Disclosure to Promote the Right To Information Whereas the Parliament of India has set out to provide a practical regime of right to information for citizens to secure access to information under the control of public authorities, in order to promote transparency and accountability in the working of every public authority, and whereas the attached publication of the Bureau of Indian Standards is of particular interest to the public, particularly disadvantaged communities and those engaged in the pursuit of education and knowledge, the attached public safety standard is made available to promote the timely dissemination of this information in an accurate manner to the public. इंटरनेट मानक “!ान $ एक न’ भारत का +नम-ण” Satyanarayan Gangaram Pitroda “Invent a New India Using Knowledge” “प0रा1 को छोड न’ 5 तरफ” Jawaharlal Nehru “Step Out From the Old to the New” “जान1 का अ+धकार, जी1 का अ+धकार” Mazdoor Kisan Shakti Sangathan “The Right to Information, The Right to Live” “!ान एक ऐसा खजाना > जो कभी च0राया नहB जा सकता ह ै” Bhartṛhari—Nītiśatakam “Knowledge is such a treasure which cannot be stolen” IS 397-3 (2003): Method for Statistical Quality Control During Production, Part 3: Special Control Charts by Variables [MSD 3: Statistical Methods for Quality and Reliability]
Transcript
Disclosure to Promote the Right To Information
Whereas the Parliament of India has set out to provide a practical regime of right to information for citizens to secure access to information under the control of public authorities, in order to promote transparency and accountability in the working of every public authority, and whereas the attached publication of the Bureau of Indian Standards is of particular interest to the public, particularly disadvantaged communities and those engaged in the pursuit of education and knowledge, the attached public safety standard is made available to promote the timely dissemination of this information in an accurate manner to the public.
इंटरनेट मानक
“!ान $ एक न' भारत का +नम-ण”Satyanarayan Gangaram Pitroda
“Invent a New India Using Knowledge”
“प0रा1 को छोड न' 5 तरफ”Jawaharlal Nehru
“Step Out From the Old to the New”
“जान1 का अ+धकार, जी1 का अ+धकार”Mazdoor Kisan Shakti Sangathan
“The Right to Information, The Right to Live”
“!ान एक ऐसा खजाना > जो कभी च0राया नहB जा सकता है”Bhartṛhari—Nītiśatakam
“Knowledge is such a treasure which cannot be stolen”
“Invent a New India Using Knowledge”
है”ह”ह
IS 397-3 (2003): Method for Statistical Quality ControlDuring Production, Part 3: Special Control Charts byVariables [MSD 3: Statistical Methods for Quality andReliability]
IS 397 (Part 3) :2003
mi+iwmma?FvJmlFT4-mT$mJl-Tad7imrafa?mf%5mwFdi
( WFi’7 $p%wl)
Indian Standard
METHODS FOR STATISTICAL QUALITYCONTROL DURING PRODUCTIONPART 3 SPECIAL CONTROL CHARTS BY VARIABLES
(First Revision)
ICS 03.120.30
0 BIS 2003
BUREAU OF INDIAN STANDARDSMANAK BHAVAN, 9 BAHADUR SHAH ZAFAR MARG
NEW DELHI 110002
Al(~USf2003 Price Group 7
Statistical Methods for Quality and Reliability Sectional Committee, MSD 3
FOREWORD
This Indian Standard (Part 3) (First Revision) was adopted by the Bureau of Indian Standards, after the draftfinal ized by Statistical Methods for Quality and Reliability Sectional Committee had been approved by theManagement and Systems Division Council.
The efficacy of control charts in regulating production is quite well known. Part 1 of this standard covers traditionalcontrol charts for variables. This Part of the standard dealing with special control charts by variables has beenprepared for use in those circumstances wherein the traditional control charts are not applicable, less efficient ormore time consuming.
Since the basic philosophy for the use of control charts in manufacturing operations remains unaltered irrespectiveof the type of chart used, this Part should be read along with Part 1 for obtaining an integrated approach to thetheory and practice of control charts. Part 1 of the standard is therefore necessary adjunct to this standard sincemany of the basic principles in the construction of control charts and their interpretation explained in Part 1 havenot been repeated.
This standard was originally published in 1980. In view of the experience gained with the use of this standard incourse of years, it was felt necessary to revise this standard so as to make the concepts more up-to-date. Followingchanges have been made in this revision:
a) Many editorial mistakes have been corrected, and
b) At certain places the concepts have been made clearer.
In addition to this Part, IS 397 has the following four parts:
L’SNo, Title
397 Methods for statistical quality control during production:(Part O) :2003 Guidelines for selection of control charts (first revision)
(Part 1): 2003 Control charts for variables (second revision)(Part 2): 2003 Control charts for attributes (third revision)
(Part 4) :2003 Special control charts by attributes (/hf revision)
-r
The composition of the Committee responsible for the formulation of this standard is given in Annex B.
IS 397 (Part 3) :2003
Indian Standard
METHODS FOR STATISTICAL QUALITYCONTROL DURING PRODUCTIONPART 3 SPECIAL CONTROL CHARTS BY VARIABLES
(First Revision)1 SCOPE
This standard (Part 3) describes the following controlcharts with examples:
a)
b)
c)
d)
e)
0
Group control charts,
Sloping control charts,
Moving averages and moving range charts,
Control charts for extreme values,
Control charts for coefficient of variation, and
Cumulative sum control chart.
2 REFERENCES
The following standards contain provisions, whichthrough reference in this text constitute provisions ofthis standard. At the time of publication, the editionsindicated were valid. All standards are subject torevision and parties to agreements based on thisstandard are encouraged to investigate the possibilityof applying the most recent editions of the standardsindicated below:
IS No.
397 (Part 1) :2003
7920(Part 1):
(Part 2):
994
994
Title
Methods for statistical quality control
during production: Part 1 Controlcharts for variables (.secomiwvision)
Statistical vocabulary and symbols:Probability and general statisticalterms (second revision)
Statistical quality control (secondrevision)
3 TERMINOLOGY
For the purpose of this standard the definitions givenin 1S 7920 (Part 1) and IS 7920 (Part 2) shall apply.
4 GROUP CONTROL CHARTS
4.1 In industrial production it frequently so happensthat the data presented for the purpose of controllingthe quality comes from a number of sources, say frommulti-spindle machine with same standard output, orseveral workers or several machines. In such casesunless proper steps are taken in choosing the sample,the quality engineer is hard put to single out the trouble-yielding source when control chart shows lack ofcontrol. One obvious way is to maintain a separate chart
for each possible source, which is rather uneconomicaland time-consuming. Group control chart, first devisedwith a view to controlling the dimensions on multiple-
spindle automatics, which has wide applicability is achallenging answer to the problem.
4.2 The group control charts are valid only when there
are enough reasons to presume that the means of each
source of data as also the variability of each source areuniform. Instead of maintaining a pair of mean and
range charts for each possible source like machine or
worker, only one pair of mean and range charts are
maintained. In the mean chart, the highest and lowest
average values are plotted along with suitable sourceidentifying indications (such as, serial number of
machines/workers) and the largest range is plotted on
the range chart. In the mean chart, the highest values
are connected by a line, so also, the lowest values inorder to avoid confusion. The underlying idea is that,
if corresponding to a particular sample the highest
value is below the upper control limit (UCL), the others
are necessarily so and similarly if the lowest value is
above the lower control limit (LCL), others are
necessarily so. The identifying number attached to
highest value that is beyond the UCL or to a lowest
value that is below the LCL at once detects the trouble-yielding source.
4.3 For drawing group control chart for means, the
various lines are obtained as follows:
CL = ~ =X Fi/k
UCL = ~ +Az~, where ~=~~/kj=,
LCL = ~-Az ~
where k is the total number of sources and X, and Ri
are the gverage and range corresponding to the ithsource. X and ~ are the averages of the homo-genizedaverages and ranges respectively.
Similarly for group control chart for range,
CL ‘F
UCL = D, ~
LCL = D, E
The values of the factors A2, D3, Dq are given inAnnex A for various sample sizes [see IS 397 (Partl)].
1
1S 397 (Part 3) :2003
4.4 An advantageous use of group control chartsrequires that there should be several subgroup sourcesthat yield approximately equal number of sub-groups
at approximately the same rate, such as, differentspindles on one automatic machine, several identicalmachines, or several operators each doing the same
operation. There should be no difference between theaverages or dispersions of various subgroups whichcannot be corrected. For example, if there are tenmachines on the same job but two of the machineshave better process capability, then the group controlchart cannot be applied for all the ten machines. Thetwo machines, which give better process capability,
should be treated separately.
4.4.1 The advantages of using a group control chartare:
a)
b)
c)
It involves less work in plotting;
A compact presentation of all information ona single chart makes the interpretation easier;and
It is easier to find out whether a particularsource is giving consistently big-h or lowvalues on average or range chart. If there isno real difference among the sources, thenumbers corresponding to the various sourcesshould occur on the charts almost equally inthe long run.
4.5 Example
Table 1 gives two measurements of the diameters oftwo pieces produced on each of six spindles of anautomatic screw machine. The values given are in units
of 0.001 mm in excess of 12 mm.
4.5.1 The highest and lowest average values areindicated in co] 6 of Table 1 as H and L respectively.The highest ranges are also indicated by H in CO17 of
Table 1.
4.5.2 Control Lirnitsfor Range Chart
CL = ~ = 35/36= 0.97
UCL = D, ~ =3.267 x 0.97=3.17
LCL = D,~=O
Since all range values are less than Dq ~, the rangesare homogeneous.
4.5.3 Control Limits for Average Chart
CL = ~ = 195.5/36= 5.43
UCL = ~ +A2 ~ =5.43+ 1.88 x 0.97=7.25
LCL = F-A, ~ =3.61
4.5.4 The average chart and range chart are plotted inFig. 1. In Fig. 1, the average chart, the highest and
lowest average values along with suitable source
.
identifying indications (spindle number) are plotted.Similarly in the range chart, the largest range alongwith suitable source identifying indications of spindlenumber is plotted.
5 SLOPING CONTROL CHARTS
5.1 In certain industries, the process level changessystematically during the course of production. Thetypical examples are that of tool wear in machine shop,
changes due to fall in pressure as a tank empties, theeffect of slowing down in reaction as a batch becomesweaker, etc. In the machine shop, it is desirable tosharpen the tools before the production of too manynon-conforming items. On the other hand, it may not
be advisable to unduly interrupt production forreplacing or resharpening the tools. The problem is to
economically minimize the total of the two costs – thecost of producing non-conforming products and thecost of replacing or resharpening tools.
5.2 In such cases ordinary ~ – R charts are not usefulto control the process because the variation in theprocess is not only due to chance causes alone but alsodue to assignable causes such as tool wear. In thissituation sloping control chart is found to be useful.
5.3 In sloping control chart samples are collected insuch a manner that the:
a) production between iwo successive samplesmust be more or less constant; and
b) individual items of a sample must beconsecutive items from the production so thatthe trend will have minimum effect on therange of the sample.
5.4 The average of the kth sample ( ~~ ) may be
expressed as ~~ = a + bk where a and b are constants.To set the control charts, the ranges are homogenized
in the beginning and let ~ be the average ofhomogenized ranges.
The central line of the average chart is a + bk where:
where
b=:,(~, -~)(k-~)/~(k-~)2,
z is the grand average.
=6~2~, (k-~) /n(n+l)(n–l), andk=l
a.~–b ~
The control limits are:
For Average Chart:
CL = a+bk
UCL = a+ bk+A2~
LCL = a+ bk-A2~
L
IS 397 (Part 3) :2003
Table 1 Diameter Measurements (Microns in Excess of 12 mm)
(Clause 4.5)
SI No. Sample No. Spindle No. Diameter Average Range Remarks
Piece 1 Piece 2 x R
(1) (2) (3) (4) (5) (6) (7) (8)
1 6 7 6.5 H I
2 4 6 5.0 2H1} I 3 6 4 5.0 2H
4 5 4 4.5 L 1
5 6 5 5.5 16 4 5 4.5 L 11 6 6 6.0 H o2 6 6 6.0 H o
Ii) 2 3 5 6 5.5 14 5 5 5.0 L o5 5 6 5.5 16 7 5 6.0 H 2H1 5 6 5.5 lH2 6 6 6.0 H o
iii) 3 3 5 5 5.0 L o4 6 5 5.5 lH5 5 5 5.0 L o6 6 6 6.0 H o1 5 6 5.5 12 6 5 5.5 1—
Iv) 4 3 5 5 5.0 04 4 4 4.0 L o5 6 4 5,0 2H6 5 7 6.0 H 2H1 5 6 5.5 12 5 4 4.5 L 1
UCL = D, i? All the range values are less than Dd R. Hence they are
LCL = D, ~ all homogeneous.
where the values of A1, D3 and Dd are given in Annex A
()
20
for various sample sizes.—
? = ~xk /20=39.504/20= 1.9752
5.5 Example F = (n+l)/2=10.5
Suppose there are 20 (equal ton ) samples of ‘Starter’machined with new tools, collected at regular ~2~, (k-~) =1.896
,.—!(approximately) intervals of production and recorded ‘=’
b=6x~2~k (k–~)/n(n+l)(n–1)=in order of production. Each sample is of size 5. Table 2
k=l
gives the average and range of ‘Head thickness’ of the20 samplesinCO13 and 4 respectively. The control limits
(6X 1.896)/(20 x 21 X 19)= 0.0014
for the sloping control chart are calculated as follows: a=~–b~= l.975– 0.0014X 10.5 =1.9605
3
IS 397 (Part 3) :2003
12.0081
12.007
12.006
a)~ 12,005
212.004
(1) (5) (4)
(2,6)A (g~
(4,6)
CL
12.003-.,
12.002-
1 2 3 4 5 6
0.004
1UCL
0.003-%)
*
‘0”002- ‘2’%!40.001- . CL
0,000-
1 2 3 4 5 6
Subgroup Number
FIG. 1 GROUPCONTROLCHART
A,~ = 0.577 x 0.0165 ‘0.009 5
5.5.1 For Average Chart:
CL = 1.9605 +0.001 4k
UCL = 1.9700+0.001 4k
LCL = 1.9510 -O.0014k
For Range Chart:
CL = 0.0165
UCL = 0.035
LCL = D3~ =0
To use the control chart during production, the sample
shall be drawn at equal intervals of production and ~~and R~ will be plotted in the chart against the corres-ponding k of the X axis. The criteria for out of controlsituation are the same as that in ordinary ~ – R chart.
5.5.2 The data given in Table 2 have been plotted onsloping control chart (for average) and range chart inFig. 2.
6 MOVING AVERAGES AND MOVING RANGECHARTS
6.1 In certain cases of industrial production it takes
considerable time to produce a new item, so it is
inconvenient to sample frequently. But in the mean
time process average or dispersion may shift and this
may incur some appreciable loss to the producer.
6.1.1 To obviate this, use of moving averages and
moving ranges instead of ordinary averages and
ranges has been suggested. Moving averages of k
items are obtained as follows. Initially, the first k
values are averaged. Then in the second step the first
value is dropped in favour of the (k+ 1)th value and
an average obtained. Next, the second value is
dropped and the (k+2)th value included and these
values are averaged, and so on. In a similar manner
moving ranges are obtained.
4
IS 397 (Part 3) : 2Q03
Table 2 Computations for the S1oping Control Chart
(Clauses 5.5 and 5.5.2)
S1No. Sample Average Range 2 (k-k) 2(k-Oxk RemarksNo. ‘k Rk
6.2 Suppose there are 150bservations. In the table, 150 bservations, the number of 5 items movingwhich follows the 5-item moving averages and moving averages and ranges would be 15 + 1 – 5 = 11.ranges, are calculated. It may be noted that for
S1No. Observation Moving Average Moving Range Remarks
(1) (2) (3) (4) (5)
1
2
3
4
5,
6
7
8
9
10
11
12
13
14
15
8
7
8
9
8
5
6
7
5
9
9
8
8
7
6
— —
— —
(8+7+8+9+8)/5 = 8.0
(7+8+9+8+5)/5 = 7.4
(8+9+8+5+6)/5 = 7.2
(9+8+5+6+7)/5 = 7.0
(8+5+6+7+5)/5 = 6.2
(5+6+7+5+9)/5 = 6.4
(6+7+5+9+9)/5 =7.2
(7+5+9+9+8)/5 = 7.6
(5+9+9+8+8)/5 = 7.8
(9+9+8+8+7)/5 = 8.2
(9+8+8+7+6)/5 = 7.6
9-7=2
9–5=4
9-5=4
9-5=4
8-5=3
9-5=4
9-5=4
9-5=4
9-5=4
9-7=2
9-6=3
—
—
—
—
—
—
—
—
—
—
6.3 Control charts are set up in exactly the same manneras in the case of ordinary averages and ranges. Usuallymoving average and moving range control charts areemployed for_the purpose of current control. Theoverall mean X and the average range ‘~ are obtainedfrom supplied past data and these are utilized to setupcontrol limits as follows:
Moving Range Chart
CL = ~
UCL = D, ~
LCL = D3~
Moving Average Chart
CL = ~
UCL = ~+Az~
LCL = p -Az~
where A ~, D, and D~ have been tabulated in Annex Afor various sample sizes.
6.4 When the control charts have been set up, the operationproceeds with the plotting of moving average and movingranges in chronological order. In a k item moving average(k -1) values are common between any two successive
avemges (or ranges), (k – 2) values are common betweenany two alternate values and so on. So unlike the case ofordinary control charts, here successive points are notindependent. In fhc~ in a k-item moving average seriesonly values (average of range) which are (k + 1)intervalsapart are independent. Hence, in moving control charts,runs on either side of the central line, do not tell the samestory as does an ordinary control chart. However, a pointout of control limits here has the same significance as inan ordinary chart.
6.5 The advantage of a control chart for moving averageor moving range is that it gives a warning signal earlier.It is not necessary to wait until an entire new sample isaccumulated. This may be important if the product iseither expensive or the rate of output is small.
6.5.1 Since the probability of obtaining a run of anykind is much larger with control chart for movingaverage ormoving range as compared to the orthodoxtype of charts, the traditional interpretation of runs isnot valid for these type of charts.
7 CONTROL CHARTS FOR EXTREME VALUES
7.1 Itmay at times be desirable to use a control chartfor largest and smallest values or the high-low controlchart, as it is popularly called, in place of the
6
IS 397 (Part 3) :2003
conventional charts for the averages and ranges. Thesetype of charts are extremely simple since nocalculations are needed for plotting the points. Besides,only one chart needs to be maintained in this case inplace of the two conventional charts since informationconcerning both central tendency and dispersion areprovided on the single chart.
7.2 Constants for Determining Limits
Let L and S denote the largest and smallest valuesrespectively in a sample of n pieces, and let ~ and j bethe average of these values fork samples.Then ( ~ + ~)/2 and ( ~ – ~ )/d2 and are unbiased estimates ofpopulation mean and standard deviation respectively, inthe case of a random sample tl-oma normal population.
7.3 Mean and Standard Deviation not Known
When the values of the process average and dispersionare not known from the past data, they are estimatedwith the help of the initial data collected and the centrallines and control limits are computed as follows:
CL = (z+~)/2=M
UCL = M+Hz~
LCL = M–H, E
Where the values of Hz are given in Annex A forsample sizes 2 to 6 and ~ = ~_ ~
NOTE — Since the efficiency of M decreases rapidly with theincreasing_sample size, it may be better to substitute grsndaverage X for M in computing both the central Iine and thecontrol Iimits.
7.4 Mean and Standard Deviation Known
When control charts for extreme values are used tocontrol a current process, and if the values of processaverage and dispersion are known as P and orespectively, then to plot both the extremes values ona single chart, the control limits and the central lineare set up as:
CL ‘PUCL = p+Ho
LCL = p–Her
where the value of H for sample sizes 2 to 6 are asgiven in Annex A.
7.5 [t has been found that undermost condition, L andS chart is nearly as good as the Z and R charts fordetecting lack of control. Like in the case of individualchart, the specification limits may validly be drawn inthe case of a control chart for extreme values.
7.6 Although an upper control limit for L and a lowercontrol limit for S are usually drawn, it is possible tohave an upper and a lower control limit for epch L and S
separately. The control limits for L and S are given by~ + (Hz – 1/2) E and j * (Hz – 1/2) R respectively.In such a case, a shift in the process; is indicated if bothL and S are above the respective upper control limits orbelow the lower control limits. On the other hand, if Lis above the relevant upper control limit and S is belowthe relevant lower control limit, this is enough evidenceto conclude the increase in process variability.
7.7 Example
7.7.1 Suppose there are 25(n) samples of ‘bolts’machined on Turret lathe, collected at regular intervalsof production and recorded in order of production. Eachsample is of size 5. Table 3 gives the largest (L) andsmallest (S) values of head diameter of 25 samples inCOI3 and 4 respectively. The control limits for theextreme value chart are calculated as follows:
~=~~lk = 99.61/25 = 3.984 andi=!
~=~Si/k = 98.98/25 = 3.959iel
Unbiased estimate of mean (M) = (~ + @/ 2 = 3.97
Unbiased estimate of standard deviation
7.7.2 Central line and control limits for the control chartfor extreme values are:
CL = M= 3.97
UCL = M+ Hz ~ =3.97+ 1.36 x 0.025 =4.00
LCL = M–HZ ~ = 3.94
7.7.3 To use the control chart during production, thesample will be drawn at regular intervals of productionand the largest value (L) and the smallest value (S)will be plotted in the chart against the sample numberk on the X-axis. The criteria for out of control situationis same as that of ordinary ~ – R chart.
7.7.4 For the purpose of illustration, data of Table 3have been plotted on a control chart for extreme valuesin Fig. 3.
8 CONTROL CHART FOR COEFFICIENT OFVARIATION
8.1 Coefficient of variation (V) may give a useful
characterization of variability in cases where samplesare drawn from populations with different means andstandard deviations but with the same ratio of thesetwo measures as is, for instance, the case witfi thestrength of concrete. In fact, the mean compressivestrength of cement mortar cubes increases with thecuring time, and the standard deviation increases in
7
IS 397 (Part 3) :2003
the same proportion, but the ratios /Y being practicallyconstant. Other examples are provided by the crushingstrength of bricks, the sliver thickness of jute-fibres atdifferent preparing stages (carding and drawing, etc).
Table 3 Largest and Smallest Values ofHead Diameter
(Clauses 7.7.1 and 7.7.4)
S1 No. Sample Largest Smallest RemarksNo. K Value L Value S
8.2 l%e relative variability may then be controlled bycomputing the sample coeftlcient of variation andplotting these on a control chart. High values of thishybrid statistic will result from high variability ofquality or poor mean quality both of which are takento be indicative of unsatisfactory products.Correspondingly low values of the coefficient ofvariation are considered favorable. Therefore, incontrol charts for coefllcient of variation, it is onlynecessary to red-flag the advent of a manufacturingtrouble causing a too high value of the samplecoefllcient of variation. Thus, one is concerned withthe upper limit mainly and this limit should be so setthat a sample value larger than this comes from thespecified population only with a probability less thanthe pre-assigned quantity.
8.3 For setting up a control chart for coeftlcient ofvariation, the various lines are obtained as follows:
CL
= v - [v=(w)”]lJCL = B, V
LCL =B3~
where the l%ctorsBq and Bd are given in Annex A forvarious sample sizes.
8.4 If the value of the process average (p) and processstandard deviation (o) are known either from pastexperience or records, then the control charts forcoeftlcient of variation are set up as follows:
wh~e the values of factors Cz,B1 and Bz have beentabulated in Annex A for various sample sizes.
8.5 Example
In the jute industry, uniformity of linear density ofsliver is an important criterion which affects thesubsequent operations of spinning and weaving. It is,therefore, desirable to control this property properly.With a view to installing a control chart for coeftlcientof variation weight of five 10 m lengths of sliver weredaily collected at finisher card stage. The records of25 days at a particular reference moisture regain arepresented in Table 4 along with the average and CV
percent. (The observations in a particular samplecm-respond to one machine only at a particular time. )
8.5.2 The data of Table 4 along with control limitshave been plotted on a control chart for coeftlcient ofvariation in Fig. 4.
IS 397 (Part 3): 2003
9 CUMULATIVE SUM CONTROL CHARTS
9.1 A standard control chart for controlling the meanof a normal population consists of a plot of observedsample means in sequence which are then comparedagainst a set of fixed limits drawn parallel to the centraliine representing the ‘aimed at mean’. Such controlcharts suffer from the disadvantage of the sampleresults being viewed individually. Consequently thetraditional control charts are leas sensitive for detectingmoderate shifts in the mean value. One way ofremedying the situation is through the increase ofsample size which has many practical limitations. Analternative is to make use of the cumulative sum controlcharts (or cusum charts as it is popularly known) whichare quite sensitive to detect small shifts in the processaverage. In cumulative sum control charts, the pointsplotted do not represent a single observation or statisticcalculated from one sample only. Starting from a givenpoint, all subsequent plots contain information fromall the observations up to and including the plottedpoint. Thus the pointa on a cumulative sum controlchart have coordinate (m, TJ, where m is the serialnumber of the sample and T. represents the value ofthe statistic computed from all the m samples takencollectively.
Table 4 Weights of 10-Metre Lengths of Slivers
(Clauses 8.5 and 8.5.2)
SI Sample Weights of 10-Metre Lengths Average Cv RemarksNo. No. (g) at 25 Percent Moisture
0.00 I , , , r , 1 LCL1 2 3 4 5 6 7 8 91011121314151617181920 2122232425
Sample Number
FIG. 4 CONTROLCHART FORCOEFFICIENTOFVARIATION
9.2 In a cumulative sum control chart’ for means,cumulative totals are plotted against the samplenumber. Denoting the mean of the rnth sample by Xmand the target mean asp, the mth point plotted has co-ordinates (m, T’J.
9.2.1 For the sake of simplicity, division by ax isavoided so that the mth point plotted has co-ordinates(m, T~) where
9.2.2 The points are plotted on a graph with an ordinatescale specially adjusted (see Note under 9.6).
9.3 The chart is interpreted by placing a V-shapedmask(see area in Fig. 5) over the chart, with the point Ocoinciding with the last point plotted on the chart andthe line OP horizontal. Lack of control is diagnosed ifany of the previously plotted points are covered by themask, that is, if any points lie below the straight line.4 ‘B’, or above the straight line A ‘B’. A point belowthe former line indicates an increase in the processaverage, while a point above the latter line indicates adecrease.
9.4 me dimensions of the mask are defined by theangle (28) between All and A “B’ and the distance dfrom O to the vertex P of this angle. The quantities dand (3should be so determined that :
a) a lack of control will be indicated when theprocess is in control with a known smallprobability, and
b) a departure in process mean to a specifiedextent will be detected with a high known
probability.
9.5 For the application of the cusum cha@ it is essentialto know the value of the following parameters beforehand
Po =
Cr=
D=
n .
satisfactory process average of the char-acteristic to be controlled
standard deviation of the process when itis in a state of statistical control;
change in the mean (absolute) which is de-sired to be detected with fair certainty; and
sample size.
9.6 From the known values of the parameters, calculateD/k and DfiO and enter Table 5 twice, once forvalue of D/k and again for D@~ to obtain the cor-responding values of (3 and d respectively for thepreparation of the mask.
NOTE— k is the numberof units of the.ordinateto one unitofthe abscissm which corresponds to the horizontal distancebetween srrccc~ive plotted points. It is preferableto choose ksuch that it is approximatelyequal to ~i~
9.7 Using the above values of ft and d, a mask(transparent) may be prepared. From the successivesamples of size n, the sample mean shall then be calcu-lated. From each of the sample mean, is deducted andthe resultant deviation (q -P) shall be summed andplotted against the serial number of the sample. At eachplotting, the mask shall be used as described in 9.3 tofind out whether the process is in control or not.
9.8 If more than one point is covered by the mask atany instant, then the fmt point (from the left) coveredby the mask indicates the time at which the shift in theprocess average started. If none of the points arecovered, it is an indication that the process is in a stateof statistical control.
9.9 Since the operation of the cusum chart assumes aconstant standard deviation of the process, it isnecessary to maintain a range chart along with the
cusum chart. Also, if the process standard deviation(o) is not known, it can be estimated by jjfd, where~ is the average range and dz is a“factorto be obtainedfrom Annex A for the different sample sizes.
9.10 Example
9.10.1 In a textile mill, it was decided to install acumulative sum control chart for means for controllingthe weight of drawing sliver. For the purpose, theweights (in g) of 5 m of drawing sliver for 15 sampleseach of size 5 are given in Table 6. The dimensions ofV-mask are computed with the following informationgiven:
P .
Cr=
D=
n=
satisfactory process average = 2ogto be attained
standard deviation of process = 0.4 g
shifl in the mean which is = 0.3 gdesired to be detected withhigh probability
the size of sample . 5
9.10.2 To calculate the dimensions of the mask, initiallythe mean of each sample (see co] 8 of Table 6) isobtained and the deviations of the means from theprocess average (see CO19of Table 6) and cumulativesum of deviations (Z’J (see COI10 of Table 6) are alsocalculated.
Gz = ~/&=().4/&=().18
11
.,
IS 397 (Part 3) :2003
k = the scale factor of ordinate (obtainedby rounding offthe value of a, J = 0.2
D/k = 0.3/0 .2=1.5
Dfio = 0.3 x&/04 = 1.7
From Table 5, the dimensions of mask are obtainedas:
d = 4.6 (corresponding to value 1.7of Dfi i ~ )
0 = 36°52’ (corresponding to value 1.5 of D/k).
9.10.3 For the purpose of illustration, T~ values (see COI10 of Table 6), are plotted against the various samplenumbers in Fig. 6. At each plotting, the suitablyprepared V-mask (with 6 = 36°52’ and d = 4.6) issuperimposed on the graph with the point O of the maskcoinciding with the plotted point and OP parallel to
the horizontal axis. It maybe seen that at the last point(15) plotted, the V-mask superimposition covers thepoint corresponding to eleventh subgroup, indicatingthat there has been a significant upward shift in theprocess average at that time.
Table 6 Weights of 5 Metres of Drawing Slivers (in Grams)
(Clause 9.10.1,9 .10.2 and9.10.3)
Sample Individual Weighings Total Mean Deviation T.= RemarksM g Xm from Target Z(% -p)
Statistical Methods for Quality and Reliability Sectional Committee, MSD 3
Organization
Kolkata University, Kolkata
Bharat Heavy Electricais Limited, Hyderabad
Continental Devices India Ltd, New Delhi
Directorate General of Quality Assurance, New Delhi
Laser Science and Technology Centre, DRDO, New Delhi
Escorts Limited, Faridabad
IIMT Ltd, R & D Centre, Bangalore
Indian Agricultural Statistics Research Institute, New Delhi
Indian Association for Productivity, Quality & Reliability, Kolkata
Indian Institute of Management, Lucknow
Indian Statistical Institute, Kolkata
National Institution for Quality and Reliability, New Delhi
Powergrid Corporation of India Ltd, New Delhi
SRF Limited, Chcnnai
Standardization, “resting and Quality Certification Directorate,
New Delhi
‘rata Engineering and I.ocomotive Co Ltd, Jamshcdpur
University of Dt]hi, Delhi
In personal capacity (IJ-109, J4alviya Nagar, New Delhi 110017)
In PCI$,mal capacity (2(Yf, Krishna Nagar, Safdarjang Enclave,\<,II lklhi 1/0029)
131SDirectorate General
Representative(s)
PROFS. P. MURHERJEE(C/rahrrurr)
Woo S. N. JHA
SHFUA. V. KRISHNAN(Alterna[e)
DR NAVINKAPLJR
Smu VIPULKAPUR(Alternate)
Stuw S. K. SRIVASTVA
LT-COL P. VUAYA~ (Alternate)
DR ASHOKKLJMAR
SsuuC. S, V. NARENORA
SHRIK. VIJAYAMMA
DR S. D, SHARMA
DR A. K. SRIVASTAVA(A[ternafe)
DR B. DAS
PROFS. CHAKRABORTV
PROFS. R. MOHAN
PROFARVINOSETH(,4herna?e)
SHRIY. K. BHAT
SHRIG. W. DATEY (A[terrrafe)
DR S. K. AGARWAL
SHRID. CHAKRABORTV(,41/errrafe)
SHRIA. SANJEEVARAO
SHJUC. DESIGAN(Alternate)
SHRIS. K. KIMOTHI
SHRIP. N, SRIKANTTi(Alternate)
SHRIS. KUMAR
SHRISHANTISARUP(A1/errrafe)
PROFM. C. AGRAWA~
PROFA. N. NAmANA
SHRID. R. SEN
SHIUP. K. GAMBHIR,Director & Head (MSD)
[Representing Director General (Ex-o&io)]
Member SecretatyStnuLALITKOMARMEHTA
Deputy Director (MSD), BIS
KolLata University, Kolkata
Laser Science and Technology Centre, DRDO, New Delhi
Indian Agricultural Statistics Research Institute, Ncw Delhi
Basic Statistical Methods Subcommittee, MSD 3:1
PROFS. P. MUKHERJEE(Convener)
DR ASHOKKUMAR
DR S. D. SHARMA
DR DEBABRATARAY (Alfernate)
(Continued on page 15)
14
..
--!
IS 397 (Part 3) :2003
(Canlinued,fivm page I4)
Organization
Ind]an Association for Productivity, Quality and Reliability, Kolkata
]ndian Institute of Management, Lucknow
Indian Statistical Institute, Kolkata
National Institution for Quality and Reliability, New Delhi
Powergrid Corporation of India Ltd, New Delhi
Standardization, Testing and Quality Certification, New Delhi
Tata Engineering and Locomotive Co Ltd, Pune
University College of Medical Sciences, Delhi
University of Delhi, Delhi
In personal capacity (B-109, kfalviya Nagar, New Delhi 110017)
In personal capacity (20/1, Krishna Nagur, Safdaq”ungEnclave,Newilelhi 110029)
Representative(s)
DRB. DAS
DRA. LAHIRI(Alternate)
PROFS. CHAKRABORTY
PROFS. R. MOHAN
SHRIY. K. BHAT
SHRIG. W. DATEY (A/fernate)
DR S. K. AGARWAk
SHRIS. K. KIMOTHI
SHRISHANTISARUP
DR A. INDRAYAN
PROFM. C. AGRAWAL
PROFA. N. NANKANA
SHRID. R. SEN
Panel for Process Control, MSD3: l/P-2
In personal Capacity (B-109, h4alviya Nagar, New Delhi 110017) PROFA. N. NANKAN.4(convener)
Nationa] Institution for Quality and Reliability, New Delhi SHRIY. K. BHAT
Povmrgrid Corporation of India Limited, New Delhi DR S. K. AGARWA~
Standardization, Testing and Quality Certification, New Delhi SHRIS. K. KIMOTHI
Tata Engineering and Locomotive Co Ltd, Pune SHRISHANTISARUP
III personal capacity (20/1, Krishna Nagar, SafdaV”ungEnclave, SHRID. R. SEN
New Delhi 110029)
15
Bureau of Indian Standards
B[S is a statutory institution established under the Bureau of Indian Standard Act, 1986 to promote harmoniousdevelopment of the activities of standardization, marking and quality certification of goods and attending toconnected matters in the country.
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the standard, of necessary details, such as symbols and sizes, type or grade designations. Enquiries relating to
copyright be addressed to the Director (Publication), BIS.
Review of Indian Standards
Amendments are issued to standards as the need arises on the basis of comments. Standards are also reviewedperiodically; a standard along with amendments is reaft%med when such review indicates that no changes areneeded; if the review indicates that changes are needed, it is taken up for revision. Users of Indian Standards
should ascertain that they are in possession of the latest amendments or edition by referring to the latest issue of‘BIS Catalogue’ and’ Standards: Monthly Additions’.
This Indian Standard has been developed from Dot: No. MSD 3 (219).
Amendments Issued Since Publication
Amend No. Date of Issue Text Affected
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