8/2/2019 IS 2 - 1960

1/14

BIS 2002B 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 MARG

N E W D E L H I 110002

I S : 2 - 1960(R e a f fi r m e d 1 99 0)

E d i t i o n 2 .3(2000-08)

P r i c e Gr o u p 3

Indian S tandard RULES F OR ROUNDING OFF

NUMERICAL VALUES

( R evised )(In corp ora tin g Amen dm ent Nos. 1, 2 & 3)

UDC 511.135.6

( Reaffirmed 2000 )

8/2/2019 IS 2 - 1960

2/14

IS : 2 - 1960

B 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 MARG

N E W D E L H I 110002

Ind ian S tandard RULES F OR ROUNDING OFF

NU MERICAL VALUE S( R evised )

En gineering Standa rds Sectiona l Comm ittee, EDC 1

Chairman

D R K. S. K RISHNAN Council of Scientific & Industrial Research,New Delh i

MembersS HRI P RE M P RAKASH ( Alternate t o

Dr K. S. Krishnan )ASSISTANT D IRECTOR Research, Design & Standardization Organization

(Minist ry of Railwa ys)S H RI B ALESHWAR N ATH Centr al Board of Irr igat ion & Power, New DelhiD EPUTY D IRECTOR G ENERAL OF

O BSERVATORIESDirectorate General of Observatories (Ministry of

Tran sport & Commun icat ions), New DelhiD IRECTOR Engineering Research Department, Government

of Andh ra P ra desh

S H RI S. B. J OSHI Inst itut ion of Engineers (India), Calcut taS H RI R. N. K APUR Indian En gineering Association, Calcut taD R R. S. K RISHNAN Indian Inst itut e of Science, BangaloreS H RI S. R. M EHRA Council of Scientific & Industrial Research, New

DelhiS H RI S. N. M UKERJI

S HRI K. D. B HATTACHARJEE ( Alternate )

Governm ent Test H ouse, Calcut ta

D R B. R. N IJHAWAN Council of Scientific & Industrial Research, NewDelhi

S H RI V. R. R AGHAVAN Centr al Wat er & Power Comm ission, New DelhiB RI G J . R. S AMSONL T -C OL R. J ANARDHANAM

( Alternate )

Controller of Development (Armaments) (Ministryof Defence)

( Contin ued on page 2 )

8/2/2019 IS 2 - 1960

3/14

IS : 2 - 1960

2

( Continu ed from page 1 )

Members

S H RI R. N. S ARMA Directorate General of Supplies & Disposals(Minist ry of Work s, Housin g & Supply)

S H RI J . M. S INHA Engineering Association of India, CalcuttaS H RI J . M. T REHAN

S H RI T. N. B HARGAVA ( Alternate )

Ministry of Transport & Communications (RoadsWing)

L T-G E N H. W ILLIAMS Council of Scientific & Industrial Research,New Delh i

D R LAL C. V ERMAN ( Ex-officio )S H RI J . P. M EHROTRA

( Alternate )

Director, ISIDeput y Director (En gg), ISI

S ecretaries

D R A. K. G UPTA Assista nt Director (Stat ), ISIS H RI B. N. S INGH Extr a Assistan t Director (Sta t), ISI

8/2/2019 IS 2 - 1960

4/14

IS : 2 - 1960

3

Indian S tandard RULES F OR ROUNDING OFF

NUMERICAL VALUES( R evised )

0 . F O R E W O R D0.1 This Indian Standard (Revised) was adopted by the IndianStandar ds Inst itu t ion on 27 J uly 1960, after th e draft final ized by theEngineering Standards Sectional Committee had been approved by theEn gineerin g Division Coun cil.

0.2 To round off a value is to reta in a certa in n um ber of figur es, counted from th e left , and dr op th e oth ers so as to give a more ra tiona l form to thevalue. As the result of a test or of a calculation is generally rounded off for th e pu rp ose of rep ort ing or for d ra ftin g specificat ion s, it is necessar yto prescribe rules for rounding off numerical values as also for decidingon th e n um ber of figur es to be r eta ined.0.3 This sta nda rd was originally issued in 1949 with a view to promotingthe adoption of a uniform procedure in roun din g off numerical values.However, the rules given referred only to unit fineness of rounding( see 2.3 ) and in course of years the need was felt to prescribe rules forroun ding off num erical values t o fineness of roun ding oth er t ha n un ity.Moreover, it was also felt th at th e discussion on th e nu mber of figur es tobe retained as given in the earlier version required further elucidation.The pr esent revision is expected t o fulfil th ese needs.0.4 In preparing this standard, reference has been made to thefollowing:

IS : 787-1956 G UIDE FOR INTER -C ONVERSION OF VALUES FROM O NES YSTEM OF U NITS TO ANOTHER . Indian Stan dards Inst itution.

IS : 1890 (P ART 0)-1995/ISO 31-0 : 1992 Q UANTITIES AND U NITS :P ART 0 G ENERAL P RINCIPLES ( F IRST R EVISION ).

BS 1957 : 1953 P RESENTATION OF N UMERICAL VALUES (F INENESS OFE XPRESSION ; R OUNDING OF N UMBERS ). British StandardsInstitution.

AMERICAN S TANDARD Z 25.1-1940 R ULES FOR R OUNDING O F FN UMERICAL VALUES . American Standards Association.

ASTM D ESIGNATION : E 29-50 R ECOMMENDED P RACTICE FORD ESIGNATING S IGNIFICANT P LACES IN S PECIFIED VALUES . Amer icanSociety for Test ing an d Mat eria ls.

J AMES W. S CARBOROUGH . Numerical Mathematical Analysis

Baltimore. The J ohn Hopkins Pr ess, 1955.

8/2/2019 IS 2 - 1960

5/14

IS : 2 - 1960

4

0.5 This edition 2.3 incorporates Amendment No. 1 (Feburary 1997),Amen dm ent No. 2 (October 1997) and Amen dm ent No. 3 (Augu st 2000).Side bar indicates modification of the text as the result of incorporat ion of th e am endm ent s.

1 . SC OP E1.1 This st an dar d pr escribes ru les for round ing off num erical values forth e pur pose of reporting resu lts of a t est, an a na lysis, a m easu remen t ora calculat ion, a nd th us assisting in dra fting specificat ions. It also mak esrecomm enda tions a s to the n um ber of figur es th at should be ret ained incou rs e of comp ut at ion.

2 . TER MINOLOGY2.0 For the purpose of this standard, the following definitions shallapply.2 . 1 N u m b e r o f D e c i m a l P l a c e s A value is said to have as manydecimal places as there are number of figures in the value, countingfrom the first figure after the decimal point and ending with the lastfigur e on t he r ight .

N OT E 1 For the purpose of this sta ndar d, the expression 10.32 10 3 shouldbe taken to consist of two parts, the value proper which is 10.32 and the unit of expression for the value, 10 3 .

2 .2 N u m b e r o f S i g n i fi c a n t F i g u r e s A value is said to have as m an ysignifican t figur es as th ere a re nu mber of significan t digits ( see Note 2 )in the value, counting from the left-most non-zero digit an d ending withth e right-most digit in t he value.

Examples :Valu e Decim al Places

0.029 50 521.029 5 4

2 000.000 001 6

291.00 210.32 10 3 2( see N ote 1 )

Examples :Valu e S ign ifican t Figu res

0.029 500 50.029 5 3

10.029 5 62 000.000 001 105 677.0 5

567 700 656.77 10 2 4

0 056.770 5

3 900 4( see N ote 3 )

8/2/2019 IS 2 - 1960

6/14

IS : 2 - 1960

5

N OT E 2 Any of the digits, 1, 2, 3,............9 occurring in a value shall be asignificant digit(s); and zero shall be a significant digit only when it is preceded bysome other digit (exceptin g zeros) on its left. When a ppea rin g in th e power of 10 toindicat e th e ma gnitude of th e un it in t he expression of a value, zero shall not be a

significan t d igit.N OT E 3 With a view to removing any ambiguity regarding the significance of

the zeros at the end in a value l ike 3 900, it would be always desirable to write thevalue in t he power-of-ten n otat ion. For example, 3 900 may be writ ten a s 3.9 10 3 ,3.90 10 3 or 3.900 10 3 depending upon t he last figure(s) in t he value t o which itis desired t o impart significan ce.

2 .3 F i n e n e s s o f R o u n d i n g The u n it t o which a va lue is roun ded off.For example, a value may be roun ded to th e nearest 0.000 01, 0.000 2,0.000 5, 0.001, 0.002 5, 0.005, 0.01, 0.07, 1, 2.5, 10, 20, 50, 100 or an yoth er u nit depending on t he finen ess desired.

3 . R U L E S F O R R O U N D I N G3.0 The rule usually followed in rounding off a value to unit fineness of rounding is to keep unchanged the last figure retained when the figurenext beyond is less than 5 and to increase by 1 the last figure retainedwhen the figure next beyond is more than 5. There is diversity of practice when the figure next beyond the last figure retained is 5. Insu ch cases, some comp ut ers roun d u p, th at is, increa se by 1, th e lastfigur e ret ain ed; oth ers r oun d down, tha t is, discar d everyth ing beyondthe last figure retained. Obviously, if the retained value is alwaysroun ded u p or alwa ys roun ded down, th e su m an d t h e aver age of aseries of values so rounded will be larger or smaller than thecorresponding sum or average of the unrounded values. However, if roun ding off is car ried out in a ccorda nce with th e ru les st at ed in 3.1 inone st ep ( see 3.3 ), the sum an d th e average of th e rounded values wouldbe more near ly corr ect t ha n in th e previous cases ( see Appendix A ).3 . 1 R o u n d i n g O ff t o U n i t F i n e n e s s In case the fineness of rounding is unity in the last place retained, the following rules (exceptin 3.4 ) shall be followed:

Rule I When the figure next beyond the last figure or place to bereta ined is less th an 5, the figur e in th e last place retained sh all be leftunchanged.

Rule II When the figure next beyond the last figure or place to beret ained is m ore t ha n 5 or is 5 followed by any figur es oth er t ha n zeros,th e figur e in th e last place retained sh all be increased by 1.

R ule III When th e figure n ext beyond th e last figur e or p lace to beret ain ed is 5 alone or 5 followed by zeros only, th e figur e in t he la st placeretained shall be (a) increased by 1 if it is odd and (b) left unchanged if even (zero would be r egarded a s a n even n um ber for th is pur pose).

8/2/2019 IS 2 - 1960

7/14

IS : 2 - 1960

6

Some exam ples illust ra ting t he applicat ion of Rules I to III a re given inTab le I .

3.1.1 The r ules for roun ding laid down in 3.1 ma y be extended to applywhen th e finen ess of r oun ding is 0.10, 10, 100, 1 000, etc. Forexample, 2.43 when roun ded t o fineness 0.10 becomes 2.40.Similar ly, 712 and 715 when round ed to the finen ess 10 become 710an d 720 respectively.3 . 2 R o u n d i n g O f f t o F i n e n e s s O t h e r t h a n U n i t y In case thefineness of rounding is not unity, but, say, it is n , the given value sh allbe rou nded off accordin g to th e following r u le:

R ule IV When r oun ding to a finen ess n , oth er th an un ity, th e givenvalue shall be divided by n . The quotient shall be rounded off to thenea rest whole nu mber in accorda nce with t he r ules laid down in 3.1 forunit fineness of rounding. The number so obtained, that is the roundedquotient , shall th en be mu ltiplied by n to get t he fina l roun ded value.Some examples illustrating the application of Rule IV are given inTable I I.

N OT E 4 The r ules for r oundin g off a va lue t o any fineness of roun ding, n , mayalso be stat ed in line with th ose for u nit fineness of rounding ( see 3.1 ) as follows:

Divide th e given va lue by n so tha t an int egral quotient and a r emainder areobtained. Round off the value in the following manner:

a) I f the remainder i s less than n /2, the value sh all be roun ded down su ch t ha t

th e rounded value is an int egral mult iple of n .

TAB LE I E XAM P L E S O F R O UN DI N G O F F VAL U E S T O UN I TF I N E N E S S

VALUE F INENESS OF ROUNDING

1 0.1 0.01 0.001

RoundedValue

Rule Rou ndedValue

Rule Roun dedValue

Rule RoundedValue

Rule

7.260 4 7 I 7.3 II 7.26 I 7.260 I

14.725 15 II 14.7 I 14.72 III(b) 14.725

3.455 3 I 3.5 II 3.46 III(a ) 3.455

13.545 001 14 II 13.5 I 13.55 II 13.545 I

8.725 9 II 8.7 I 8.72 III(b) 8.725

19.205 19 I 19.2 I 19.20 III(b) 19.205

0.549 9 1 II 0.5 I 0.55 II 0.550 II

0.650 1 1 II 0.7 II 0.65 I 0.650 I

0.049 50 0 I 0.0 I 0.05 II 0.050 III(a )

8/2/2019 IS 2 - 1960

8/14

IS : 2 - 1960

7

b) If the remainder i s grea ter th an n /2, the value sha ll be roun ded up such th atth e rounded value is an int egral mult iple of n .

c) If the rema inder is exactly equal to n /2, that rounded value shall be chosenwhich is a n int egral mu ltiple of 2 n .

3.2.1 Fineness of rounding other than 2 and 5 is seldom called for inpra ctice. For t hese cases, th e ru les for r oun ding ma y be stat ed in simplerform as follows:

a) Roun ding off to finen ess 50, 5, 0.5, 0.05, 0.005, etc. Rule V When rounding to 5 units, the given value sha ll be

doubled an d r ou nd ed off to twice th e requ ired finen ess of roun ding inaccord an ce wit h 3.1.1 . The value t hu s obta ined sha ll be ha lved to getth e final round ed value.

For example, in roun ding off 975 to th e near est 50, 975 is doubledgiving 1 950 which becomes 2 000 when r ounded off to the nearest 100;when 2 000 is divided by 2, the resulting number 1 000 is the roun ded

value of 975.b) Roun din g off to finen ess 20, 2, 0.2, 0.02, 0.002, et c. Rule VI When rounding to 2 units, the given value shall be

halved and rounded off to half the required fineness of rounding inaccord an ce with 3.1 . The value th us obtained sh all then be doubled toget t he fina l roun ded value.

For examp le, in round ing off 2.70 to th e nea rest 0.2, 2.70 is halvedgiving 1.35 which becomes 1.4 when roun ded off to the n ear est 0.1;when 1.4 is doubled, th e resulting num ber 2.8 is the roun ded value.

TAB LE I I E XAM P L E S OF R O U N DI N G O F F VAL UE S T OF I N E N E S S O T H E R T H AN U N I T

VALUE F INENESS OF R OUNDING , n

Q UOTIENT R OUNDEDQ UOTIENT

F INAL R OUNDEDVALUE

(1) (2) (3) = (1)/(2) (4) (5) = (2) (4)

1.647 8 0.2 8.239 8 1.62.70 0.2 13.5 14 2.82.496 8 0.3 8.322 7 8 2.41.75 0.5 3.5 4 2.00.687 21 0.07 9.817 3 10 0.700.875 0.07 12.5 12 0.84

325 50 6.5 6 3 10 2

1 025 50 20.5 20 1 0 10 2

8/2/2019 IS 2 - 1960

9/14

IS : 2 - 1960

8

3 .3 S u c c e s s iv e R o u n d i n g The fina l rou nd ed value sh all be obta inedfrom the most precise value available in one step only and not from aseries of successive roun dings. For exam ple, th e value 0.549 9, whenroun ded to one significan t figure, sha ll be written as 0.5 an d n ot a s 0.6which is obta ined as a resu lt of successive roundings t o 0.550, 0.55,an d 0.6. It is obvious t ha t t he m ost pr ecise value available is near erto 0.5 and n ot t o 0.6 and t ha t t he err or involved is less in th e form ercase. Similar ly, 0.650 1 sha ll be rounded off to 0.7 in one st ep an d n otsu ccessively to 0.650, 0.65 an d 0.6, since t he most pr ecise value a vailablehere is near er to 0.7 tha n to 0.6 ( see also Ta ble I ).

N OT E 5 In those cases where a final rounded value terminates with 5 and itis int end ed t o use it in fur th er comput at ion, it ma y be helpful to us e a + or signafter t he fina l 5 to indicate wheth er a su bsequent r oun ding should be up or down.Thus 3.214 7 may be writt en as 3.215 when r oun ded to a finen ess of

rounding 0.001. If fur th er roun ding to three significan t figur es is desired, th isnu mber would be rounded down a nd writ ten a s 3.21 which is in error by less t h a nhalf a unit in the last place; otherwise, rounding of 3.215 would have yielded 3.22which is in error by more th an h alf a u nit in th e last place. Similarly, 3.205 4 couldbe written as 3.205+ when r oun ded to 4 significan t figures. Fu rt her rounding to 3significant figures would yield th e value a s 3.21.

In case t he final 5 is obtained exactly, it would be indicated by leaving the 5 assu ch with out u sing + or sign. In s ubs equen t r oun ding t he 5 would t hen betr eat ed in accorda nce with Rule III.

3.4 The rules given in 3.1 , 3.2 and 3.3 should be used only if no specificcriteria for the selection of the rounded number have to be taken intoaccount. In all cases, where safety requirements or prescribed limitsha ve to be respected, roun ding off should be done in one d irection only.

4 . N U M B E R O F F I G U R E S TO B E R E TAI N E D4.0 Pertinent to the application of the rules for rounding off is theun derlying decision a s to th e nu mber of figur es th at should be reta inedin a given problem. The original values requiring to be rounded off mayar ise as a result of a test , an an alysis or a measur ement, in oth er words,experimental results, or they may arise from computations involvingsevera l steps.

4 .1 E x p e r i m e n t a l R e s u lt s The number of figures to be retained inan experiment al resu lt, either for th e pur pose of reporting or for guidingthe formulation of specifications will depend on the significance of thefigur es in th e value. This asp ect h as been d iscussed in d etail un der 4 of IS : 787-1956 to which r eference ma y be ma de for obtaining helpfulguidance.4 . 2 C o m p u t a t i o n s In computations involving values of differentaccuracies, the problem as to how many figures should be retained atvarious steps assumes a special significance as it would affect theaccuracy of the final result. The rounding off error will, in fact, be

injected into computation every time an arithmetical operation isperformed. It is, therefore, necessary to carry out the computation in

8/2/2019 IS 2 - 1960

10/14

IS : 2 - 1960

9

such a manner as would obtain accurate results consistent with theaccur acy of th e data in ha nd.4.2.1 While it is n ot possible t o prescribe det ails wh ich ma y be followed

in computations of various types, certain basic rules may berecommended for single arithmetical operations which, when followed,will save labour an d at th e sam e time ena ble accur acy of origina l data tobe norma lly ma inta ined in th e final an swers.4.2.2 As a guide to the n um ber of places or figur es to be reta ined in t hecalculations involving arithmetical operations with rounded orap pr oximat e valu es, th e following procedur es ar e recomm end ed:

a ) Addition The more accur at e values sha ll be roun ded off so as t oretain one m ore place than the last significant figure in the leastaccur at e value. The resu lting sum sha ll th en be round ed off to the

last significan t place in t he least accur at e value.b) Subtraction The m ore a ccur at e value (of the t wo given values)shall be rounded off, before subtraction, to the same place as thelast significan t figure in less accur at e value; an d th e resu lt sha ll bereport ed as su ch ( see also N ot e 6 ).

c) Multiplication and Division The number of significant figuresretained in the more accurate values shall be kept one m ore t hanth at in th e least a ccur ate value. The result sha ll th en be roun dedoff to the same number of significant figures as in the leastaccur at e value.

d) When a long compu ta tion is car ried out in severa l steps, th eintermediate results shall be properly rounded at the end of eachstep so as to avoid the accumulation of rounding errors in suchcases. It is recommended that, at the end of each step, one moresignificant figure may be retained than is required under (a), (b)and (c) ( see also Not e 7 ).N OT E 6 The loss of the significant figures in the subtraction of two nearly

equal values is the greatest source of inaccuracy in most computations, and itform s th e weakest link in a cha in compu ta tion where it occur s. Thus, if th e values0.169 52 and 0.168 71 are ea ch corr ect t o five significan t figures, th eirdifference 0.000 81, which h as only two significan t figures, is quite likely tointr oduce ina ccur acy in su bsequent compu ta tion.

If, however, the difference of two values is desired to be correct to k significantfigur es an d if it is known beforeha nd t ha t t he first m significan t figur es a t th e leftwill disappear by subtra ction, then t he n um ber of significan t figur es to be retain edin each of th e values sha ll be m + k ( see Example 4 ).

N OT E 7 To ensu re a great er degree of accur acy in t he compu ta tions, it is a lsodesirable to avoid or defer as long as possible certain approximation operationslike that of the division or square root. For example, in the determination of

sucr ose by volu met r ic meth od, t h e expression ma y be bet t er

evaluat ed by tak ing its calculat ional form as 20 w 1 ( f 2 v1 f 1 v2 )/ w 2 v1 v2 whichwould defer th e division un til th e last opera tion of th e calculat ion.

20 w 1w 2

-------------- f 2v 2------

f 1v1------

8/2/2019 IS 2 - 1960

11/14

IS : 2 - 1960

10

4.2.3 Examples

Example 1

Required to find the sum of the rounded off values 461.32, 381.6,76.854 and 4 .746 2 .

Since th e least accur at e value 381.6 is kn own only to th e first decimalplace, all other values shall be rounded off to one more place, thatis, to two decima l places an d th en a dded a s sh own below:

The resu lting sum sh all then be reported to the sa me decimal place asin th e least a ccur at e value, tha t is, as 924.5.

Example 2

Required to find the su m of th e values 28 490, 894, 657.32, 39 500

and 76 939 , assuming that the value 39 500 is known to thenear est hu ndr ed only.

Since one of the values is known only to the nearest hundred, theoth er values shall be rounded off to th e near est ten an d th en a ddedas sh own below:

The sum shal l then be reported to the nearest hundred as 1 465 100or e ven a s 1.465 10 5 .

Example 3Required to find t he difference of 679.8 an d 76.365, assu ming t ha t

each n um ber is kn own to its last figure but no far th er.

461.32

381.6

76.85

4.75

924.52

2 849 10

89 10

66 10

3 950 10

7 694 10

14 648 10

8/2/2019 IS 2 - 1960

12/14

IS : 2 - 1960

11

Since one of the values is known to the first decimal place only, theother value shall also be rounded off to the first decimal place andth en t he difference sha ll be foun d.

The differen ce, 603.4, sha ll be report ed as su ch .

Example 4Requ ir ed to eva lua te cor rect t o five sign ifican t figu res.

Since = 1.587 450 79

= 1.577 973 38

and three significant figures at the left will disappear onsubtraction, the number of significant figures retained in eachvalue sh all be 8 as sh own below:

The result , 0.009 477 4, shall be report ed as su ch (oras 9.477 4 10 3 ).

Example 5

Required to evaluate 35 .2 / given that the numerator is cor rect toits last figur e.

Since the numerator here is correct to three significant figures, thedenomina tor sha ll be t aken as = 1.414. Then ,

= 24.89

an d th e result sha ll be report ed as 24.9.

Example 6

Required to evaluate 3.78 /5.6, assuming that the denominator istr ue t o on ly two significan t figu res.

Since th e denomina tor h ere is corr ect t o two significan t figu res, eachnu mber in t he n um erat or would be taken up t o three significan t

679.876.4

603.4

1.587 450 81.577 973 40.009 477 4

2.52 2.49

2.52

2.49

2,

2

35.21.414---------------

8/2/2019 IS 2 - 1960

13/14

IS : 2 - 1960

12

figures. Thus,

= 2 ,08.

The result sh all, however, be report ed as 2.1.

A P P E N D I X A

( Clause 3.0 )

VALIDIT Y OF RU LE SA-1. The va lidity of th e ru les for roun ding off nu mer ical valu es, as givenin 3.1 , may be seen from the fact that to every number that is to beroun ded down in a ccord an ce with Ru le I, th ere cor resp on ds a nu mberth at is to be roun ded u p in a ccord an ce with Rule II. Thu s, th ese tworu les esta blish a bala nce between r oun ding down an d up for allnumbers other than those that fall exactly midway between twoalternatives. In the latter case, since the figure to be dropped isexactly 5, Rule III, which specifies th at th e value should be round ed toits nea rest even n um ber, implies th at roun ding sha ll be up when t hepreceding figures are 1, 3 , 5 , 7 , 9 and down when th ey are 0, 2 , 4 , 6 , 8 .Rule III h ence advocat es a similar bala nce between r ou nd ing up an d

down ( see also Note 8 ). This implies th at if th e above rules are followedin a la rge group of values in wh ich r an dom distr ibut ion of figures occur s,th e nu mber roun ded u p an d th e nu mber r oun ded down will be nea rlyequal. Therefore, th e sum an d th e average of th e rounded values will bemore nearly correct than would be the case if all were rounded in thesa me dir ect ion , th a t is, eith er all up or a ll down.

N OT E 8 From purely logical considerations, a given value could have as wellbeen r ounded t o an odd num ber (and not an even nu mber as in Rule III) when th ediscarded figures fall exactly midway between two alternatives. But there is apractical aspect to the matter. The rounding off a value to an even numberfacilitates the division of the rounded value by 2 and the result of such division

gives the correct rounding of half the original unrounded value. Besides, the(rounded) even values m ay genera lly be exactly divisible by ma ny m ore n um bers,even as well as odd, tha n a re t he (roun ded) odd values.

3.78 3.145.7

-----------------------------

8/2/2019 IS 2 - 1960

14/14

B u r e a u o f In d i a n S t a n d a r d sBIS is a statutory insti tution established under the Bureau of Indian Standards Act , 1986 topromote harmonious development of the activit ies of standardization, marking and qualitycertificat ion of goods and a tt endin g to connected ma tt ers in t he coun tr y.

C o p y r i g h t

BIS has the copyright of all its publications. No part of these publications may be reproduced inan y form with out th e prior perm ission in wr iting of BIS. This does not preclude th e free us e, in t hecour se of implemen ting th e sta nda rd, of necessar y details, such as symbols and sizes, type or gradedesignat ions. En quiries relat ing to copyright be addr essed to the Director (Pu blicat ions), BIS.

R e v ie w o f I n d i a n S t a n d a r d sAmendm ents ar e issued to standa rds as th e need arises on th e basis of comments. Stan dards a realso reviewed periodically; a standard along with amendments is reaffirmed when such reviewindicates t hat no cha nges are n eeded; if the r eview indicates tha t chan ges are needed, it is ta kenup for r evision. Users of Indian Stan dards should ascerta in th at they a re in possession of the latestamen dment s or edit ion by referring to th e latest issue of BIS Catalogue and Stan dards : Mont hlyAdditions.This Indian Sta nda rd h as been developed by Techn ical Committ ee : EDC 1

Am e n d m e n t s I s s u e d S i n c e P u b l ic a t i onAm e n d No . Da t e o f I ssu e

Amd. No. 1 Febu ra ry 1997

Amd. No. 2 October 1997

Amd. No. 3 Au gu st 2000

B UR E AU O F I ND IAN S TAN DAR DSHeadquar ters:

Mana k Bhavan , 9 Baha dur Sh ah Zafar Marg, New Delhi 110002.Telephones: 323 01 31, 323 33 75, 323 94 02

Telegrams: Manaksan stha(Comm on t o all offices)

Region a l Offices: Teleph on eCe nt r a l : Ma n a k Bh a va n , 9 Ba h a du r S ha h Za fa r Ma r g

NEW DELHI 110002323 76 17323 38 41

Eastern : 1/14 C. I . T. Scheme VII M, V. I . P. Road, KankurgachiKOLKATA 700054

3 37 8 4 9 9, 3 37 8 5 6 13 37 86 26 , 337 91 2 0

Nor th er n : SCO 335-336, Sect or 34-A, CH ANDIGARH 160022 60 38 4360 20 25

South ern : C. I . T. Cam pus, IV Cross Road, CHENNAI 600113 2 35 02 16 , 235 04 4 22 35 15 19 , 235 23 1 5

Western : Manaka laya , E9 MIDC, Marol, Andher i (Eas t )MUMBAI 400093

8 32 92 95 , 832 78 5 88 32 78 91 , 832 78 9 2

Bra nch es : A H M E D A B AD . B A N G A L O R E . B H O P A L . B H U B A N E S H WA R .C O I M B AT O R E . F A R I D A B A D . G H A Z I AB A D . G U WA H AT I .H Y D E R A B A D . J A I P U R . K A N P U R . L U C K N O W . N A G P U R .N A L AG A R H . P AT N A . P U N E . R A J K O T . T H I R U VAN A N T H A P U R A M .V I S H A K H A PAT N A M .