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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 interestto the public, particularly disadvantaged communities and those engaged in the pursuit ofeducation and knowledge, the attached public safety standard is made available to promote thetimely dissemination of this information in an accurate manner to the public.

!"#$% '(%)

!"# $ %& #' (")* &" +#,-. Satyanarayan Gangaram Pitroda

Invent a New India Using Knowledge

/0 )"1 &2 324 #' 5 *)6 Jawaharlal Nehru

Step Out From the Old to the New

7"#1 &" 8+9&") , 7:1 &" 8+9&") Mazdoor Kisan Shakti Sangathan

The Right to Information, The Right to Live

!"# %& ;

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Indian StandardRULES FOR ROUNDING OFF

NUMERICAL VALUES

RevisedSixteenth Reprint JULY 2007

(Including Amendment Nos. I 2 3)

UDC 511.135.6

IS : 2 1 9 6(Reaffirmed 2000)

REAFFi iviED

Ini J

G r 3

Copyright 1960B U R E A U O F I N D I A N S T A N D A R D SMANAK BHAVAN. 9 BAHADUR SHAH ZAFAR M AR G

NEW DELHI 110002

September 1960

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IS: 2 96

(. lIli,l/Ir from flllgt: J )

JIembersSI I RI

R. \;.SARM.\

S R I J M. SINHAS J l R I J M. TIU .OAN

Uin:ctoralt: l ~ n l r a l

uf Supplics at Dispoaals Ministry of Works, Houllng at Supply)Engincering Association of India , Calcut taMinistry of Transport at Communications (Roads

Wing)SURI T. N. ~ H R O i.Ullmate

LT-GBN H. Wn.I.IA)IS Council If Scientific IndultriaIRc.nan:1i,New Delhi

DI I LAT. C. VERMAN J : . ~ i o Director, lS I. SURI J. P. MElIllO,.R. , Deputy Director Engg ), IS r

~ l l e r n Q l l

S,crrtari, sDit A, K. G U P TA Assistant Director Stal ), lSI: 1111.1 B. N. StKl ll i Extra Assistant Director Stat) , lS I

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AMENDMENT NO. 1 FEBRUARY 1997TO

IS 2 : 1960 RULES FOR ROUNDING orsNUMERICAI.. VALVES

Revised

Page 3, clause 0.4 - Insert the following after first entry:

IS 1890 PARTO : 1995/1S0 31-0: 1992 QUANTI11FS AND UNITS: PART 0GENFRAL PRINCIPLES FIRST REVISION

Page 8, clause 3.3 Insert the foJlowing ne w clause after 3.3:

3 .4 The rules given in 3.1, 3.2 and 3.3 should be used only if no specificcriteria for the selection ) f t h l ~rounded number have to be taken into account, Incases, where specific limit ( Maximum or Minimum ) has been stipulated orwhere specif ically mentioned in the requirement, it may he advisable always toround in on e direction.

Examples

I The requirement of leakage current for domestic electrical appliancesis 21 0 tA rms maximum,

The rounding may be done in on e direction. For example, if a test resultis obtained as 210.], it will be rounded up and reported as 211 tA.

2 The requirement for cyanide (a s e for drinking water is specified asO < ~ 5IllgIJ, maximum beyond which drinking water shall be consideredtoxic.

The rounding may be done in one direction. Fo r example, if 8 test resultis obtained as 0.051 mg/l, it will he rounded up and reported as0.06 mg/l.

3 The requirement for minimum thickness of the body of LPG cylinder is2.4 mm.

The rounding may be done in one direction. For example, if It le st resultis obtained as 2.39 mill, it will be rounded down and reported as 2.310m.

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Amend No.1 to IS l 6

4) The requirement for Impact-Absorption for Protective Helmets forMotorcycle Riders is:

'The conditioned helmet tested shall meet the requirements, whentbe resultant acceleration RMS value of acceleration measuredalong tbree directions) measured at the centre of gr vity of tbebeadform shall be s 150 go where go 9.81 m/sec) for any 5milliseconds continuously and at no time exceeds 300 go.

The rounding may be done ill one direction. If a test result is obtained as150.1 go, it will be rounded up and reported as 151 go.

(MSD 1 )

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IS : 2-1960

Indian StandardRULES FOR ROUNDING OFF

NUMERICAL VALUES

Revised

F O R E W O R D

0.1 This Indian Standard Revised) was adopted by the Indian StandardsInstitution on 27 July 1960, after the draft finalized by the Engineering

Standards Sectional Committee ha d been approved by th e EngineeringDivision Council.

0.2 To round off a value is to retain a certain number of figures, countedjrom the li t and drop the others so as to give a more rat ional form to thevalue. As the result of a test or of a calculation is generally rounded offfor the purpose of reporting or for drafting specifications, it is necessary toprescribe rules for rounding off numerical values as also for deciding on the number of figures to be retained.

0.3 This standard was originally issued in 1949 with a view to promotingthe adoption of a uniform procedure in rounding o numerical values.:However, th e rules given referred only to unit fineness of rounding su 2.3 )and in course of years the need was felt to prescribe rules for roundingoff numerical values to fineness of rounding other than unity. Moreover,it was also felt that th e discussion on th e number of figures to be retained asgiven in th e earlier version required further elucidation. The present revisionis expected to fulfil these needs.

0.4 In preparing this standard, reference has been made to the following:IS : 787-1956 GuIDE FOR INTBR-CONVBlUION 011 VALva. FROM ONE

SYSTEM OF UNITS TO ANOTHER. Indian Standards Institution.

B.S. 1957: 1953PRUBNTATION

OFNUKltlUCAL VALUES

FINBNESS OP

EXPREISION; ROUNDINO 011 NUMBERS . British Standards Institution.

AMERICAN STANDARD Z 25.1-1940 RULES FOR ROUNDING OFFNUMltRICAL VALUES. American Standard. Auociation.

ASTM DUIGNATION: E 29-50 RECO . . . . . NDED PRAOTICB FOR DElIONATINO SIONlflCANT PUCU IN SPEOI.,BD VALuas. AmericanSociety for resting and Materials.

JAKBS W. SoAllBOROUOH. Numerical Mathematical AnaJ) IiJ. Baltimore. The John Hopkins Preas, 1955.

3

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l im l PltUu5

622

Signijieant Figures5

36

105

6454

IS: 2 .1960

1. SCOPE

1.1 This s tandard prescribes rules for rounding off numerical. values for thepurpose of reporting results of a test, an analysis. a measurement or a calculation. and thus assisting in drafting specifications. also makes recommendations as to the number of figures that should be retained in courseof computation,

2. TERMINOLOGY

2.0 For the purpose of this standard. th e following definitions shall apply.

2.1 N u b e r or D e d m . Places - A value is said to have as manydecimal places as there are number of figures in th e value, counting from thefirst figure after the decimal point and ending with the last figure on th e right.

tampllS:Valu,

0 0295021-0295

2 000 000 00 I291 00

1 32 X 1 1

S II Note I )NOTa I F o r th e PW P 1e of this I t ndard, th e expression 10 32 x 101 should be

taken to conaist of two partl. th e value proper which is 10 32 aDd the unit of lpt uionfor th e value, 10.,

2 .2 NlUDbe r or SigaiaeaDt Figures - A value is said to have as manysignificant figures as there are number of significant digits t I Note 2 )in th e value. counting from the left-moat nOlN,tTO digit and ending with theright-moat digit in the value.

Exampl,s:Valu,0 0295000 0295

10 02952000 000 0015677 0

56770056 77 X lOt

. 0056 7703900

S t Note 3 ) OT 2 - Any of the dititl, 1. 2, 3 9 occurrin, in a value than be a ailni.

fiCIIDt digit I); and zero ball be Iigni6caDt digit only whrn it is preceded by lOme

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IS : 2 -1960

other digit ( excepting zeros) on its left. When appear ing in the power of 10 to indicatethe magnitude of th e unit in th e expression of a value, zero shan nOI be a significantdigit.

NOTa 3 - With a view to removing an y ambigui ty regarding the significance thezeros at the en d in a value like 3 900, it wo uld be a vHys desirable to write th e value inthe power-often notation. For example, 3 gOO ma y be written as 3 9 x 10 3 3 9 0 X 10 3or 3 900 X 103 depending upon the last figurc(s) in th e value to which it is desired toimpart significancl .

2.3 FIDeDe o f ROIIDcUll. - The unit to which a value is rounded off.

Fo r example, a value may be rounded to the nearest 0 00001,0 0002,0 000 5, 0 001, 0 002 5, 0 005, 0 01, 0 07, I, 2 5, lO, 20, 50, 100 or anyother un it depending on the fineness desired,

3. RULES FO R ROUNDING

3.0 The rule usually followed in rounding of f a value to unit fineness ofrounding is to keep unchanged th e last figure retained when the figure nextbcyond is less than. ) and to increase by I th e last figure retained when thefigure ncxt beyond is more than 5. There is diversity of practice when thefigure nex t b eyond the last figure retained is 5. III such cases, some cornputers r ound u p , that is, increase hy I, the last figure retained: others, round down that is, discard everything beyond th e last figure retained,Obviously, if th e re ta in ed v al ue is always rounded up , or always rounded down , the sum and th e average of a series of values so rounded will

be larger or smaller tha n the corresponding sum or average of th e unrounded values. , However, i f rounding oft is carried out in accordance with therules slated in 3.1 in one step see 3 .3 ) , the slim and the average of th erounded values would be more nea rly cor rect than in the previous caSC

see Appendix A ).

3.1 ROllllcUDg Oft to UDit Finellea8 - In case th e fineness of rounding isunity in the last place re ta ined, th e following rules shall be followed:

. Rule I - - W hen the figure next beyond the las t figure Or place tobe re ta ined is less than 5, the figure in th e last place retained shal l heleft unchanged.

u e t ~ n the figure next beyond the last figure or placeto be retained is more than 5 or is 5 followed by any figures otherthan zeros, the figure in th e last place retained shall be increasedby

Rule 11/ When th e figure n ex t b ey on d the last figure or placeto be retained is 5 alone or 5 followed by zeros only, th e figure in th elast place retained shall be (a) increased if it is odd and (b) leftunchanged i f even zero would be regarded as an even number Iorthis purpose).

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b h 21960

Some examples illustrating the application of Rules I to III are givenin Table I

TABLE I EXA.MPLES OF ROUNDING OFF VA.LVES TO UNIT FINENESS

VAunt nr NUI Olr ROUNDING A _ _ _ _ .

1 01 001 0001r A ~ ~ r A ~ r - - - - - A . . . - - - -Rounded Rule Rounded Rule Rounded Rule Rounded Rule

Value Value Value Value

72604- 7 I 7.3 I i 726 7260 I14725 15 )47 I 1472 I I I (b) )4.725

34-55 3 I 35 n 346 III a) 345513545001 14- 135 1 1355 13545 I8725 9 87 I 872 I I I (b) 8725

19205 19 I 192 I 1920 II I (b) 1920505499 1 05 I 0.55 0550

0-6501 1 07 0.65 1 0650 I00 )49 50 0 I 00 005 0050 111 (a)

3.1.1 The rules for rounding l aid down in 3.1 may be extended to applywhen the fineness of rounding is 0'10, 10, 100, I 000, etc. For example,

2 43 when rounded to fineness 0 10 becomes 2'40. Similarly, 712 and 715when rounded to the fineness 10 become 710 and 720 respectively,

3.2 Roaad i ...g Oft to Fi ...e... ess O t h e r than n i t y In case the finenessof rounding is not uni ty, but , say, it is n the given value shall be roundedoff according to the following rule:

ule I V - When rounding to a fineness n o ther than unity, thegiven value shall be divided by n The quotient shall be rounded off tothe nearest whole number in accordance with th e rules laid downin 3.1 for unit fineness of rounding. The number so obtained, thatis, the rounded quotient, shall then. be multiplied n to get the final

rounded value.Some examples illustrating the application of Rule IV are given in

Table II .NOTE 4 - The ruJea for rounding oft'a value to an y linenf SS of rounding, I I ma y allO

be alated in line \\itb thOle for unitfineneas of rounding ( . 3. 1 ) as follows:Divide th e g iv en v al ue by II 1 0 that an iotqJral quotient and a rC'mainder ar e

obtained. Round oft tbe value in the following manner:a) I r the remainder islf:l l tban 11/2, the value shall be rounded down aueh that

the rounded value is an integral multiple of II

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IS 12 -1960

b) I f the remainder is greater than n/2 the value shall be rounded up suehthat th e rounded value is an integral multiple of n

e) I f the remainder is exactly equal to n/2 that rounded value shall be cbOlC D

which is an integral multiple of 2n.

TABLE D EXAMPLES OF ROUNDING OFF VALUES TO FINENESSOTHER THAN UNIT

VALUE FINENESS OF QUOTIENT ROUNDIill FINAL ROUNDKDROUNDINO, n QUOTIENT VALUE

1) 2) 3) = I / 2 4) 5)= 2) X 4)

1.6478 02 8239 8 1.6

2-70 02 133 14 2824968 03 83227 8 2 4

175 05 35 4 20068721 007 98173 1 0700875 007 12-5 12 084

325 50 65 6 3 X 1 2

1025 50 205 20 10 x 102

3.2.1 Fineness of rounding other than 2 and 5 is seldom called for I npractice. For these cases, the rules for rounding may be stated insimpler form as follows:

a) Rounding off to fineness 50, 5, 0 5, 0 05 , 0 005, etc.

u e V - When rounding to 5 units, th e given value shall bedoubled and rounded off to twice the required fineness of rounding inaccordance with 3 .1 .1 . The value thus obtained shall be halved toget the final rounded value.

For example, in rounding off 975 to th e nearest 50, 975 is doubledgiving I 950 which becomes 2000 when rounded off to the nearest 100; when2 000 is divided by 2, the resulting number I 000 is t he rou nd ed value of

975.b) Rounding off to fineness 20,2 ,0 2 ,0 02,0 002, etc.

Rule V J When found ing to 2 units, the given value shall behalved and rounded of f to half the required fineness of rounding inaccordance with 3 .1 . The value thus obtained shall then he doubledto get the final rounded value.

FOf example, in rounding off 2 70 to the nearest 0-2, 2 70 is halvedgiving 1 35 which becomes 1 4 when rounded off to the nearest 0 1; when1 4 is doubled, the resulting number 2 8 is the founded value.

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IS: 2 1960

3 3 Successive Rouading - The final r ou nd ed va lue shall be obtainedfrom the most precise value available in ant s tep o nl y and not from a seriesof successive roundings. For example the value 0 549 9, when rounded toone significant figure, shall be written as 0 5 and not as 0 6 which is obtainedas a result of successive roundings to 0 550 0-55 and 0 6. t is obvious thatthe most precise value available is nearer to 0 5 and not to 0-6 and thatthe error involved is less in the former case, Similarly 0 650 I shall berounded of f to O 7 in one step and not successively to 0 650 0 65 and 0 6since the most precise value avai lable here is nearer to 0 7 than to 0 6( s lso Table I ).

OT 5 - In those cases where a final rounded value terminates with 5 an d it isintended to usc it in further computation ma y be helpful to use a ,+, or sign afterth e Iinal 5 to indicate whether a subsequent rounding should be up or down. Thus3 214 7 may be written as 3-215- when rounded to :1 fineness of rounding 0 001. further rounding to three significanr figures is desired this number would be roundeddown an d written as 3-21 which is in error by l ss than hal f a unit in the last place; ,other ise, rounding of 3-215 would have yie lded 3 22 which is in error by mort than halfa unit in the last place. Similarly, : 1 : 105 4 could be wri tten as 3 205+ when roundedto 4 significant figures. Further rounding to 3 significant figures would yield th evalue as 3-21.

In case th e final 5 is obtained exact ly, it would he indicated by leaving the 5 as suchwithout using or sign. In subsequent rounding the 5 would then be treated inaccordance with Rule I l l .

4 NUMBER OF FIGURES TO BE RETAINED

4 0 Pertinentto

the application of therules for

roundingoff is

the underlying decision as to the number of figures that should be retained in a givenproblem. The original values requiring to be rounded of f may arise as aresult of a test, an analysis or a measurement in other words, experimentalresults, or they may arise from computations involving several steps.

4 Experilnental Results - The number of figures to he retained in anexperimental result, either for the purpose of reporting or for guiding theformulation of specifications will depend on the significance of th e figuresin the value. This aspect h as b een discussed in detail under 4 of IS : 7871956 to which reference may be made for obtaining helpful guidance.

4 2 Computations n computations involving values of differentaccuracies the problem as to how many figures should he retained at varioussteps assumes a special significance as it would affect th e accuracy of th efinal result. The rounding off error will, in fact, be injected into computationevery time an arithmetical operation is performed. I t is, therefore, necessaryto carry out the Computation in such a manner as would obtain accurateresults consistent with t he accuracy of th e data in hand.

4 2 1 While it is not possible to prescribe details which may be followedin computations of various types, certain basic rules may be recommended

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IS : 2 9 6

for s ingle arithmetical o pe ra tio ns w hi ch , w hen followed, wil l save labourand at the same time enabl e accuracy of original data to be normallymaintained in th e final answers.

4.2.2 As a guide to th e number of places or figures to be retained in thecalculations involving arithmetical operations with rounded or approximatevalues, th e following procedures are recommended:

a) Addition - The more accurate values shal l be rounded off so as toretain one more pl ce than the last significant f igure in th e leastaccurate value. The resulting sum shall then be rounded off toth e last significant place in the least accurate value.

b) ubtr ction The more accurate value of the two given valuesshall be rounded off, before subtraction, to the s me pl ce as th elast significant figure in less accurate value; and the resul t shall bereported as such see also Note 6 ).

c) Multiplicatiofl and Dioision The number of signijicant figuresretained in th e more accurate values shall be kept on mor thanthat in the least accurate value. The result shall then be rouudedoff to the same number of significant figures as in the least accuratevalue.

d) When a long computation is carried out in several steps, th e intermediate results sha ll be properly rounded at the end of each stepso as to avoid the accumulation of rounding errors in such cases.I t is recommended that, at th e end of each step, one more significant figure may be retained than is required under a , b) and c) see also Note 7 .

NOTr. 6 --- Tile loss of the significant figures in the subtraction of two nearly equalvalues is the Krratcst soirce of inaccuracy in most computat ions, and it forms th eweakest link ill a chain computation where it occurs. Thus, if th e values O 1 6 ~ 152 an d ) 168 71 arc each correct to five significant figures, their difference O O()O 81. which hasonly two significant figures. is quite lik.e1y to i ntroduce inaccuracy in subsequentcomputation,

If, however, the diffcrcnct: of two values is desired to be correct 10 k i ~ n i l j c a n tfiguresand if it is known beforehand that the lirst m significant figures at th e left will d isappearby subtraction, then the number of significant figures to hI retained ill earh of thrvalues shall be m + se Example -1).

on 7 - To ensure a greater degree of accuracy in th.. computations. it is alsodesirable to avoid or defer . I long as possible certain approximation operations likethat of th e division or square root. For example, in the determination of sucrose by

volumetric method, the expression ~ l f _1 1may be better evaluated by takingIt . VI Vl

iu calculational form as 20Wl it Vl - f t V: 1411 1 1 I . which would defer the divisionuntil the last operation or the calculation.

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IS: 2 1 9 6

4 3 Examples

Example 1

Required to find th e sum of th e rounded olT value s 461-32. ~ 8 676 854 and 1-7462. . ,

Since the least accurate value 381 -() is known only to the first decimalplace, al l other values shall be rounded off to one more place,that is, to tw o decimal places and then added as shown below:

461 32381 6

76 8fi4 7.-,

924 :;2

The resulting sum shall then be reported to th e sam e decimal placeas in th e least accurate value, that is, as 924S.

tample 2

Required to find th e sum of th e values 2B 190, (J H, 6.57 32, 39500and 76 939, assuming that th e value l . i00 is known to the:

nearest hundred only.S in ce o ne of the values is known only to t he nea re st hundred, th e

other values shall be rounded to thr- nearest tC 11 and thenadded as shown helow:

2849 X IO X 1066 X 10

395 X In

7694 X ) J

4648 x 10

The sum shall then be reported to the nearest hundred asJ 46 5 X 100 or even as 465 X lOG.

Example 3

Required to find the difference of 679 8 and 76 365, assuming thateach number is known to its last figure but no fnrth( l .

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IS: 2 196

Since one or the values is known to the first decimal place only, theother value shall also be rounded of f to th e first decimal placeand then the difference shall be found.

679 876 4

603 4

The difference, 603 4, shall be reported as such.

Example

Required to evaluate y 2 52 - V2 49 correct to live significantfigures.

Since V ~ F 5 = 1 587 450 7 ~

V2 49 = 1 577 973 3Band three significant fig-ures at th e left wilJ disappear on subtraction, t he n um be r of significant figures retained in each valueshall be B as shown below:

1 587150

}.577 i

)o )09177 4

The result, OoO -77 4, shall be reported as such o r as9 477 4 )< 10 - 3 ) .

H ample i

Required to evaluate 35-2/\12 given that th e numerator is correctto its last Iigure-.

Si nce t he numerator here is correct to three significant figures, thedenominator shall be taken as v i = 1 414. Then,

35 2 _-= H ~J Hland the result shall he reported as 2 J ~

Esampl

Required to evaluate 3 78TCj5 6 assuming that the denominator istrue to on ly t wo significant figures.

Since the denominator here is corrrect to two significant ligures, eachnumber in the numerator would be taken up to three significant

204 Dcptt. of HIS/200?

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IS: 2-1960

figures. Thus,3 78 X 3 4

5 7 = 2 08.

The result shall, however, be reported as 2 1.

P P E N I X A

l use 3.0)VALIDITY OF RULES

A-I. The validity of the rules for rounding off numerical values, as givenin 3.1 , may be seen from the fact that to every number that is to be roundeddown in accordance with Rule I, there corresponds a n um ber tha t is tobe rounded u p in accordance with Rule II. Thus, these two rules establish a balance between rounding down a n d up , for all numbers otherthan those that fall exactly midway between two alternatives. In the lattercase, since the figure to be dropped is exactly 5, Rule III , which specifics thatthe value should be rounded to its nearest even number, implies that roundingshall be up , when the preceding figures are 3, 5, 7, 9 a n d down whenthey are 0, 2, 4, 6, 8. Rule III hence advocates a similar balance between

rounding up , a n d down s lso Note 8 . This implies that if the aboverules are followed in a large group of values in which random distribution offigures occurs, the number rounded u p and the number rounded downwill be nearly equal. Therefore, the sum and the average of the roundedvalues will be more nearly correct than would be the case if all were roundedin the same direction, that is, either all up or all down .

;- ;OTE Is- From purely logical considerations, a given value could have as well beenrounded to an od d number and not an even number as in Rule III ) when the discarded figures fall exactly midway between two alternatives. Hut there is a practicalaspect to the matter. The rounding off a value to an even number facilitates thedivision of the rounded value by 2 an d the result of such division gives the correctrounding of half the original unrounded value. Besides, the rounded even values maygenerally be exact y divisible by many more numbers, even as well as odd, than arc the rounded odd values.

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