi, x'2, x'1 , and x4)-the intercept, bo of the line in a' (e.g., V' 00 + O2 a + 0 27" + 0 3XI), I her,making appropriate allowance for th posiblt eary :7l has- found that Berksoui's conclusionof bias of the analytical procedure resulting in a carries over to the extent that the usual least-non-zero blank. squares estimnates (given in Chapter 6) of the

Without going into the merits of the foregoing c!oeflicients of the two highest powers of x (i.e.,Nas an analytical procedure, let us not!e a number of 0, and P3 here) retain their optimumn proper-of featLures that are common to controlled expci i- ties of unbiusedness and muinimium variance, but

5-29

AMCP 706-itO ANALYSIS OF MEASURLMENT D~ATA

the coi,1idcnee-inturval an~d tvst~s-of-signxificarlLL' It sh~ouild be notcldhat the use of thesc Umi s-p! ocediureq require 0 odificaiion. formations is cert~kir to accorinplish onv thing

only -ixe., lo yield a relationship in straight-line

5-4.4 SOME LINEAPIZING fornil The transforrind data will 110L 1-cosTRA.Nsr3RMATIONS s'ilri satisfy certaini Lssumpt ions whi~chl are

It 11.e fc. ni of a non-linear rvlationship be- theoretically necessary in ordi r to apply thfetwet tw vaiabes s knwnit s smetmes Procedures of Paragraph 5-4.' 1, fim- ex-ample,

POwsbie two vara ble s a tr nsornmia i s mtionefs the assumptI ion that th c! variability- of Y give.n

both variab~les such that the relationship) be- purois and uwiti the rang of diwce, fua prctonatween the trai-qformned variables can be ex- purpeses anhewithjii the range oftAen doatll cin-pressed as a strai~uht Une. For exmpe, we sdrd h rnfrmtosotnd epi

iniýh ki, that the relationship is of tile fo, n tinsieadY' - 0). If we take lo~s of both sides of tl,:sequation, wI? obtain Thus far ' our diiscussion has centered oil the

use of transformations to con vert a A-uoici rela-iog Y - log a + X log b, tionsinip to linear form. n~,existence of such

which will be recognized to b~e a straignt 'line linearizing. transformations also makes it p1o.,-whose intercept onl the log 1' scale is equal to sible to determnine the form of a r-elationship emn-log a, -.nd whosle slopex is equal to lcg b, The pirically. The following possibilities, adaptedprocctdure for fitting the relationship is given in fromn S&arborough,"0 11 are suggested in thisthe following steps. regard;

(1) Make the tr.mnforniatiorr YT - og Y'(i.e., take logs of all the observed Y (1 Plo I'aant~ n riayg-val les). paper. If the points lie on a straight line, the

relationship is(2) Use the p'-ocdure of Paragraph 5-4.1.1 b

---to fit the line Y7- - l'o + bjX, subst~i- a +

tuting Yreverywhere for 1'.

(9Obtain the conatants of the original (2) Plot ~ gis nodnr rpA~quation "by substituting tL2 calculated paper. If the points lie on a straight linc, the-alues of bo and b, in the following relationship is

§ -vquatioml:1

- bo - loga orbxb, -log b,1

and taking the required antilogs.+

Some relationships between X and Y which (3) Plot X agains,-t Y' on seruilog paper (X on

can easily be transfornied into straight-line the arithmnetic scale, Y on the logarithiviic scale).

the appropriate change of variable for each rela- x.re related in the formtionihip, and gives the formulas to convert the Y =. aclx, orconstants of the resulting straight line to the Y = ab'r

*constants oi'the relationship in its original form.*In addition to the ones given in Table 5-4, some ()Po.YaantXo o-o ae.I

rnore-cotnpliea2'ed relationships can be handled (4) Plots on agstainst Xlone loglo papr.at arfby using special 'xicks which are not described rlthed pin the fon saigh ie h vrakhei-e, but can be found in Lipka,(") Rietz i') and lav n h ormScarborough.,- Ya - a=

5-30

LINEAR RELATIONSHIPS BETWEEN TWO VARIABLES AMCP 700-110

TABLE 5-4. SOME LINEARIZING TRANSFORMATIOKS

PNut the Iraonsformned Cvnvel Straight LineVariable, Constants (b,, and bi)

If tiae kolationship Fit the Straight Line Tv Origina! Coaiitants:Is of tie form: Y_ -4 b bbjr

- Xr

b 1 b1I - a -4- X .- Use the pi-ocdires of

-'ar'graph 5-4.1.1.I

a- t-bX, or _X all formulas given a b1 Y thvre, :Cubstitute values

of Yr for Y and valuesof X7 for X, as appro- -

y = .-k X nite, a b

S=b log Y X log a log b

Y ae"X log Y X log a b log e

Y = aXb logY 11, X log a b

S-a+ bX+ , Y a bwher n is known

5 .5 PROBLEMS AND PROCEDURES FOR rATI Rd.AT!Ors5! oraf

5-5.! Sl RELATIONSHIPS anee of X giver Y) which is constant for ELI`,

In this case, we are interested in an a.socia- values of Y * Tbken together, tL.-ase tI a actstion betwveen two variables. See Paragraph of assumpions imply that X ane Y are jointly6-3.2 and Table 5-1. distributed according to the bivariate normal

We us':ally mrke the assumption that for distribution. In practical situations, we usuallyje value mfke the csspon valuesr have only a sample from all the possible pairsany fixed value of X, the eorrespox~ding values of valuei X and Y, and ther-efoit we cannot Sof Y form a normal distribution with means deteroine either of the true freressio t ies,

Fx - go 4- #i and variance ,y.x (read as de ,=-n either of tru rvgexac im%"variance of Ir given X") which is constant for fi = 6 +i#X or I -9- 6,'1 exactly. if

we have a random sample of n pairs of valuesall values of X. * Similarly, we usually as.SUnlIe | 3 ,( 3 ,,.. (. ew a s

that for any fixed vaiue of Y, the corresponding ( I te Y r)I (XlIn Ye ) ,o both. ( O. e tod), we can esti-

values of X forin a normal distribution with mate either line, or both. Our iethod of fittingmean •r -- , +,,Y and variance a-.y (vari- the line gives us best predictions ir, the sense

n that, for a give X - X' our estimate of thecorrespondiig value of Y = Y' will:"2-trictly. we should write (a) ou the average equal fx, tte mean value

tiX-. = , + 0, X o" Y for X - X' (i.e., it will be on the true Iinaand ix = O0 + OIX); and

-nxv - 0. 4 C Y - (b) have a smaller variance than had we usedFee FootnutA ;n Paragraph 5-,3.2. any other method for fitting the line.

AMCTP 706-110 AMýALYSIS OF MEASUREMENT DATA

Soo

o 4U0-

0

I5--

tUU

S5300-W0

w0

o 0

S•00 0

w 0S100

z

Alu

* U-

,,. I _ _ _ __--

fr- 0 0 0 Z0C ,300 400 600

TREAD LIFE ( HUNDRED5 OF MILES ) BY THE WEIGHT METHOD

_________ " nter_prroove ?nethod of c~stil4t~tingl tread life -

•- --- • '2•-=• -__ -__ . . •- 'G---1 SI T- -W - - .

'1

5 -32

LINEAR RKLATIONSHIS F•TWEVtJ TWO VARIABLES AMCP 7(1-110

Data Sample 55.11 -- r-'timated T,.'ad Wots ~o flre&x I

-Tread Lif., Proad ' U1,1. dita usli fol illustratiol "e fri a ih idred, " o t . a (11',lred of f.r.hoa

st.udty of tNo Incthwt, I of estiuiikting tread wt.,Jof cm'mire'cMi tires .Stihltr and otrners'"). Weight M•,thod Ccil r 2:t.. Meti•'j

The d&1,i are shown here and JhaLted Iln Figare 7==1 =-: - .-

5-10. The vavible 'hlu h is taken as the inde- 459 357

tp,1ndelt 7aiiable A is the estii ,oitc trv mu 41V 1192

life i luudnlrvds of r•1 i;s by the ,'eivh-I.oss 2b811"method. TIle associated vai iable I is tho 1 281SI10 2,46mated tread life by the vroovdlcpth imct.od 30w 287(center g:i'ove•;). The plct.. t'.t; to indclate a 0 259relationship bI twceik X and Y, b,,t thlu rehtion- Z311ship is statistical rathnr than fulnetiollLd k.r 341 231exact. The sc.ttur of the points stems pr:- 273 037niarily from pr-dtuct variability and variation 2041 201)of tread wea: under normal operating condi 24,5 161tioas, vather than from enrors of lieasurcmmunt 209 199of weight h,;s ur groove depth. Descriptions 189 152

and predictions :vrc applicabie on',y "on the 137 115average." 114 112

5-5.1.1 What Is th Bet Line To Be Used for Estimcing Y.V for Giv~ii Value, of X?

Pwfeoduro

'The procoudre is identica. to that of Paragraph 5-4.1.. Using Baiic Wor" heet (seeWorkshect 5-5.1), compute the lieti

V b0 + b•X.

This is an estimate of the true rces,'esion ;ine

"V -'. I- to 1X. :'

SliSjT!g D)aLa Sample 5-5.1, the equation of the fitted line iýs

Y - 13.506 + 0.7902-2 X.

nh Figure 5-11, the line is drawn, and coufiducne limits for the fine '&,. Paragraph 5-5.1.2)i-"are Rhown.

15-33 q

A.-JA 7O6-ii0 ANALYSIS OF MEASUREMENT DATA (

WORKSHEET 5-5.1

EXAMPLE OF SI RELATIONSHIP

X dcnotes Tread Life Estimated Y denGtes Tread Life EstimatedbjL Weight Method _by Center Groove Method

zX = 4505 2-Y = 3776

= 281.5625 Y = 236 _ _

Number of points: n = 16

Step (1) IXY 1,170,731

(2) (ZX) (2;Y)/n = 1,063,180

(3) S. - 107551

-(4) 2X 2 = 1,404,543 (7) ZYI = 985749

, (5) (=X)2/n 1,268,439.0625 (8) (ŽyY)2/n = 891136

(6) S. - 136103.9375 (9) S 94604

"=1.0) bi = .790212 (14) -(S- - 94988.119

S. 3 San

-(11) ? 236 (15) (n-2)# - 9615.881

--(12) b49t - 222.494 (16) . - 686.819

N(1) bo=- b,9. - 13.506 ,y 23.21

Eqrption of the line: EstimRted varianoe of the slole.

.! - 18.506 +I .790212 X17Estimated variance of interoept:

0.0710,%7 -- , + -

a - 21.04.8 .... . 002

5-84

V LINEAR RELATIONSHIPS BETWEEN TWO VARIABLES AMCP 706-110

400- Y1-aX .oioix.

300

Szoo *-

0too 310040nTA!AO LWI(OdUh2REDS OF MILII) SYW1f E40

Figure 5-11. Reloatio7ship hetwee-n weight method and centergroove method--the line shown with its confiden~ceband is for estimating tread life byj center groove

metihod from tread life byj weight method.

Usting thG UK.gres~sion Line for Prediction. The equation of the fitted line may be used to predictYx, the average value of Y associated with a value of X. For example, using the fitted line.Y - 13.506 + 0.790212 X, the following are some predicted values for ?x. I

X ?

200 172260 21130)0 261350 290

0 i30

4, 86Q

AM~CP 706-110 ANALYS.15 OF PAMEASUM61TIL1 DATA

-5-5.1.2 What arc the. Confil(,,tco Interval ksinvoics for: the Line as a jimolt. v Poin? an !hz Ulnw;a Singirs Y' Crrosponding to u N4ew Value of X?

Read the diwiusa±i~i ýjf the intterpr-etation of three types of c. fideiie-, intervais in iFaragr-.phB-4. 1.2, in order to de~tide which is the appropr~ate kind of eoi'Aidecr(e interval.

The ý1olufiorns Rrc, identical to thow3 given in Paragraph 5-4.1.2., and ar illustrated for the tread---.*ear of eonunerc, .al tirves example (Data S$amplc 5-b. 1).

5-1.. What Is the (0-- a) 'enfiddrtce band for the Line cis a Whoia?

Piocodure Example

( Cbco*e the desiredi confidence lmvel, 1 (1) Let: 1I a .95

.05

(2) QktaiA sy. fromy Work3heet. (2) =26.21

(3) Look up P1, for ý2, it - 2) degreei of fre-ý (13) 16doin io Table A-b. F.,1 (2, 14) 37

- 4) Cboooe a namfer of values of X (within the (4) 4et: X =2~aneof the data) a:. which to compute

* points for drawing the confidence bajnd. X OX - 350(X = 40W, . *

for example.

(5) At each selected value of X, compute: ý(5) See Table 5-5 for a canvenient computa-~ ?+b (r-~)- tional arrangement, and the exgample cal-

cidations.I

_()A (1 - a) confidence band for the whole (6) See Table 5-5, 4

line is detoei-mined by

-j7' Tr, draw the line and its eonfidence band, (7) seeW Figure &-11,~ot Y. at two of the extreme aelactedvý_alues of X, Connect the two points by asatraight line. At each selected value of X,plot alroY. + Wand Y. - W1. Connectthe upper meries of points, and the lowerseries, of voiata,, by smooth curves.

curves, mnotU that, bcAut 3 of syrrmetry, the cal- valuo at X m~25 0L.s;. .~ 1.6)t~culation of 14, at-e% values of X actually gives X 8 18.12 (i.e., t ~l$.-

Wat 2%values of X.

S• ,• - . :- : 1,:. :, • -•• , .- :

LINEAR RELATIONSHIPS BETWEEN TWO VARIABLES A1rMCP 7 I6-110

TABLE 5-5. COMPUTATIONAL ARRANGEMEN' FO.1 PROCEDURE 5-5.1.2.1

(X-X)~~ - 4,X (X - ) Y, n .__. _ W, Y.+ W1 Y.- W1

2 0 -81.56 171.6 0.111375 76.50 8.746 23,9 195.5 147.72(0 -,31. 5CI 211.1 0.06981, 17 95 6.925 18.9 230.0 192.2$0W +18.44 250.6 0.064998 44.64 6.681 18.3 268.9 232.8350 68.44 290.1 0.096915 66.57 8,1S9 22.3 312.4 267.84W0 118.44 329.6 0.165569 113.72 10.66 29.2 358.8 A00.4

X- 2,81.5625 4 1- 686.849 Y. - F + bl. (X - X)- 236 1 3625

"b, -0.790212 Ž -S.. 136103.9375 -

2.735: •;Y•: W•= VZF -s -

. 3-S.1.2.2 Give a (1 - a) Confidence Interval Estimato For a Single Point On the Lino, I.e., the Mean7 "- Valve of Y Correisponding to X X'.

Procedure Example

(1) Choose the desired confidence ievel, 1 - a (1) Let: 1 - a - .95a f .05

(2) Obtain er from Worksheet. (2) s- 26.21 L

(8) Look up ti,/, for n - 2 -greesoffreedom (3) 16in Table A-4. t..9% for 14 d.f. 2.145

(4) Choose X', the value of X at which we (4) Let X' -250,want to make an interval estimate of the for example.mean value of Y.

(5) Comwute: (6)

1 -W +- (2.145) (26.21) (.2642)

and

Y. - F + bi (X' - ,) Y, -211.1

(6) A (1 - a) confidence Interval estimate for (6) 9, 95% confidence ititerval estimate for thethe mean value o! Y correWpondini to mean value of Y currsponding to X - 260X - X' is given by is

S¥Y, f-*. Wo 1. * 14.8 ,:

the interval from 1W.8 to 226=9

"57


S-5.1.2.3 Givei (1 -v ) Confidence Interval Estimate For a Single (Future) Value of Y Cowespondingto a Chosen Value of X - X'.

Procedure Examplo

(1) Choose the desired confidence level, 1 - a (1) Let- 1 - a .95 aIa - .05

(2) Obtain sy from Worksheet. (2) s- 26.21

(3) Look up -it for n - 2 degrees of ireedon, (3) n - 16in Table A-4. t.c for 14 d.f. - 2.145

(4) Choose X', the value of X at which we (4) Let X" - 250,wput to make an interval estimate of a for example.single value of Y.

(5) Compute: (M)Lr 1 (7' .- , l

_W - h_-[/ By 1 + + , Ws - (2.145) (26.21) (1.0343)- 68.1

* and

-- Y.- + b, (X' - ) Y. - 211.1

(6) A (1 -- a) confidence interval estimate for (6) A 95% confidence interval estimate for a- , (the single value of 1' correspconding to single value of Y corresponding to X' = 250

*•_x) is is 211.1 =i= 68.1, the interval from 15.0 to" • L269.2

&5.1. 3 Give a Confiden ,e Interval Estimate For 0., the Slopo of the True Regression Line,V1 - P. + oix.

,,'ho solution is identical to that of Paragraph 5-4.1.3 and is illustrated here for Data Sample 5-5.1.

Procedure example

(.) Choose the desired confidencc level, 1 - a (1) Let: 1 - a - .95

_' o ook up ti-./, for n - 2 degrees of freedom (2) - 16 ,ýVn,6 M=..IA A A

)Obtain a. from Worlmheet. (8) a,,- 0.0710387

J1 (1 a! - i,•, rkes interva! atimtm for (6) b, - 0.7902127Z (- W MU

rA 96% conftdence interval estimate tor 0,' -• :'= is the interval 0.1•90212 -L 0.162878, i.e.,• .

Su • ~the interval from 0.K7$U to 0.942590.

5_U

--eT

LINEAR RELATIONSH IPS BETWEEN TWO VARIABLES AMCP 706 -410 [s.ýs1.4Wha is he estLineForPrvidir iv b,'281.5625 - (1.133355) (2M6)

From Given Values of Y? -1.61

For this problem,. we fit a line X - b,' - b,' Y(ani estimate of the trut line IZr -? P± + 91 Y). Theqainoteftedlei.

That is, the fitted line will be:X -b.' + b, Y 7In ord( i to obtain confidence intervals, we

where~~~~~~~ -ne h 4olwn omls

Xr' S"_

'S.- --- - -

1.* 1385S.

13 2 + 1.3 85

FiglVure 5.1f. Rqiatioý.ip between weight #wthod and ccntergroow m4W-ehowsg the £two r grmioi 16we.

-Mi

A C?# , _M- I I-NAIYSlS OF MFASUREMENT DATA

W-5.1.5 What h the Degree of Relationship of the Two Variables X and Y as Measured by p, Gin

Correletion Coefficient?ProcoWure Example

(1) Compute (1) Using Worksheet 5-5.1,

, _- 107651

* N/1 163.94 N/k04107551

(368.92) (807.68)- 0,95

(2) A 95% confidence interval for p can be ob- (2) n = 16tained fromn Table A-17, using the appro- r - 0.95

4riate r. and r. If the confidence interval-does not include p - 0, we may state that Framate of17 i the inte iroSterval etimate of p is the interval from

the data gi-ie reason to believe that there is 0.85 to 0.9•. Sinct this interval does nota relationship (measuled by o rw 0) ma include p - 0, we niny state that the data

tweengive reason to believe that there is a rea-state that the data are consistent with the tionsoip betw ete two theds o ei-tionship between the two methods of esti-postsibility that the two variables arc un- mating tread wear of tires.

* correlated (p = 0).

"8-5.2 S11 RELATIONSHIPS Data Sample 5-5.2--Esth oated Tread Wear of Tires

lii this case, we are interested in an associa- For our exampli, we use part of the data used_lion between two variables. This case differs in Data Sample 5-5.1 (the SI ea.xnple). Sup-

SArom SI in that one variable h-s been measured pose that, due to some limitation, we were only4A. nnlv nry.slwe•:d values of the other variable, able to measure X values between X - 200 and(See Paragraph 6-3.2 and Table 6-1.) X - 400, or that we had taken but had lost the

data for X < 200 and X > 400. From FigureFor any given value of X, the corresponding 5-19, we use only the 11 observations whose X

values of Y have a normal distribution with values are between these limits., The "se-mean Px - pa + PýX, and variance .1y.x which lected" data are recoried in the following table.is independent ou the value of X. We have n_ auof values (XI, YO), (X2, YO),., (X,, Y.), X Y

An which X is the independent variable. (The - Tread Life - Tread LifeX values are selected, and the Y values are (Hundreds of Miles) (Hundreds of Mile)k

ik.... .. ~Wa urlab fn d~cwirýh, tb. +A ~,~tziatcul IRV

line which will enable us to make the bect esti- Weiplht Method Center Groove Methdl.mate of values of Y correbponding to given ---- , _ _ _

values of X. -75 a11a84 281

We have seen that for SI there are two lines, $10 240one for predicting Y from X and one for pre- $06 287dicting X from Y. When we use only selectld 809 259vaues of X, however, the only appropriate line $19SutatoI Y - b, + th X. e04 28127827

It should be noted thMt SII is handled com- 209putatiowaly in the sArne manner as, Fl, but both 246 161the underlying asumptios and the interpreta- 249 199

tion of the end results are different.

S •-!-_ ..-. 40

-=~ -4-7 -r

I ~.LINEAR RELATIONSHIPS BETWEEN TWO VARIABLES AIICP 706-11010)

5-5.21 What Is the U.st Line. To So Used for Using~ Data Sample 5-5.2, the fitted line isEstimating Yx From Given Valuesof X? Y - 8.96 + 0.661878 X.

PFroedurv

Using Basic NN orkshcct (see Wcrksheit The fit ted line is, shown in FIgure 5-1'8, anA the6-5.2), compute thc line Y' - ba + b1X. This confideice band for the line (see the procedtirsis an e~tixate of the2 true line fX + P,-I~X. of Paragraph 5-5.2.2. 1) also is ebo wn

WORKSHEET S-5.2

EXAMPLE OF Si1 RELATIONSHIP

X denotes Trea~d Life Estin'ated Y denotes- Tread Life Estimated byby Weight Method Centei Groove Method

I8 187 2;Y 2648

- 289.727 P240.727 ___

Number of points: ns - 11

AStep (1) ZXY 786369

(2) (2;K) (T-Y)/ 767197.818

(3) S. 18171.182

(5)(XgI/n E, 92330111 (8) (2ZY)'/n 637441____118 _

(1) i S0,6,87 (4 12027-015

(11 E 240,727__ (15) (n -2)4 EV m 6281.16:

() a bg48.965 2f._418

Equation of the line: Estimnated variam*ce of the slope.

Y N b + U~ _ _

- 48.965±+0.661878 X Sam.

O.1689 Etimated mviu-axce of i-atercept:

1 2197.818

46.83 n X1

- - -,

AX20P 70 -110 ANALY51 OF MEASUREMENT DATA

SYx •' 48.965 + 0.661873 Xa ( *

200 0

* ~ 2OO-

.. - TREAD LIFE (AUNDkEDS OF MILES) BY THE WEIGHT METHOD

"reen welht mt and enter, groove method when tI range off u.ight .method

Aw been re trted---an SI re"tionshi..

54.2.2 Wh0t mre the Confidence Interval Iestimates Uor. the Lin* as a Whole; a Point con the Line;a Single Y Carresponding to a Now Volvo of X?

ReeA the diciuIon of the inLerpretation of these three types of confidence intervalr in Faragraoh5-4.1.2 in order to decido which i4 the •ppropi ;te kind of confidence interval.

542

&~ 4Z

- - s...f .. * Atj-t I:~ te ......

LINEAR REIATIONSHIPS BETWEEN TIO VARIABLES AnC1, M-1..

5-5.2.2.1 What Is the (1 - a) Confldenco lknd For the Line as a Whole?

The solution is identical to that, of Procedure 5-4.1.2.1 and is illustrated here for Data Sample5-5.2.

Procedure Example

(1) Choose the desired confidence level, 1 a I) Let: 1 - a - .95a - .05

(2) Obtain sr from Worksheet. (2) From Worksheet 5-5.2sy - 2C.418

(3) Look up F1, for (2, n -2) degrees of free- (8) VS 11

dona in Table A-6. F.m (2, 9) - 4.26

(4) Choose a number of values of X (within the (4) Lpt: X 200 -range of the data) at which to compute X -250

points for drawing the confiaence band. X S00X - b50X - 400,for example.

(5) At each selected value of X, Compute: (5) Sec Table 5-6 for a convenient comput-a-

1Y. - f + b, (X - T) culations.

and+

(6) A (1 - a) confidence band for the whole (6) See Table 5-6.line is determined by

Y., =i W,

(7) To drrw the line and its confidence band, (7) See Fgure 5,-4 3.

plot Y, at two of the extreme selectedvalues of X. Connect the two points by astraight li-ie. At each ielected value of X,also plot Y. + W, and 1Y. - W1. Con-nect the upper series of points, and thelower series of points, by smooth curve.

If m(,. e po-*.nts are needed for drawing the curvea For example: 4', (but not Y.) has the "aMI.

f for the band, note that, because of symmetry value at X - 250 (i.e., 1 - 39.78) as at

-• the calculation of W, at n values of X actually X - 829.5 (i.e., t - 39.73).givfw W1 at 2n va'-%es of X

S.. ..... . . ." *-: . . .---

AMECP 706-110 ANALYSIS O7 MEASUREMENT DATA

TABL 5-6. C'OMFUTATIONAL ARRANGEMENT FOR PROCEDURE 5-5.2t2.1

I -X+k( - -X Y . + W ,X (X - k) Y. n S.. . , 1 Y - W,

200 -89.73 181,3 0.384179 258.12 16. 37 47.8 229.1 133.5250 -39.73 214.4 0.148404 103.67 10.ICI 29.7 244.1 184 7800 +10.27 ?A7.6 0.094751 66.127 P.132 23.7 271.2 223.8860 60.27 280.6 0.223219 155.79 12.48 36i.4 317.0 244.2400 110.27 313.7 0,53810 372.55 19,30 56.3 370.0 257.4

., -. 289.727 aV -697.9074 Y,- + b, (X -A)240.727 - - s--- 0.0909091 s.- +

b, - 0.661873 L

S.. - 27454.182 "/2F - V 8.52 2.919

54.2.2.2 Give a (1 - a) Confidence Interval For a Single Point On the Line, i.e., the Mean Valut-of Y Corresponding To a Chosen Value of X (X'), ,

Procedure ExI.

(1) Choose the desired confidence lexel, 1 - a (1) Let: 1 - a -95

- .05

(2) Obtain 81, from Basic Worksheet. (2) From Worksheet 5-5.2- 2C.418

(3) Look upt _t4,forn - 2degreesotfreedom (3) n - 11in Table A-4. c.7, for 9 d.f. - 2.262

(4) Choo% e X, the value of X a' whic. we war.t (4) bet X' . 300,to make. an irterval estimate of the mean for example.value of Y.

(6) Compute: (5)

WS - 4-0 SY -+ -- j) W2 - (2.262) (26.418) (0.3078)-18.4

and

Y. - f• + b, (X' - £) Y. - 247.5

(6) A (1 - a) confidence ititerval estimate for (6) A 95% confidence interval estimate for thetht: mean value of V --irrespondinjg to mean value of Y at X - 300 ia the inter.,valX - X' is given by 247,5 i= 18.4, i %' he interval from 229.1

to 266.9.i l•~~1 + b, (X - X) :t: W, t 6.

- Y. :1 W,.

6544

LINEAR kELATIONSHIPS BETWEEN "IO VARIABLES AMCP 706-110

5-5.2.2.3 Give u %'I - a) Confldetice Interval Estimate For 9 Sing, (Future) Valve of Y Con srondiricTo a Chosen Valie of :f - X'.

Procodure Example

(1) Choo" the desired confidence level, 1 - c (1, Let: I. - a .95a- .05

(2) Obtain sy from Worksheet. (2) From Woi]sheet 5-6.2-' 26.418

(3, Look up fur n - 2 degivee of freedom (3) ý.17: for 9 d.f- 2.262in Table A-4.

(4) Choose X', the value of X at which we want (4) Let A' - 300,to make an interval estimato of P single for examnple.value of Y.

(5) Compute: (5)

(1-/1 8, [1 + 4 W& - (2.262) (26.418) (1.0463)- -62.5

and

YI. f V + b, (X' - .,$) Y, - 247.5

(6) A (I - .Y) confidence interval estinaate for (6) A 95% confidence intervrl estimate for YY' (the single value of Y corresponding to at X - 300 is thc interval 247.5 --L 62.5,X') is given by i.e., the interval from 185.0 to 810.0

p + b6 (X' - X) W,= V :j-4J1

5-5.2.3 What Is the Confidence Inte-val Estimato for #I, the Slope of the True Unv. Yx += #,f - 151X?

jroctdvr* Example

(1) Choose the desired confidence level, 1 - a (1) Let: 1 - a - .95,x -= .05

(2) Look up t,,/, for n - * degrees of freedom (2) V - 11in Table A4. t..m tor 9 d.t. = 2.262

(3) Obtain so from Worksheet. (3) From Worksheet 5-5.2a =0.159439

(4) Compute (4)

W4 -C te-,, st. W, - 2.262 T0.159439)=- 0.860651

(5) A (i - a) confiderce interval estihate fur (5) 1, ,- 0.66i873A, is W, 0.3(W0651

b. :- 7, .A 9.,)% confiden-e interval e~timate for Ois the interval 0.661873 - 0 360651, ie.,the interval from 0.301222 to 1.022524 .

5-45


REFERENCES1. C. Ehiniurt, "The Interpretation of Cer- Jour ofa the Arcqiik S*atiical Asso-

tein Regression Methods and Their Use kieion', Vol. 45, pp. 164I. 80, 1960.in Biological and Industrial Research," 7. R. C. Geary "Non-linear Fu•nctiojial Rela-AnWaOl of M&Aematkal S~ksiestic. Vol. tionships Between Two Variable- When10 No. 2,pp. 162-186, June, i989. One is Controlled," Journao of th Am.er•-

2. M. Euskiel, M-ethAV of Correlation A•asoljis cas Statistical Asociatinm, Vol. 48, pp.(2d edition), Chapter 20 Jo , Wiley & 94-103, 1963.Sm Imnc Newi ork, N.Y., 1941. 8. J. Upka, Graphica n d Mecaniceal Corn-

8. J.Mandel, 'FittingaStraightUnetoCer- pu*o John Wiley & Sons, Inc., Newt, u Types c/Cumulative Data," Jour- x'ork, N.Y., 1918.fW of the A •ericm Sxd"ea Aaws,vs- 9. H L. Rietz, (ed.) Readbook of Ma•jemt-N Vol. 52 pp. 56•-666, 1367. ical 0kisetcs, Woughton Mifllin Com-

SF. S. Acton A w4 si s of Stft igA -Line Data, Eany, Boston, M aes , 1924.John Wiley & Son%, Ine, New York, 10. J. B. St'rborough, Numerical MaOthematiclN.Y. 1969 A% sis (2d editior), The Johns Hop-

6M. " Bartlett, "Fitting trh n kins Press, Baltimore, Md., 1950.Wba Both VariablN are Subject to 11. R. D. Stiehler, G. G. Richey and J. Man-

roik ," pBioWric, Vol. b, 1o. 8, pp. del, "Measeurement of 'readwear of27-212, 1149. Commercial Tires," Rubber Age, Vol. 73,

&. J.r merka, "Am There Twe Reageicns?", No. 2, May, 1953.

(54)

5-46

AXCP 706-110

CHAPTER 6

POLYNOMIAL AND MULTIVARIABLE RELATIONSHIPS

ANALYSIS BY THE METHOD OF LEAST SQUARES

6-1 INTRODUCTION

In this Chapter, we give methods for estimating the coefficients of, and for answering variousquestions about, multivariable functional relationships of the form

! I• - • •e + 0, X, +.. + 04._, x,_ 6-1

between a dependent variable if and a number of indepetdea variabs x%, xt., xk-,. We reatriet

our discussion, however, to the aein which the values of the independent variables z% z,.. , -are known exactly, and errors oi measurement affect only the obs mad values Y of y, that i*, tomany-wariable analogs of the F1 functional relationships considered in Paragraphs "-I. and "-I..

Methods for the analysis of many-variable relztionship. in which errors of measuremment affeot thevalues of the x's involved as well as the obsmed Y's, i.e., the multivariable analolg ot the FIIstructural relationships conaideied in Paragraphs 5-4.1 and 5-4.3, are vot dicumed por as in this

Chapter. If, however, the ep rors that affect the x's are not errors of mewirement, but rather areerrors of control in the sense of Paragraph 5-4.3.2, i.e., are errors made in attempting to setX% X..,..., Xb-,, equal &o their respective nwminal vaiw.s z0 z4 ...... z;-, then the methods ofthis Chapter arm applicable, prov that the errom made in adjusting Xt, X1,..., X&-, to thurreaptive nominal values are mutually independent (or, at leat, &re uncorrelated).

The techniques pr Iented in this Chapter are general. They are applicable whenever we knowthe functional form of the relation between y and the x's, and are primarily cocesrned with seti-mating the unknown values of the coefficients of the respective terms of the relatiomship. Tbhu,taking x. - 1, z, - x, z* -t,... , x. - zx, the methods of this Chapter enable us to setimate the Icoefficients of, and to answer various questions about, an ruth degree polynomial relationship

M'.Wth &psdant j =da -; Te -. 4-J;e-dm!w x A't~N"?"ily. takingE Eao 1.i-, • -, x, = , x. - , z, = z, and A, - z', the techniques of this Chapter can be used to imvsti-gae the nature of a quadratic aurface relationship

I - 04 + (Piz + 00) + (Ag, + 04 + PIZ)(-)

between a depcd&t variable 1 and two independmi variable• x and z. For example, we may wiSto test the hypothesis that the srfac, actually is a Jdae, i.e., that ps, 4 and A. in Etquation (64)

are equal to zero, and so forth.

6-i

AMCP 706-110 ANALYSIS OF MEASUREMENT DATA(

Muitfvarate' Statistical reilationships analogous to the Si and Sii situations considered inParalraphs 5.3.2 and 5-5.1 are not cor idered per as in this Chapter. If, however, Y and X,X .,- * X&- t, have a joint multivariate frequency (probability) distribution in some definitepopulation, and if a sample of size n is drawn from this population, with or without selection )rrestrictions on the values of the X's but without seleetion or restriction on the Y'V, then the methodsof this Chapter, taking X, a 1 throughout, are directly applicable to estimating the coefficientsof, and to answering various questions about, the multivariate regression of Y on X 1, X9,..., andX", namely,

?w(g - +10 + PIXI + X + P-lX,--,, (6-4)

where Pg,3 is shorthand for mv.x,x,.. . ,, the mean value of all of tho Y's that are associated iithe population with the particular indicated combination X1, X2... Xk_1, of values of the X's(nme footnote of Par. 5-3.2)-and, where

A- Mr- - -Pmx - ... - amznmx,, (6-5)

i i,. . ., ,, are the population means of Y1. X1,. I, Xh-1, respeclively. The fitted regres-sion, yielded by the application of the methods of this Chapter to observational data of this kind,will be of the form

?w - be + biXi + bX +... 4-. bXi- (6-6)

with b.=? - bil, - b k14TIk-• (6-7)

where 1, X9, ,,... , _,, are the means of Y, X1, X,,..., and Xk-l, in the sample; and each bwill be a Ust (i.e., minimum variance unbiased) estimae of the corresponding true 6.

When, as in all of the previously mentioned situations, the relationship between y and the z's isUsear in the coeffients whose values are to be determined from the data in hand, the Method ofIeast Squares is the most generally accepted procedure for estimating the unknown values of thecoefficients, and for answering questions about the relationship as a whole. A widely applicableLeat Square. Theorem is given in Paragraph 6-2; and its application to a general linear situationU nisrmated in detail in Paragraph 6-3, with worked examples. Special applications to polynomialand other situations are discussed in subsequent pringraphs of this Chapter.

The numerical Calculations required for least-squares analysis of multivariable relationshipsoften are lengthy and tedious Hence, this Chapter is directed toward an. ngement of the workfor automatic computation on modem electronic computers. Consequently, basic equationb calledfor in the calculations are written both in traditional and in matrix forms. This Chapter concludeswith a diseusion ol matrix operations that are useful both in formulating and in carrying out therequisite calculations, Paragraph 6-9.

In most instances, related Procedures and Examples appear on facing pages in this Chapter.

'The important ditinetion 3n statistieal work between a r(able and a sariate is drawn in the Kendall-BucklaadDkt•mary of S&W6oao Tes(l') as follows:

Varienerally, any quantity which varies. More precisely, a variable in the mathematical sense, iLe., .aquantity which may taeke any one of,,a "s*, .iffed set of .values. It is conyenent to apply- the, sam~e word to denote

noa-ameitmble eharaeteristic, e.g., "sex is a variaLle in this sense, sine any human individmut may take one ofWe "Vfthe", male or female.

It k useful, but tar from being the general practice, to distinguish between a varlatlo as so defined and a randoravariable or variate (q.v.).Variek-In •utrlds etion to a variable (q.v.) a variate is a uantit which may take any of the values of aSedfied ret with a specified relative frequency or probability. he v rate is therefore often known as a randomvaiable. It is to be regarded as defined, not merely by a set of permisible values like an ordinary mathematicasvariable, but by in asmociated frequency (probability) function expremsing how often thore values appear in thesit ation under discussion.

6-2

POLYNOMIAL AND MULTVAAI.E C•!!LAT!Q SPS AEPOw-.1o

6-2 LEAST SQUA.RE THFORIEM

If the ni measurement. Y, Y,, ... , Y. am statistically independent with cowm vu*ma63und have expected values E(Y,),

E(Yi) - pool + IX1u A u A- ... + •,-iXi.iE(Ys) - AP% + Pzi, + A#X +... + - (- -)

E(Y,,) - X.+ PA:. + "Is + -- + ~~X~then the best linear unbiased estimated &, 1, A?, ,... , O,_, of the unknown cos•cients are givau by

the salution of k simultaneous equations, called the twi ad eqwitioss,

&, z Z,: + ý',I-.a +.. + z,- z ,=,,-1 z me YZ.4 0.z e + 21-1 Z XY

Sz, + 0, T =,_,=, +... + O,_, z 4, - z Z*-, Ywhere the sumation is over all of the x values of the vadibls invrlv-i; e.g.,

and the estimate of r is gien by

If no unique solution to Equation (6-9) exists (which will oeur when one or morm of the x's amlinearly dependent, for example, if z1 - ax, -I- bz,), then not all k coefleients can be esmatedf~trui the d-ta. V -'abh! may be delete or several variablva may be replaced by a lnear funetionof those variables so that a solvabie systeim involving fewer equations result-.

In situations where the Nwriance of the Y's in not the same for all Y's and/or there is correlatimamong the Y's, a transformation of variables is required. The methods for tLese casm are dia-mdlater in thir Chapter.

This theorem can be restated using matrix notation a follows:

Let, Y YI]X XO- Z1... ,_ij and -[P

Y.I L 7,.% .. zX,-,,nJ L Pi,-

The expected viues af the Y's then is expressed as

£(Y) - X0, (6M)

and the condition of independence and common variance is expressed byVar (Y) - V - o,1I.

Under these conditions, the minimum variance unbiai-od estimates $ of p are given by the solutionof the normal equations

X'X = X'Y. (6-9M)

AMCP 706110 ,ANALYSIS OF MEASMREMENT DATA

The wtimate o., I i giver, by

1

1 -- (Y'(Y - ,•,ZY).

Equations (&-8), (6-9\, and (6-1.0) are given in Vbe usual algebrasC notation, and the correspondingequstions in roatrix notation are (6-8M), (&-9M), and (6-10M).

-3 MI$LTIVARIASLE FUNCTIONAL RELATIONSHIPS

"bI1 t'W ^W AS5UM.IW D

least-squares methods for estimating the coefficients of a functional relation of the formi - "z + Pix, + i't q,. 4-) • , (6-1)

amused in a num-ber r'~~'~os(a) when it is nucwa fhaw theoretical considerationa in the ubject matter deld that the rela-

tionmhi o ifetwo fn'm;(b) when the enac, expresion relating V and the es either is unknown or in too complicated

to be umed directly and it is muned that an approximation of this type will be satisfactory.

lu the otter cam, the proximrationi often can be justified on the grounds that, for the limitedra=V of the :es eowider, the urface rep"eenting jr a a function of the z's is very nearly theb wan sies by Fqzti~on (6-1). '1Uh. method is strictly valid in (a), b-it in (b) there is danger .,

of obtaining minled&&._ remita, anal4Vur. to the bias arising in the straight-line came from thewumnption that the funithmal relation in,i#Wved is lihio hen in fct it in not linear.o

In addition to t'e vaihdity of Equation L-l), the following asumptions must be a.tiahed:*

(a) the random erors in the Y'ra have me m zero and P- common variance l";a.) tM t-"ndorn " i t.- Y's wre mutimd.y idpedet th st-stiatit•l• sense.

For striet validity of the usual tesut of signifiimce, and confidence interval estimation proceduresin Prarph 6.&8 (Step 8 and 9), an additiotal absuption mus b satisfied.

(c) thi random errors affeting the Y's are zrmoally distributed."" , variables may be powers or other functions of some bade variables, and several different

tunxlons of the sme z v&rae may .e used. (See, for example, Equation (6-2) or (6-8)).The data for analyis conist of the % ;ws (x•, z,, . ... , zh-,.,, Y,) (z, z,. ... ,,.2 Y,) ... ,

(go. as..., x--.. Y.), and usually we Tapreented in tabular form as:

KX ., X, X1-, Y

R ,•ft 4-1..1 YX

SWhe then meftMMm m anot a"Jied, fo rSUt9~ $ the am of ilqualiQt of vafand Prg ( )fer the eav d• .veatlamo u the Y'O

IiWPOLYNOMIAL AND MULTIVARIABLE RELATK)NSHIPS AMCP7 70&110I)I

Alternatively, the data may be expressed in the form f o•o4epgisd tadkEiop

+ "•" i +II "&•' 41 +o - "- Ji-- I~ + Yi - W1l + #I

~ 1 +~Z~ip.Z~ +..+p.~lX--.1 1 m 1 ~(8-11)

06aX 4 I. + PA. + " ... + b_,, * a . -. + en

where e1, e, ... e. denote the errors of the Y's as measured valuefs of the corresponding true V'sWhen the number of observational equations exceeds the nurnl.er of unknown coefficients, i.e.,when a > k, the observatina equa.tions ordinarily are mutually contradictory; thae is, the valuesof 4, Ot. . . . , and 0, -, imyrP'-v b , mny chosen solvable selection wf k of the equations do not satisfy(,Ie or more of the reniahi. ig n - k equations. Hence, there is a need for bees estimates of the p'sbaed on the data m a whole.

For a unique least-squares solution, is must not be less than k, and the normal equations (6-9)must be uniquely solvable. If not, some variables must be deleed or suitably combined with othervariables.

64L2 D0SCUSSION OF PROCEDUIES AND EXAMPLES

In setting forth the fteps in the 3olution, the formulas are given in the usual algebrdc notationand also in matrix notaition where appropriate.

Data Sample 6-3.2, selecad for arithmetical simplicity, serves to illustrate the worked examplesoV numerical procedutwv involved in estimating the coefficients of, and in 4naweriag various questionsabout, multivari.&blh fIt•trmnal relationships.

1 8 2 22 81 7 42 6 0 48 1 2 4

2 7 3S5 1 3

We assume that these data zorrespond to a situation in which the functional d pendence of p onzi, xb and xj is of thr forn.

p - AX, + 02MS + " (6-12)

which is a special case of )quation (6-1) with the term "Sr. omitted; i.e., with Po taken equal tozero. Equation (6-12) impli" tli-t the functional dependence of p on zx, ,b, and xz, takes the formof a hyperpiane* that pmw; 4'.hrough the origin (0, 0, 0, 0) of the four-dimenszýnal Euclidean apac

* A fiat smrace in four or mome imensions in termed a hkpporlaim when it Is 'he loumm of points t•ht vary in morethan two dimensons.


whom coordinates stre zb Z,, xe, and V. If we wiahed to allow for the po&Sibility that the dependenceof V an z., 2, and r.. may take the form of a hyperplane that intersects the v-axis at some point(0, 0, 0, Q), not necomaily ths origin, then we would substitute

for Equation (6-12), and take z = 1; i.e., amend Data Samp'e 64-.2 by adding an ze column cf 1's.

By analog with Equations (6-1i), Data Sample 6-8.2 and the assumed functional relationshipEquation (6-12) can be summarized pstiy*lc by obewuss equo'a"t of the form

(6-13)

Arm. + OXM+ Agn I's

Wo.tutlcm of the rluss of the z's and Y's of Data Sample 6-8.2 in Equation (6-18) givesA.1 +: A,4 + Pa.1 -2Al + P.8+ + . -2

4,4 + A..6 + P.0 - 4 (6-14)t8 +A-.1 + %.2 -4

A.4 + 1.2 + A. - 8+.4+ A5 + -1 -8

as the obewrational equations corresponding to Data Sample 6-8.2.

64L. PlOCIDURES AND XAMP.ISsh,. I Ps=c. • !i.n en=----W-o• w Equasemns. The or*W eqatiotw are foimed from the

muns of sqmuame and cross products as follows:

POZr + AIZOXIz +.. - A- ZcZ- -+ my

p.Zxz, + IXA + ... + •_•-1ZrXZXA1 - • (6-9)

ApZxa-,xo + flZzk-ixi + -.. + Zxb-, -i

or in .atrix form

x'xO - X'Y - (6-9M)

where Q' - %, q,,- q

an q1 - z, Y, (j 0, 1, ... - 1).

.jjidPOLYNOMIAL AND /ULTIVARIABLE RELATIONSHIPS AMCP M06110 1

Stpo I Example-Formatlor @t Normal Equations. The sormul equyions (See Equations (6-9))corresponding to the observatir.nal equations (6-13) are

fi 1A, + P,, Xzaxi 4, 0, lz Zz1Y0, 2X 1X1 + 5, ZZ: + P, 2,T, - z2,Y (6-16)

01 ZZAx + 2 Z;XIXS + #%, Ze, - ZXSYor in matrix form

.1 A- . A

* ~~where 5 F'

Numerical evaluation of the requi.ite sums of squares and sums o1 cross products for Data SWdple 46-3.2 and substitution in Equation (6-15), yields

0.-50 + 01'67 + #.a53 - 540-67 + %U,.194 + , -85 - 97(6-16)

0, .53 + 4%.65 + #,-104 = 62

and the matrices involved in Equation (6-15M) become

(X'X) - [Zxl Z~XI 2 ZzXX 1 - [610 61, 5M]~X~X1X -*z ZxýW: [67 194 86

LZXX, ZXX,, Z4 - _3 85 104]

W)Y)=IZXIj -A - q. (6-16M)LJL6.J Li 6-7

Af -M..#Dfl AAi.AIVIW CE t'M: PAC,•: t• MlJT DATA

S3 l 2 en of Nrw. I Equefons. Equations (6-9) can be solved by a number ofmethods giving valuest for•% 0,, 0% .... ,which can be expresed as

•,-Coo% + cfq, + • .+ c,.,-sq,-,

c# - 61., + ¢11q, + ... + cj.i-:q#.-. (6-17)

~b1 C...jVj+ Ch.. ,i1j + +c&,hq.

A solution for the 0,'s chJ be arrived at without explicitly computing the cj's, of course, but inthe following computtntions the cj's are ne•ded. The values of the c,/s depend only on the sumsof squares and crow prmducts of the indepeni vakiables X-% z1,..., z, so that the estimates ofthe #,'s can be expressed as a linear function of the Y's.

In matrix notation, this step is given by computing the inverse of the matrix of normal equations,i.e.,

CIO 1[ e .11 . C€1 .-1

and Rquat•o. (6-17) become

S- (X'X)-'X'Y (6-17M)- (X'I)-'Q.

(

IiPOLYNOMIAL AND MULTIVARIABLE RELA TiONSHIPS AMCP 706-110

Step 2 Examp1--S.Iution of Normal sqijtions. The J dues Ab, 03 and $x, that constitute thesolvtions of the normal equations can be expressed (See Equations (6-17)) in the form

Al - l q, + cel (i + cl ql- cl'*54 + clo"97 + cls*62

- ell q + c q, + cu q(6-18),-,C.

54 + et. 17 + c.. 62

A, c,-,c q, + c .q9 + ca q- c.1.54 + c,--97 +- es,'62

where the r's are the elements of the inverse matrix

(X'X) -1 - [C1 2 C1, C

LCS1 Ca Cul

(X'X)-l may be computed in many ways. * The exact inverse of the matrix (X'X) de~ermined by

the first equation of Equations (6-16M) is

1 12951 -2463 -4587"]

(X'X)-, - - -2463 2391 -6!•9 (6-18M)4587 -699 5211_]

where the factor in front of the matrix is to be applied to the individual terms in the matrix.Using the first equation of Equations (b-IS), we get

S129q-418 1(12951) (54) + (-2463) (97) + (-4587) (62)1

1

j699354 238911 - 2W141

176049239418

- 0.735 320 652.

The other ccefiicients are obtained similarly:

9, = 0.232 175 526B3 - 0.031 664 286.

"The prediction equatin, therefore, is

, 0.735 320 652 z, + 0.232 175 52t; x, + 0.031 664 286 x,.

•'The advent of automatic eleetroniz digital computers has reduced the inversion of matrices of even moderate isteto a matter of seconds. Routines for matrix inverrion are standard tools of automatic computation. In wontrast,

matrix inversion by desk ealculators is a time-consuming and tedious affair. Detailed ilzustrotion at f" ncture ofany one of the common Ancthods of matrix inversion by desk calculator would not only constitu'e 9. disti interrup-tion to the orderlv presentation of the esential features of this Chapter but would lene-hen it considerao,%. The twomost common methods of matrix inversion by desk cnlculator--the DoiiZie methoi, and the a W-eri JtoolUit mtLWod(also calied the Giujs-DooliU4l mel1 od)-are dev'cribed and illustrated by "uynerical examples in various statisticaotextbooks, e.g., in Chapter 15 of Anderson and Beacroft.'l Details of the square-root method, favored by Some aom- zputers, are given, with a numerical illustretion, in Appendix IIA of 0. L. Davies' book.(3 All of the con.mon met rodsof mitrix iuvermor. by desk calculators are described in co.sidt-rable detail, illustrated by numerical exanples, andcomn.pared .witt -espe.t to advan.tages and disadvantages in a pr; er by L. Fox, PraeAicWl Solution of Limar vg'q,'wio"wand Inversion of Matrices, included in Tauasky.A) Reference aulo may be made to the book of Dwyer.(') The readerof this Handbook who is faced with matrix inversion by desk calculator is referred to thes standard sources for guidau-eand details.

6-9--- -- mu

AMCP 7M-1 I-A ANALYSIS OF MEASUREMENT DATA

Sftp 3 Procedur--Calulaflon of Deviation Between Predicted and Observed Value of the Y*s.The prdicted value I, au a given poirt (4,, x,,, . , xk..,, Y,) is given by substituting the valuesof - in the prediction equation, i.e.,

?, - A, + Az,, + 0,",, 4... + - ,,_+

and the reaidu.iR r, Yi -Y, are given b)r, - Y, - ki Y1 - AX0, + 4,X,, + .. + g,_ ,z,_ ,.,)I' - Y2 - f,- Y1 - (bol + Ax1 + ... + k-1x •)

(6-19)

ra - Yf - Y%- ( sZ + Izf, +... +4._1x _..)

or in matrix notation

r - Y - X0 (6-19M)

where

r -K

In classical loast-squares ,uialysis, ?, is termed the a4usted vsae of the observed value Y,. It isimportant to distinguish between the errors of the Y, with respect to the corresponding trute valuesyo and the r e..--sGz o0 the YI with respect to their agtwesd or predwte vlue3 Y, ; that W, betweenthe e, of E4uations (6-11) and the r; Or Equatione (6-19).

Dep 4 Proceduvs--Ulimation of as. The estimate a' of " is computed frona.

, • ,_, )(6-20)pa' - 04r'

or in matnx notation1

t(r'r) (6-2oM)

- h Y' -

6-10

I i

POLYNOMIAL AND MULTIVARIABLE RELATIONSHIPS AMCP 706-110

Step 3 Exampl--Calruleflon of Deviation Beween Pmtddicbod Wnd Observed Vol.e ot the Yr$. Thepredicted or adjusted values Vi corresponding to the observations Y, apn obtained by substituting,.he values of the x's iiio tCe pret'etion equatiou. For the first observation, s,.batitutivg xi - 1,zt - S, x$1 - I leads to

0- 0.73h 320 662 k1) + 0.232 175 5W 2)+ 0.03i W64 286 (1)- 2.624 388 146.

The eoiresponding residual is

ri YI - 1-1

-2 - 2.624 389 146- .624 389 146.

The full data, the corresponding , .d;ected values (1',) and their r-sid-ala (r), arm

Residuals

1 8 1 2 2.6231 389 i6 -. 624 In9 1462 2 8 7 4 3.549 Gf ?'514 .456 N4 4863 2 6 0 4 2.W3 694 460 1.1.q6 306 5404 3 1 2 4 2.501 46C 064 1.438 S33 94i5 4 2 7 3 3.627 283 662 -. 627 282 6626 4 5 6 3 4.1&3 Si 524 -- 1.133 824 524

..t-! 4 ,of A he est.im,.1'p PI of f2 may I*- ,-omputed dircetly i.omm the sumc.' squared reiiduals. Thas,

-k '

13 (5.808 473 047)

- 1.936 157 692

where n is the number of observatio.ial points (herv 6) and k is the number of coefficient-% estimatedfrom the Gata (here 3). Alternatively. s8 myv be eval,tated from

6 3 {70 (0.735 320 652) (Eý) - 1.232 175 526) (97) - (0.031 6u ) (62)j

1 (5.808 473 038)

S1.936 157 679.

Exixacting the square root Liveb

a -• 1.391 458W.

6-11

AXCP 706-110 ANALYSIS OF MEASUREMENT DATA

Sfep S Posed&um-..Ilemaed Sa.lonrd Deviations of the Clefficenks. The estimated standarddeviation I i is given by ,/,i, whri ithth ith dional term of the inverse of the matrix ofnormLi equations

ust, s~d.ofe 8est. sd. of 8 -V

S.....(6.-21)

•st. a d. Of . bl.

lStP 6. Pmesedv,.-S0snded Dovias of, a FPuncfln of the P'. The standard deviation of£ , aA + aAL + a, +... + in- estimated by

esLt ad. of L - aiaicii(622

or in matrix form

est. &d. of L - &vi',(X5'X)- (6-22M)

where V aej .. ., a-

Cames of special interest are:(a) estimate of a single coefficient, i.e., L 0= j, in which caie Equation (6-22) reduces to Equa-

tion (6-21);(b) estimate of the difference of two coefficients, i.e., L - 0, - Aj, in which case Equation (6-22)

becomes

est. s.d. of (0, - •) - SVCI, + Cj - 2c,. (6-23)

I..~ -. - * w .. . mn.1. .a- - .6 ~~ -19W~~,bC

dicted yield Yk. at any chosen point %,• zl,... *,-.), is given by

-•, A.x. + #i,, + ... + 04-10,-1.h

which ia a linmr fWction of the OL Application of Eq)ation (6-22) leads to

est. .d. of , (6-24)


est. s 'd. ot P - (Mr... ,N

where z- .--- "k

11 POLYNOMIAL AND MULTIVARIABLE RELATIONSHIPS AMCP 706-110

Step 5 U•xmple- vmId Sbandud Dovkfiens of th Coollidaft Th values of the estimate1 ý

standard deviations of the 0's are:

EstimatedStandard Error oF

Coefficient Coefficient, sv'-;

21 .282 581 An2 6272, .099 984 .139 054

.147 531 .201 288

Ste 6 Example-Standard Devkuion of a Unsor Function of WMo CoA*ICkns. For illustrative pur-poses, consider L , - 10 2,.

By Equation (6-22), or Equations (6-22M),

tst. s.d. of fL - sV-e + 100 ca - 2o enin matrix notation

est. s.d. of £ -with 1,- (0, 1, - 10).

Numerical evaluation in this instance gives

est. &d. of L - 1.391 45M8 (537471) 1

- 2.0848.

II

Slop 7 Example-S-ondoad Deviation of a Predicted Point. By Equation (6-24), or Equation(6-24M), the eitimated standard deviation of the predicted idid ?A, correaponding to any chosenpoint (zl., &., xsA), ij given by

est. s.d. of t - s Xi'(XXY1


where I' - (x,.%, X,%, z,•).

Thus, the estimated standard error of ?,, the predicted or adjusted 06W corresponding to thefkst observational point (1, 8, 1), is

est. sd. of - ste, + 8e,, + ell + $&,2 + 64Wv + &.n + es, + 8c. + call

1- 0.9 49 8

-35.

(\239418)- 0.949 23.

6-13

AMC1P 706-110 A 4ALYSIS OF MEASUREMENT DATA

Re a FPmc*x.--Andysl, of Valance Tod Sof n 3 l4ance of a Gmup .1 p < k o. ts C.eftienhs.To test the statistical significance of a set ol p of the A's (for simplicity the last p), start with areduced set of normal equations, omitting the last p rovs and columns, and repeat Steps 2, 8, and 4,as a problem with (k - -)) variables:

(a) The equations in Step 2 then are reduced to

(6-25)

zx~&.4.+. + Z~$...

and its solution becomes

+ +

(6-26)

+ h-p-1 +.. +cp1 q,..

(b) Thse values c,. will, in geners!, be different from the ei, for the original equations, so thattOw Coeftients

a:, a:, ,...,•-,w'll resdt.

(c) A new vhlue of s, say s*', is computed from

ZYI -p) - (6-27)

These operations can be handled conveniently by matrix methods. Paragraph 5-9 contains afurther discussion of "Matrix Methods."

An An*#u of Vantece table is formed as follows:

d.f. Sum of Squares Mean Square

Total z ZY'

Rduetioncue to k constants k A, q, K

Residual (after k constwts) 0 -k z Ys - ,q,

Reductio due to k - V constants k - p q, A

ReuU u-aiaftrk - Vcomtants 0 - (k - p) ZY'- $? q, *'

R•dction due to additional • constants P q, - $ q_ P

If the Iu' are normally diatzibuted about their expected vaLies, then

6-14

POLYNOMIAL AND MULTIVARIASALE RELATIONSHIPS AXCP 7W6-110

i.p 3 rtne-p-Aml*s of Veries. Toed of 5Imome of Lc * Cedemkb. The required

ArsLpiaof@ Ven..... table is:

dU. Sums ol Squares Mean Squar

Yotal 6 70.900 000

Roduetiondueto oomtstats L,Aband .) 8 64.191 527 21.897 176 -i i

R'Iiduals (sifter 8 constants) a 5.8ON 478 1.986 166 e1.. ......................................................... ............................ ............................. ............................

RsduethmduetoDand.only 2 64146 461 207 780 - A

mbiduai (aftr 01 and $) 4 5.864 589 1, 46 685 -

R.duetion due to A, 1 .046 066 .04066 - P

As implied by Equation (6-27), the sum of sqrs for W reducwio due to 1 avnd O ol -• q, + 0: q where A,* and 0 are the estimt~ms of 0 and A that are obtained when A in taken equal

to zero: i.e., when the undiefying functional relation is taken to be V - piz, + pA.

The steps required to evaluate 0r and a•

_ [50 67(X'X) 67 194]

FXl 541 _[qj

' '' Ir- #.*J - kXX)-' [I j

They yield

H- 0-.768 198 245"- o.26 422 951.

Heice, reductiou due to #*, and 0: only is given by(0.768 193 245) (54) + (0.286 422 951) (97) - CA.4,145 461 .

as shown in t~he Analysis of Variance table.

A .LrI-1^F^uL1rT0 OFr ME SIREMENTi DAiA

(L) F - K is dixtrbuted a F with d.f. -k, - ,and ser-es s a test of whether all L constants

account for a signi•icant reducticn in the error variance.P

(b) F -j is distributed as F with d.f. - n,, - k, and serves a a test of whether the addiLor,

of the oetffcients accounts for a significant reduction in the error variance over that accountedfor by U• frst k - constants

NOTE; In caes where a constant tei-m is involved (i.e., zo, - 1) we wculd use

F-M

-. which idistributed as F with (k - 1) and (n - k) degree of freedom o a test for the efficacy of theprediction equation.

im 9 !•. &M-.C.O--., ce I vwd immew. L, and L, constitute a 100 (1 - a) % conf•lJent*

interval estimate for:

(a) a acefcient p,,

wha LA1 - +- t.•, (est, s.d. ofLs- •,÷•., rt.ad i•)

(b) a predicted point on the curve 9,wh," I-, t • .- &,.. (est. a.d. of 1?,)

A - +t.4. (est. s.d. of P.);

(P) a diffrenep of two eweffirients A -- A..

when L, -- ( 0, -- 0 -t._. (est. .d. ofj•, - ),- ( - $ + t+ . (est. Ld. of • --

In the above, t,_.,. is the value of Student's f for (i - k) degrem of freedomo exteded Nrith

probeality 2.

£

6-4 I

.OLY.INO.IAL AND MUL'IVARIABLE RELATIONSHIPS AMCP 705-10 Jslop F Examuple (Ceuit)The test of significance for i% W

P

.046 060ON.96 158

-. 024, d. - 1, 8.The -alve of F(1, 3) exc•ede,•e, probability .05 is 10 13. The observed F dcs not exsd

this iricsd value, so that A# is 7,11. ,•,,pried as being statistically •s•ificantly different from mro.

Step 9 Axempie -csdew* ktderIs isi-s. Fjr 01. the 95% confidence interval estimateL2 : P_ :5, L is determined by

- - tL.., (est. &d. of 0)-0.735 320 652 - 3.182 (.823 627)- -. 294 460

1A• - $1 + ts, (est. s.d. of Oili ~- 3).785 320 662 + 3.18-2 (.328 eV2)

- 1.765 102

where. to.-. -. 8.182 is the value of 'Student's t distribution for three degrees of freedom exceededvv--4 p Ca, , .026 (or exceded in absolute value with probability .05). 1

t

1I

6-4 MULTIPLE MEASUREMENTS AT ONE OR MORE POINTS 4

More than one m=zuritnieI y ke• . am or at all of the valueu of the independent• ~variable xc. '1his ubually is done when the random errors are suspected of being comloe•! of two

copoetson omonn awsociatod with the variation of the points about the curve•, &ad the,|other component associated with the variation of repeat determinations. The ith measurement at

the ith point then can be reprsented asy o, +- -, "t -,+ + is,-A ,-l., + 41 + , (6-28)

where the e's and vy's are independent and have variances o' and a. respectively.If a number p, of repeat determinations are made at each of the n points, the estimation of or,

and e, follows from a modification of the Anvalysis of Variawe table:

6-17

t


Sum of Squares d.f. Mean Square

Reduction due to fitted constants dzz ,Y) - C k C/k

R sadrl (riter fitted cons': nta) T - C - R Xp, - k R/(zp, - k)

Repeat determinatitna (Yj - 1•, - E zp, - a E,/(Zp, -n

Vhatido of averages about the curve R - ,- E1 - - k El/l( - k)

The exp.ted value of E./(Zp. - n) is c', and the expected valie of E1/(a - k) isd; + pe,when all the I, awe eiual to V.

The quantity [E1/(% - k)j/[f./(Zp, - w)I - F is (under the assumption of a normal distributionfor the a's and's) distributed as F, if o - 0, with m - k and Zp, - % degrees of freedom, and may

be used to test the statistical signiflance of the component of variance associated with the e's byeWmperhw the observed F value with tables of the F distribution.

If a) the A are equal, the proper variance estimate to use in calculating the standard errors, orConfidane interva of the estimated constants, is E1/(* - L).

6-5 POLYNOMIAL FITTING

If it C" be asumed that the relatioi between the deps"dwvarb Y and the independmiWWd. x is

Y, - A P4, z + M +..,-+-4 +, (6-29)

and thbat the errorm of measurement 4, are independent avd have the same variance 0', then theechniques for multiple regression carry over without change, by setting:

The nc=Wal equations are

Za-ps + Sz', + Ze+,01 + ... + Zxv,-,2 -Y-

Note that if the constant term is assumed tc be zero, variable xp is dropped, and the first row and (cohumn are dropped from th3 normal equations.

6-18

POLYNOMIAL AND MULTIVARAKE RELAWi)hNSiiP5 ANMCr 71V•_A&V4

In using a polyno.aial as ai approziniutui to s-me unk•,w-r ffunctioc, r a -formula, the correct degree for the )olynomiial usually is not known. The followingi procedureusually is applied:

(a) Carry through the steps in fitting polynomiabs of 2nd, 8rd, 4th, 5th,..., degrees.(b) If the reduction irt the error sum of square due to fitting the kth degree term is statisetallv

significant on the basis of the F-tAst, whereas the similar test for the (k A- 1) degree to ý, is not.then the kth degree polynomial is accepted as the bedt fitting pol n .ni' t.

In thih procedure, the degree of the polynomial is a random variabi, And ropetitions of the txperi-ment will lead to different decree polynomials. When the law is truly polyoW, the computedcurve will either be of correct degree and hence will give unbiased estimates of the coeficienata or,ij o of correct degree, will lead to bihaed estimates.

Whea the law iL not exactly a polynomial, the error distribution for the Y's will be centeredaround a value off the curve, and it will be difficult to assess the effect of such systematic errosIn the limiting case, where the variance of the Y's is nearly zere, these systematic errors will betreated as the random error in the mecaurements. Usually, it will not Le valid to assume that thesesystematic errors are uncorrelated. On the cther hand, if these systematic errors are small relativeto the measurement error, their effect, probably can be neglected.

6-6 INEQUALITY OF VARIANCE

V*SK-AO OF -- ZMRE A- 0- nIIDISW)~ I4r FR 1LaUUKUA m

When the measurements Y, have different precision, i.e., when V(YJ - a and 4r', 0',% for atleast one pair of subscripts 1 :< ii <i2 _< it, the conditions of the least squarme theorem of Parm-graph 6-2 are not satisfled. However, the transformed variates

have a common variance V(Y') - 1. Often, we have information on the relative magnitudes ofthe variances ,r• only, and not on their aboolute magnitudes. If the variances $ are expressedin the form

4 *" -• (6-1),

thez k. is termed dhe reative weighS* of the measurement Yj, and the quantities YIf - ,/w-, Y,h'it cummon variance a', the magnitude of which may be unknown. In other words, equality ofvaw ;, is achieved through weightng the observations by quantities proportional to the reciprocalsof ts'• qm sr andard dpviations.

);The abýtýue iighd of a meaurement Is, hy de•l-. ton, the reciprocal of its variance.

6-19

AMCP 706-110 ANALYSIS Of MEASUREMENT DATA

""2 PRFACHUR9S AND EXAMPLUS

Prece~dwm-The equations rnpreeenting the expected values o( the Y,* am

E(Yr) - V/"NE(Y,)

~ .'0.2.+ PINV~ X~i~ +.. + ~- - + (6-82)

whare z: - zd, j.

The nor ial equations for the estimiation of the O's ae

:4 1%. + XW Z4~10+.. + 1;W zwbah.A1 4 wwyW XZ.ZL + Z W 48+.. + 2WZIP X4~-Ls&-a - E2Z 1 (648)

Z1U1~+2 W io &.h- +.. +io-A- z W W z6.Y

U estimate sa of o in given by the usual formula

YO- - 6-

which may be written, in terms of the original wvriable, as

efl

Note that in the case where the value of ao is known, we may perform a test of significanee of

the closenesm of the observed estimate to the known value by forming the ratio F - and

comparing this value with the 100 (1 - ,) percentage point of the F d~stribution for n - k and ®

degrees of freedom; or, equivalently, we may compare x - (n -. k) 8 with the 100 (1 - a)

pereen"ae point af the x, distribution for it - k degrees of freedcm. Restatement of the foregoing.

using matrix notation, goes as follows:

If Var (Y) - Diag (@, o, .. )

then the trandcrmed variates

Y* - Dig (V- v...lv' .)Y -W Y andX. - Diag (v•i , ... vL .)X -WX

satisfy the requirements of the least squares theorem of Paragraph 6-2, aid the normal equations are

(X*)'(X*)- (X*)'Y* or,X' W' X0- X' W' Y.

6-20

POLYNOMIAL AND MULTIVARIAk[LE RELATIONSHIPS AMCP 7Wl6-!0

The estimate of c, is given byas (y ,

- L~. f k Ye - ' (X*)'Y*l or,

r' Wi rI-k

Example--Filtng SmlIght Une Relation (Variance of Y Prepo.'end to bsciss). Conside" theestimation of the coefficients of a line where

V a + AX,,

and whern Var(Yi) - a'zi t - 1, 2 .... ,. The eosations 4f ekpecrtation are

E(YI) - a + Az1E(Y,) .- a + fz, (6.6)

-C,. a + aX3

Transforming to Y.* - Y,/V',, gives

E(jr,*) - + #V',

E(Y.*) -~ + V'

and the normal equations for estimating u and • becomeS1_

z, (6-498)d Z1 + •n -- I Y. ,

Direct solution ox these equations gives

(649),- (Zx,) (z)

- Y, - Zz, Y.,/x, (6-40)

a'-(xr,) 1 (1 )and for the estimate of 0r,,,- ' 't

)(.1)

6-21

AMCP 796iO ANALYSIS -F MEASUREMENT DATA

6-7 CORRELATED MEASUREMENT ERRORS

£-7.1 DISCUSSION OF PFrOCEDURES AND EXAMPLES

If the errors of measurement are not independent but instead are correlated so that they hvvecovariances

Covar (Y4 Yj) - ý Vi, (6-42)

and variances

Var (Y0) Os,

then a triusformat-rn of the variables Y,, Y, ... Y., ýo new v&:Ables Y', )* -.. , Y, isrequired so that the method of least squares may be applied. In some simple cases, a transformationin the form of sums and differencea of the original variables immedietely suggests itself, and theupected values of the new variables are computed easily. The example usea to illustratc thetechniques presented in this Paragraph is such a case.

6.7.2 PROCEDURES AND EXAM!L.SPresdun--The var-aucen and covariances may be represented by thei X n variance-covariance

matrix

I. [ (s-42M)

Amsuming V to be of full rank, i.e., determinant of V is not zero, it is possible to factor V into theproduct

v - T P (643M)

whore T ii. lower triangular and T' ib the transpose of T. The requir.d transformation then isgiven by

Y0 -I y and X* - T-1 X (6-44M)

where (Y*)f - (Y, Y,*..., Y.*) is the vector of transformc d variables and Y' - (YI, Y., ..... Y.)is the vector of original variables. X* and X are the matrices representing the equations ofexpected v-.lues of the transfor-ned variables and of the original variables, respectively. (SeeParagaph 6-9 for the method of computing T and T-1).

The normal equations then are

(X*)'(X*) - (X*) Y* (645M)

or, in terms of the original variables,

X'V--X • - X'V-1 Y, (6-46M)

and the estimates of the A's are given by

S-[(X*)'(X*))-,(X*)'Y(7- (X'V-'X)-'X'V-'Y. (6-47M)

The variance estimate

6-22

I

POLYNOMIAL AND MULTiVARiABLE RELATIONSHIP5 AMCP -'PO6 U0

Prcdures (Car*)in Dn estimate Df unity when the variances and covariances are known. Tain may be written as

at ~ ~ ~ 1'XV-'Yj-V,-Y(648M)

"- r'V Ivj

where r is th, column vector of deviaticn, r - Y - Y7.

2', 'if, instead of V, a matrix with entries proportional to the variwnces und covariances is used, say

W - 1V, then a' is an estimate of .

E..,Amples-.orabolic keeationship with Cumulative Erors. If the errors of measurements of Y atsuccessive x w'Jues in a case of a ptlral I., law Y - 09 + DPx -. Pr amr cumulative, i.e.,

Y, + Dz-4 zi+ .

Y2 05 + Oat1 + &A4 + +i

Y. o+

chen E(Y) I- X 24

, Z: 01

If all the t's are from the awe diitribution,

then Var (Y,) - i.e'i , (y. , i <

Covar (Y, Y,)

and the variance covariance matrix becomes

1:

2 : c

Taking W - V, the necessary tre.formation is given by factoring W into W 2' T'.

6-2.3

AMCP ItY30-.110 ANALYSIS OF MEASUREMENT DATA

67.2 PROCEDURES AD EXAMPLES (CONT)E"x,,M (COW)

A little computation gives

= 1 T T'

S- (T')-0T1 - L 1 00-11 -1 0 0

1 1 0 0 -1 1

-1 0 0 . 0 11 0 0 0 -

2 -12 -1

L 1 2 -1

-1 2 -1

which, ,or the transformed variate, gives

1 YI2 Y2 -Y'

-1 0 Y0 Y, Y,

A I

W-7

POLYNOMIAL AND MULTIVARIABLE RELATIONSHIPS AMCP 706-110

xoampes (COMn)and

Jo 7-X 1

0 0 -1 1 1[ .

0 Lo 0•_ ,. _..F:~

Note that Y• - Yj - Y,_•- (zx - z,--)Po + (x" - 4') + e ,for i 2

and Y"' - Y';

hence, the Y *'s have the same variance, and have zero covariancee.

The normal equaticns become

X1 ef +4.~ i0 !+ ,(iX-)

X"e'+i - 4I 20 X +

I'~Y4ff.-, (Y• V '

AlYi + 2 (M - 4-) (y, - Y,_)

L

6-25


&7.2 PROCEDURES AND EXAMPLES (COMl)

6"I"eso (Co)

or, i terms o( the original matrices X'W-'Xf - X'W-,Y, give

1. 2 -1 1.. 2i 9• , , -1 2 -. 1 , z

1 2 --1

-1 2 -11 1

-Ii1

L

.1

which, upon multiplicatioat, will be eemn to give the same normal equations as above. If the analysisis carried out in tervas of the transformed variables, a, is estimated by

-_ -8-8

or equivalently, In tenns of the original variables, by

S' -i-- Y'W-IY - 0'X'W-1Yj.

C.-: USE OF ORT6%JGONAL POLYNOMIALS WITH EQUALLY SrAC;E o VALUES

"A.I DSCUSSION Of PROCIDURIS AND IXAMPULS

The fitting of a polynomial

Y " 0, + 10x + Am, + ...- + p,-,o-' (&49) 1to observations at * equally-spaced values of z (spaced a distance D apart) can be simplified bytranuforming the x's to new variables fb- i,..-,-• which are orthogonal to each other.

6-26

POLYNOMIAL AND MULTIVARIARLE RELATIONSHIPS A 'f& rl

The variables then becomee•- b.-1

ki- X16 where b -D

- s,, where C, - D ( j 12

)-X,t, where, - - (x• _ )(3n _) (6-k0)

- ', where Ed D-D-)a(851. 18) + 6 (W 1-,1 %+9)

& I x., where, . _ -( +- 4)15% 4 - 28" + 4 07\

k2 (w~ - k1)*Where fk+1 kik=& (4k!1 - 1)

The Xj are chosen so that the elements of , are inteters.

By fitting Y as a function

Y - aw0 + a,• +... + ah• 6-1, (6-51)

the estimation of the o's and the analysis of variance are simplified because the normal equationsare in diagonal form.

In order to obtain the estimates of the O's and their associated standard errors, or to m• a Equi'-tion (6-51) for predicting a value for a point not in the original data, an e..tra calculation but noinatrix inversion is requirte.

Tables of t', x, and X,(')' are given by Fisher and Yates(61 for n < 75, and by Anderson andHouseman(7 ) for n < 104 for up to 5th degree polynomials; in DeLury(8) for i R 2J for all powers;and in Pearson and Hartley('l for n <52 for up to 6th degree polynomials. Table 6-1 is a samplefrom Fishpr and Yat_(• )

To illistrate the calculations, consider the fitting of a cubic to the following (z, Y) pointa:

z Y10? .

20 11.780 87.240 80.150 151.460 2C3.270 392.6

6-27


TAIU 6.1. 1AMMU TAKI OF 011400UA1 POMVMW1S

4t

r . r. Vs I*r. i's~ ra V . 6 e I's I's I's el I's V9 I 's 6 ' a. I 's f 6 f'.

a-5 +5 -5 + - -8 +5 -1 +8 -1 r7 -7 +7 -12 +21 -5+5 - +A 1 0+ 1 + - +1 +5 - is +20

- +1 -1 1 - ,j _4 -8 -1 +7 -8 +5 - 0+ -7+4 -6 +7 -+ -17-1 +1 -0 - +8 , 01 +8 ++1 1 - -8- +9 -17• ,I.1 -1- 1 +1 _12 -0+ - -4 0 6 05

+ + +5 51 -++++1 0- - ++ +10 -5 -3 +9 +-15+- 41 +1 +S ÷ 1 +1 - 1 -- 4•I - -7 - .• aS 0 + 1 - 7 -1+ 1 + + 3 -- 3 - 11 - o + 1 7

+8 ++ +1 +8 +1 +5 +1 -5 -18 -U+7 +7 +'4 +7 +7'

3(' 6 16 1 4 1i 10 14 10 70 70 84 180 25 2 84 6 IM 84 168 168 964 621811 182 1 1~ 11 f29 * A , 2 19 #~

9 10 It TINa ' l i's I's I's Is I 's V. I's I'. I's I'm a I 's ̀ es I's I's I' 4.

0 -s 0 +18 0 +1 -4 -12 +18 +6 0 -10 0 +6 0 +1 -86 -7 +3 +30+1 -17 -9 +9 +0 +8 -2 -81 +8 +11 +1 -9 -14 +4 +4 +8 -OP -19 +12 +44+2 -8 -18 -11 +4 +b -1 -36 -17 +1 +2 -6 -3 -1 +t +5 -17 -25 --18 -t-29+- +7 -- 7. -21 -11 +7 +2 -14 -2 -14 4-8 -1 - 3-6 -1 - -7 +1+ -1 -8 -21+4 ++14 +14 +4 +9 +4 +4 +18 +6 +4 +6 -6 -6 -5 +9 +3 -8 -15 -57

+5 +15 -rlo +6 +8 +11 +56 += += +,33

3 1 ( O 990 4418 M 8,580 790 110 4,200 166 572 5,148 15,9123,7"3 2,00 its 2,8" 080 386 12,012 8,00

I 18 #11 2 # gA 1 2 8 3 '

8I I 14 Vi Ca '. a' . a '. a' . a'. a '. a' . a', a'. a'. a'. a , a'. a'. a'.

0 -14 0 +84 0 +1 -8 -24 +106 +60 0 -36 0 +766 0+1 -18 -4 +64 +20 +8 -7 -67 +6Z +-145 +1 -68 -27 +611 +d75+2 -10 -7 +11 +25 +5 -5 -95 -18 +189 +2 -44 -49 +P6.1 +1000+8 -5 -8 -54 +11 +7 -2 -96 -9 +28 +8 -29 -61 -249 +751+4 +2 -6 -96 -18 +9 +2 -66 -182 -1•2 +4 -8 -68 -"70t -"I+" +11 0 -so -82 +11 +7 +11 -77 -187 +5 +19 -85 -869 -979+6 +U +11 +99 +.122 +18 +18 +148 +lU +148 +6 +52 + 18 -429 -1144

+7 +91 +91 +1001 +1001

(f')" 12 5 4,1388 910 F;36 245,144 280 89,7?0 10,56 ,480,0028 4,r'a 728 Im.18 81,128 1,46,460

x ~ ~ ~ ~ A AT : 2• S t i ;. 4

'.

8-2t

V , P 'lM•

POLYNOMIAL AND MULTIVARIABLE RELATIONSHIPS AMCP706-110

TAU$ 4.1. SAWUt TAb.W OP 40HOWIM POIIISOMSA&S (C.9S~.4ITI

13 I? U0

+1 -21 -3 +182 +45 0 -2 0 +.6 0 +1 -40 -s +44 +gIm+3 -19 -179 -129 +11 +-I -2B -7 +$1 +6a +8 -87 v - +-- +38m+6 -15 -us +23 +181 +2 -90 -'$ +17 +0 +5 -31 -as +12 +"-m+7 -9 -301 -101 +77 +8 -15 -7 -8 +a- +7 -0 -41 -11 +m+9 -1 -97 -201 -as +4 -8 -18 -24 +6- +9 -10 -4, -36 +1÷6

+11 +9 -1" -22 -148 +5 +1 -15 -so -68 +11 +6 -OR -61 -AID+13 +21 -'91 -91 -in +6 +12 -7 -30 -104 '-18 += -18 -47 -VI1+1 +16 + U3 +V$3 +148 +7 +25 +7 -13 -91 +15 4-44 +-0 -1, -6741

+8 +40 +u -+a +104 +1., +0 +a- +6÷ +11m

I(')IMO 1,00•,71• B01,bU 4m1 8,8"/4 100,6 I''Ml 61141151"•t IM•,~5,712 470.8 7,7'2 14,796 U. 211

News In> al > I.oI k oj rjosuei mgi

No;*i In T"bl 6-., oumy Umv vinii tfr poi... 4 I,, weSim flyor 'l v Sý (%/I r for gad( 1/2

for. ito *U) mnu* bo %rlla by u'dnx the utyen rows In re-ye.* owder. vchanglus the -ism forr odi-nuubabee booe. 7 and

?~eA5h~gd. n*~. M~ U (1&.4spi t A.tb 1W F . YW%

From Table 6-1, for n - 7 we copy out:

y3 -3 5 -1 8.4

1 -2 0 1 11.71 "• -i -8 1 37.21 0 -4 0 80.11 1 -3 -1 151.41 2 0 -1 253.21 8 5 1 892.6

wheiv

C 1/

10/

.4m 6with :z-40, D- 10.

6-29

-~ -~an.M~

AMCP 706-&110 ANALYSIS OF MEASUREMENT DATA

"-.2 PROCaDMUS AND EXAMPLIS

S3p I ireodum•-Form the quantities

ZY

(6-52)

and, using the values of zj', Ze' I', ... , given in Tab'e 6-1 form the estimates of theparametwo, (' ab ... , as follows:

ihp2 Pvcedum--Calculate the deviation s r, from

r,- Yf4 - p - •,. - d•., - ... - d•-4.-,o4. (6.54)

It

6-80

- POLYNOMIAL ANDi MULTiVARiABLr RELATIONSHIPS A VCP 706-410

Uop I Exempl.-.Using the values copied from Table 6-1, the following calculaions are made:X•Y - 929.6 2-*,- 7

ztY - 1764.8 1'* - 28zMY - 1093.8 , - 84ZEýY - 33.5 11;2- 6.

IThe estimates of the coefficients in the replresentation of V as a function of the J, i.e., an

S- • ,•i+ a29• + a&6

are given by I

,- Y/Zo' 929.6/7 - 132.8 9

d - - 1764.8/28 - 63.0285 7143

- -Y/Zf, - 1093.8/84 - 13.0214 2857 1,�-� z~Y/~ - 33.5/6 5.58338 3333.

Step 2 Example-The predicted value for the point z - 10 is given by substituting its corre-sponfding values off the 4' -(r - 1, t - -$, Ej - 5, and - in the -etuntun

1. - 112.8 + 63.028 5714 &' + 13.021 4286 t, + 5.683 33 •,i.e., I0 - 132.8 + f3.028 6714(-8) + 13.021 4286(6) + 5.v433 =33(-l)

- 3.236 095

leading to a deviation between observed and :aculated of

io 3.4 - 3.238 0955- 0.161 9015.

For the entire set of purints, we get:

Observed Calculatd Residual

3.4 3.238 0955 0.161 9045 111.7 12.826 1905 -0.626 190687.2 36.290 4761 0.909 52S90.1 80.714 2856 -0.614 2856

151.4 151.180 9523 0.219 0477253.2 253.273 8095 -0.073 8095392.6 392.576 1905 0.023 8095

6-31

AMCP 706-110 ANALYSIS OF MEASUREMFNT DATA

Skep 3 Procedure-The estimate of a, is given by

x,- Y•IZ - 42 Y - -6z•Y - Y (6-55)

Slop 4 Procedure-The estimate of the standard deviations of the d's is given by

ad. (Q) - (6-46)

Step $ Precedu,--The Aftalj(si of Variatmce table becomes:

d.f. Sum of Squares

Total Xyt

Reduction due to fitting am d0 (I-Y) = R,Deviations from fit with ae. n- 1 Y -R 0

Reduction due to fitting a, &, (xf"Y) f R,Deviations from fit with ci, a n -2 ZY' - R - R,

Reauction due to fit of cx- 1 4,-i (Y ) -

Deviations from fit with 41,•MI..., - n- ZY -- R. - R •. - R&,,

PCfIIY.JfMIAI AMr AMIITIVARIAlRIF DPI;IATI(AeKIW1PC AaA' P 746-1tS

Stop 3 Example--The estimate of a! is given by

1 4r!

-3(1.676 9048)

.- 558 9683

- .7476.

Stop 4 Example-The standard deviations of the -.oefficient' are given by

a.d. (A,) - s/N/z'

e.d. (d.) - .7A76/V/7 - .2826

s.d. (Ad) - .7476/V'- .14- 3

s.d. (d2) - .7476/VA8 - .0816

s.d. (4) - .7476/VS - .3062.

stop 5 Exampei--The Analysis of Vaiance table becjmes:

d.f. Sum of Squares Meai 3qaare

Total 7 249 115.26Reduction due to coef. of • 1 123 450.88 123 450.88Residuals from 2.ý 6 125 664.38 20 P44. t6Reduction due to coef. of ' 1 111 232.822 86 111 232.82Residuals from dc + 6, 5 14 431.557 14 2 886.31Reduction due to coef. of • 1 14 242.838 57 14 242.84Residuals from d9, + &,' + 6,t; 4 188 718 57 47.18Reduction due tc coel. of ý 1 187.041 67 187.04Residuals frrmn " + + d,4 3 1.676 90 .5690

(-38


Sp 6 Procimdwre--Convert to an equation in the original x units b,' substituting the expreesiow

in Equations (6--0) into Eo latioln (6-51). By writing the Aqs as linear functions of the &'s. say

Ob bi di,

the standard deviation can be computed from

&~d. of b. - (&d. of di)

The foWowing Equations (6-57) show the O's as a function of the &'s for polynor.ials up to 5thdegree. (If a polyncmial of 4th degree is use•d, simply disregard the terms involv*ng 46; if 3rddegree, disregard the terms involving a, and t,; etc.)

As an example, if a 4th degree polyn~omial is fitted, the er.imate • is given by

and tbG Ld. of 4, in estimated by

O,/ + ( 414) (2 )V - (1)401 Z(E;)' + O D IW

So Equations ("4-67) on page 6-86.

3eU

POLYNOMIAL, AN[V MULTVARiABE 'ATiONSHS ARCrNA ? n l

Slop 6 Exampse-To oLAain the atioadn "n ulcrn Uf or.g,.-i•-i"! r .b;le. .. , ez. ma nn u

NO + Rix. + -,X2 + OWX', we s•stitute as follows:

4, o (!) + oi ( j N) + i. [( . 4-1. - ,] .. -ti( , -4 ' ._, (;,O)]0

. (ac. 4a 12x,. - 6.0) + ( -i + X +

Substituting the estimated values for the u's gives

Y = ;8.4428 5714 - .299 007 9375 x + .018 547 6191 z2 + .000 930 5556 z8.

The standard deviations of the 9's are given by

s.d. of &, s d. of (4 - 46, + 12,9, - 6Q,)

+ + 6

R V7

24 2.70

s&d. of +§ - 48)2 + + D

S.2190

d. of 4 (-2)

.006 158

Ld. of Os, - 6

- .0000 5097.

6-35

A UCP 7O6-11 I AbJAavcIC ^C &AC^CIIDEAACINJT IMATA

4

+-.

+

A JA+ 0

r-i Ai' A

+.

'm IC4

PAN MLP VPf PG I AqtfL1CLJIBC A "017Af' I I ~ A&~POLYNOMIAL "ND MULT;VARtAiA BLIC RUMAT I MON a , ',

6-9 MATRIX MiTHQDS

6.9.1 FORMULAS USING TRIANGULAR FACTORIZATION OF NORMAL 6QUATlONS

The rmatrix for the left-hand side of normal equations can be factored into (X'X) - TT' whereT is lower triangular, so that (X'X)-1 - (T')-l T' - (T-')' T-1.

Thus, • - (T-')' (T-Q) where Q - A'Y.

Denote th3 column vector T-'Q byg ,-T-•- gi].

Therefore, • (T-')'g.

This representation leads t n certain simplifications, e.g.:

(a) Thke estimate of or' L4 given by

1 (y,y _ Q)

1S(Y'Y "" -'T--Q)nk

n - (Y'Y - g)

(b) Th., variance of a linear fuact'on, L oa'A of the 4's i riven by/2 al' (T-')' I' a•l s (T-'a)' (T•r')

a ' 24~

when h - - 2''a.

(c) The reduction in rum of squares due to fitting the last p constants is

6-37


This fornmlation als permits us to make a detailed Akn,2sis of Variance table. An importantcaution is in order. The reduction due to the addition o, •, is the reduction given Owt , # .... -,have been fittd go the data. The reduction due zo given that any other set of coefficients havebeen fitted will be different.

The Ana"pas of VavIa'ne tab!e becomes:

d.f. Sum of Squares

Re d,• ,.tion due to fitting el 1

Residual (after fittin; A0) t - 1 zy'-Additionai reduction fitting 1 goReduction due to flt•.ing 01 and 9, 2 9 +91Resid.ual (after fitting 0. and 00) a -2 ZY3 - g -

Additional reduction due to 1Otting 192 1

Reduction due to fittir.g 1, B,...,. k .

Residual (after fitting n, #§.... ) - k ZY2 -

This form of analysis is especiplly useful ;n the analysis f3r polyvnomials where the ordering is bypowers of x. In tbi multiple regression case, the reduction attributed to ' is dependent upon theordering of the parameters #, #, ... , #,-, and will be different for different orders.

6.9.2 TRIANGULARIZATION OF MATRICES

The real symmetric matrix

N -, G .s . . . ,a

as, as a*:

can, if N is non-singular (i.e., if IN! 0), be factored into the product of two triangular matricesso that N - TT', i.e.,

[an C1 ]. . L a,. ci t I F ll .. Jall~ an .. oas., Lc,, %;Is 4.3, C4J

POLYNOMIAL AND MUWiIVARIA, LE RELAT.ONSHIPS AMCP 706W-10

Tle elements cj awe computed from the following (note that ej/- 0 fve j > 0:

Cs1 - Giihi/,

C. -- elc. - (an - cS, ell)/cf

ý. - (a.2 - C•, C ,-1) /C.

ii-V ai-Ci j-

As an example, consider

N [4 6 8 1

20 86 380

Applying the formulas for ci, we get

ell -6/2 -3C#.,- 8/2 - 4c41 - 10/2-5

cAS M "V25 - (3)' - 4

-~i120 -4(3)]/4 -- 2

S- 127 - 5(2)]/4 - 4€.=•/3 -2-43 -4

toy - - - 1

This Civcs

N t ][22 8 424 2 4 4 1

15 3 1 1- -

6-89

AJLCP 706-110 ANALYSIS OF MEASUREMENT DATA

The inverse of a triangular matrix

Cal Cas ]

;a given by

1'-'

7-X bitbig

Lb., .i".where::n : (C.1 b c3b +C*b. - - (bit c-,)!/c

bu - - (ou b. + ci b.t)/cis

i.l (c., bit + cs bat +-. + cIA-1 bn,-I,)/€..11

b. - C-a bu/ca,bo - (cabn +c•b)/lCMA

b.j - (c, b,, + c.a b, + ... + c.... 6.-.1.2)/c.

b~-1

bi - (cij bi, + ci.j.+i bj, + . + b 1.)cI

Examrple:

For T F213 4

4 2 4L5 3 1 1_1.

6-40

POLYNIOMIAL AND MIILTIVARIABLE RELATIONSHIPS AMCP 706-110

The elaments of T- are-

bu-- [4 (-) +•-(-)]/A +1 -

2 1

1v

b- --

[r,) G')61

b1 - -21 -2 4

be- 10 -4 16

and N-I -, (TT")-' - (T'-') (T-L) givesj

,,,1 [8 -6 -5 -i7]r 8 t- g _ F_ 414 1•f 48 -27]b- --5 -2 4 4- 2 32 -64

L Ts-7 -10 - 6 L-272 -160 -64 ,

By forming the matrix p_"oduct

[x'.] (X.,,) . FX'X X',YLi" L Y'x Yi'Y]

6-41

AMCP 706110 ANALYSIS OF MEASUREMENT DATA

and replaci is Y'X by 0 (a nu3l ratrix) &and Y'Y by I (the identity matnix), we obtain

L 0 'rY].

In this form, Y may be r single vector of observat;ona Y' - (Y 1Y,... Y.), or a met of p vectots

1 Y.,j.

Then,

whr I inp X p and 0 in p x k, givea all the values needd for the computations of thi aParagraph.

1. M. G. Kmndall and W. R, Buckland, A Dic- 6. R. A. Fisher and F. Yates, Statisticcd Tabl&,tiarw" of Staistecal Terms, Oliver and for Biclogice, Agrtdtural and MedicalBoyd, Ltd., Ediniburgh, 1954. Re.5arch (5th edition), Oliver and Boyd,

&. R. L. Anderson and T. A. Bancroft, Sta- Ltd., Edinburgh, 1957.teicW T/w in Reaarck, McGraw-HillBonk ('V'pany, Inc., New York, N.Y., 7. T. L. Anderson and E. E. Houseman, T -bin195?. qo Orthgonal Polynomial Values Ex-

3. 0. L Davies, (ed.), Tiv esign snd Aalysi.s tede to N = 104, Research Bulletinof ladustrial Eirifmla. Oliver and _, 9Q7 __iiiro P. .o-a Qto_

Boyd, Ltd., Edinburgn, 1954. tion, Iowa State College. Ames, Iowa,4. 0. Tausaky, (ed.), Contributions to th Sow- 1942.

tioe of 4steN of Li.,ae Equa ., andthe Dtnwrination of Eigenalus, (Na- 8. D. B. DeLury, Valtws and Tntegrals of thetional Bureau o Staidards Applied Orihogonal Polynomials up to n = £6,Mathematics Se.nes, No. 39), U. S. Gov- University of Toronto Press, 195Z,ernment ]tnting Office, Washington,D. C., 1954. 9. E. S. Pearson and H. 0. Hartley, (eds.),

5. P. S. Dwyev', Linear Computations, John Biometr-ýka Tables for Staisticias, Vol. IWiley & Sona, Inc., New York, N.Y., (2d edition), Cambridge University Press,1961 Cambridge, England, 1958.

6-42

AMCP 706-Iq

(AMCRD-TV)

FOR THE C4MVANuER:

OFFICIoAL: LEO B. JONESMajor General, USAChief of Staff

? .R. HORNE

Chief, Administtative Office

DISTRIBUTIOtI.Spe,,ial

ENGIN"EERING DESIGN HAkNDIBOOK.SLisf#)d below are the Mattdbooks v~hich have been published or are currently vinier preparation. Handbooks with ,ublica-

tion datia orlor t. I APagust 19652 were published as 20-seies Ordnance Corpi pemliilets. AMC Circular 310-38. 19 Jtvly1963. redesigrAstad these pulilications as 706-series AK pealo~ile's (e.g., DOPI 20-138 was redesignated A)W 706-v38l).All new. reprinted, or revisal H~andbooks art being published i.% 106-series AMC. pamphlets.

No. 1t00 NO. Title

1 00 *D'esign Guiaeoce for Producibility 202 'Rotorcraft Engineerog. Part Two. Delail104 *Value Engineering Design106 Elements of Armament Engineering, Part One, 203 'R1okorcraft Engineering. Part Three, Qualifi-

Sources of Energy cation Assurancelot E*.ments of Armmnt Lnginieering, Part Two, 2035 *Timing Systems and Cowuenent%

ballistics 210 Fuzes1oe Elemer~ts of Armamnt. Engineering, Part Th~ie'. 211(f) Fuzes. Proximity. Electrical. Part One 'U)

Weapon Systems and Componects 2121S) Fuzes. Proximity. Electrical, P art Tw3 (U)110 Experimental Statistics, Section 1. Basic Con- 213(S, Fuzes, PrOAimity, Electrical. Part Three (U)

cepts and Analysis of lMeasuremen~t Lace 214(S) Fuzes, Proximity, Electrical. Part Fcur (U)Ill Experimental Statfitics. Section 2. Analysis of 215 ,C8 Fuzes, 7ro.ii.ty, Electrital. Part Five (U)

Enviuerativ. and Classificaipry Data 233 'Hardening Weaponn Systems Against RF Enact,,112 Experimental Statistics, Section 3. Flarning rf(s) 'S~r 'I, Arms Aimanition (U)

and Analysis of Comparative Experiments 240(C) Grenades (U)11i Experimental Statistics. Section 4, Speceal 241(S) 'Land Mines (U)

Topics 242 Des Ign for Control of Projectile Flight114 Experimental Statistics, Section L. TAbles Characteristicr115 Basic Environma*.tal Concepts 244 Ainanitlon, Section 1, Artillery Apmunition--116 '8esic Envirommental Factors General, with Table of Contents, G'.ossary120 *Design Criteria for Environmental Control of and Inde.x for Series

Mobile Systeme 24!(C) Ammunitiamn, Section 2, 5e~ic'i for Teraina]121 Peckaglng avnd Pack Engineering Effects (U)123 'Hydraulic Fluisds 246 t~emunftion, Section 3, Des gn for Control of125 Electrical Wire and Cable Flight Characiteristics127 *Infrared Military Systems, Part One 2'vl Auniitior, Section 4, Design for Projection128(s) 'Infrared MilltarV System . Part Two (U) 248 tAimmnition, Section S. Inspection Aspects of130 Design for Air Transport and Airdrop of Artillery A.meinitic's Design

Vateriel 249 Ammnition, Section 6, Marufeitutre of Metallic134 Okintainabilit; Guide for Design Cvompoents of Artillery Amaunition135 Inventions, Patents. andi Related Matters 260 Guns--feneral136 Servomecheni~ee, Sec:tion 1, The-ory 251 Muzzle Devices137 Servomechanismis, Section?', Measurement and 2i52 Gun Tubes

,ignea Conv~erters 255 Spectral Characte, istics of Muizzle FlashIIA Se v.rmxch&iisms. Section 3. Ampl fication 260 'Automatic Weapons13; evezia9~ S-cti.vi 4, Power E~erne'.ts 2?t. PMonli@-It Artuatod lUevlc-S

and System Des go 280 Design of Akereit-ami" --iv Stabilized Free140 11rajactorios. Differential Efferts. and Data Rockets

145for Projectilie. G 1 yte281'S-RD) Weapon Syitem Effectiveness (U)150 Interior Sallis~is a un 283 Aerodynamics160()() Elements of Teraina' Callistics, Part One, 284(c) Trajectories (U)

Kill Mechanism$ anJ Vulnerability (U) 285 Elements if Airmraft and Missile Propulsion161(5) Elements of Terminal Ballistics, Pert Two, 286 Str-uctur,

Collection and Analysis of Data ConcernIing ?90(C) Warheads--Genersl (U)Targets (U) 291 Surftce-t3-Air Missiles, Part One, System

162(S-RI)) Eloenot, 01 Terminal Ballistics, Part Three, IntegratiunApplicativ. to Missile a" Space Targets (U) 292 Surface-bo-Air Missile,, Part Two, Weaion

165 Liquid-Fillpd Projectile Design Control170(C) Armor and lac Application t' Vehicles (U) 292 Surface-to-Alt Missiles, Part Three, Computers115 Soiij Prolpeiiants, Part Cme 294ts) Sufc-t-i !ts Ils.P -tFcr M4~is..176(c) Solid Propellents . Pqrt Iwo (U) Armament ýU177 Properties of Explosives of Military Intcrest Ps(!) Surface-to-Ai- Missiles, Part Five, Counter-178(C) tPrcpertiis un Exp'.osives of Military Interosv. measures (U)

Section 2 (U) 196 Swrface-to-Air Missiles, Part Six, Structures179 Explosive Trains and Power SourcesISO 'Principles (), Explosive Bv~eimvi, 297(S), urface-to-Air Missiles, P'art Seveii, SampleIts Military Pyrotechn~cs, Part One, Theory anid

0rotlem (U)

Appliration 321 Fire Control Systems--General186 Military Pyrotechnics, Part Two, Safety, '129 hFire Control Connotiug SysLems

Procedures and Glossary 331 Compensating Elements181 Military Pyrotechnics, Part Thrj-e, Properties 33t'4540) *Nuclear Effects on Weapon4 Syitems (U)

of Materials Used in Pyrotechnic Compositions 340 Carriages a.id Maunts--Geriprak188 *Military Pyrotechnics, Part Four, Design of 341 Cradles

Ammnition for Pyrotechnic Eff-cts 342 Rec,:i I Systems189 Military IPyrotecl-nics, Pars. Five, Dibliography 343 Trip Carriages190 'Aray Weapon System Analysis 344 bottom Carriages195 0tevelopment Go Ide ~or RtAliability, Part Use 345 Equilibrators196 *Daveloprvnt Cuide for RelI Iab ilI ty , Part Two 346 ..ievating 4echanisms19? *Development Guide for Reliability, Part Three 347 Traversing Mechanisms196 'Developeent Guide for Reliability. Part Four 350 *Wlse,-Id Aenhibians199 *Devolopmernt Guide for Reliability. Part FIvF 355 The Automotive Assembly200 'Development CGnIde for Reliability, Part 5i x 3!6 Autoaxtive Suspensions101 hNotorcraft Engineering, Paet One, Prelimi- 3S1 'Autoexntive Bodies avid dulls

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