is 9300-1 (1979): statistical models for industrial applications, … · 2018. 11. 15. · is i...
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 9300-1 (1979): Statistical models for industrialapplications, Part 1: Discrete models [MSD 3: StatisticalMethods for Quality and Reliability]
IS • 9300 ( Part I ) • 1979
Indian Standard
STATISTICAL MODELS FORINDUSTRIAL APPLICATIONS
PART I DISCRETE MODELS
UDa 519-248: 658-52-011-2
@ Copyright 1981
INDIAN STANDARDS INSTITUTIONUANAE BHAVAN. 9 BAHAllUR SHAH ZAFAR MARG
NEW DELHI 110002
March 1981
IS : 9300 ( Part I ) • 1979
Indian StandardSTATISTICAL MODELS FOR
INDUSTRIAL APPLICATIONS
PART I DISCRETE MODELS
Quality Control and Industrial Statistics Sectional Committee. EO 3
ChairmanDn P. K. BOSE
Member«SHRt B. ANANTllKUISIINANAND
SHRI R. S. GlJP'rA ( Alternate)SHRt M. G. BUAUE
DJJUl:C'!'OU
ReprlsentingUniversity of Calcutta, Calculta
National Productivity Council, New Delhi
The Tata Iron and Steel Co l.td. Jarnshr dpurIndian Agricultural Statistics !(t>sr',ar(h Inst .ture
( rcxn ), Nvw Delhi
Central Statistical Organization, New DelhiIndian Jute Industries Research Association,
Cakutta
Indian Statistical Institute, CalcuttaNational 'fest House, CalcuttaIndian Association for Productivity Q}tality and
Rel iabititv, CalcuttaSTInt B. IhMATSINOKA (Alternate) •
SlInI T. R. !JUlt I Army Statistical Organiaat ion (~linistry ofDefence), N cw Delhi
SIIRt U. DUT'rA (Alternate)SHRIP. LAKSHMANANSInH S. MONDOr.,
Dn S. P. MUKIlJo-:UJEE
DR S. S. PILLA I (Alternate)Snnt D. DU'l''1'A The Indian Tube Cu Ltd, JaTnshedpuf
SHIII O. N. AOAn,vAT" ( Alternate )SHRI Y. GHOOSJr. I~IIAN NGEF Ltd, Bangalore
SHIU C. RA.JANNA ( Alternate)SllRI S. K. GUPTASUUI A. LAUlIU
SnRI R. B. BARMAN (Alternate)SHRI RAIUESli SHANKEl.. Directorate General of Inspection, Ministey of
Defence, New DelhiSHut N. S. SENOAR (Alternate)
Snnr '1'. V. RATNAM
Du D. RAY
The South India Textile Research Association,Coimbatorc
Defence Research and Dcvelopmr nt Organization,Ministry of Defence) New Delhi
( Conl,n"'td Oft /Jtll' 2 )
@ Copyright 1981
INDIAN S'fANDARDS INS'rITUTIO~
This publication i. protected under the India" Copyright Act ( XlV of 1957) andreproduction in whole or in part by any means except with written permission of thepublisher shall be deemed to be an infringement of copyright under the said Act.
IS I 9300 ( Part I ) • 1979
( Co"tinutdfrom pa" 1)
DR. B. N. SINOH,Director ( Stat)
Members Represellti"gSURI P. R. S~NGUP'.rA. Tea Board, Calcutta
SHIU N. RAMAOURA( ( Alternate)SHRI S. SunRAMU Steel Authority of India Ltd, New DelhiSHIn S. N. VOJlltA Directorate General of Supplies and Disposals,
New DelhiDirector General, lSI ( Ex-officio Member)
SecretarySJlRI Y. K. BUAT
Deputy Director ( Stat ), lSI
Process and Product Control Subcommittee, EO 3 : 6
ConvenerDB P. K. BOSE University of Calcutta, Calcutta; and Indian
Institute of Social Welfare and BusinessManagement, Calcutta
National Productivity Council, New Delhi
DJltECTOR
DR P. K. J)UT'fA
DR ~f. N. GOP\LAN
Snnt C. Y. KU.ISlINA MURTIDR S. P. MUKIlJ:I:RJEE
MembersSHIU B. ANANTIJKRfSlINANAND
Sum H.. S. GUP'rA (Alternate)Snur A. K. BIS\VAS The Consultative Committee of Plantation
Association, CalcuttaIndian Agricultural Statistics Research Institute
( leAR ), New Df'lhiPieco Electronics and EJectricals Ltd, BombayIndian Institu te of Technology, BombayIndian Statistical Institute, CalcuttaIndian Association for Productivity Quality and
Reliability, CalcuttaSnRI B. K. CHOWOUURI (Allernat,)
SURI RAM~8H SHANKER Directorate General of Inspection, Ministry ofDefence, New Delhi
SHUI N. S. SESOAR (Alternate)SIIRI J. S. SANOIIVI
Suur P. M. R \0 ( Altemate )DR D. RAY
SHRtS.SUDUAMU
Hindustan Petroleum Corporation Ltd, Bombay
Defence Research and Development Organization( Ministry of Defence ), New Delhi
Steel Authority of India Ltd, New Delhi
2
A.1'I.IEN.I>~EN"'r .N"O. 1 JULY 199ZTO
IS 9300 ( Part 1 ) : ..979 STATISTICA.L ~ODELS FOR.INDUSTRIAL APPLICA T.ON"S
PART -J DISCRETE IVIODELS
C.xi-f- !
I .LV [ 1117, ,'"up ( I ~ P ) ] d..x
~I-'
( Page 5, c~al.lse 3.1.1, fasr r,ne ) - Subs"t.l"tut:e Cor t:ncll- collec"tlon IS knovvn as event:' for Careknovv."1 as even1:s'
( Page 5, c1ause 3.-:1.2 ) - Substlt.ute ~he 1'oJlo"'Ing £or the eXisting clause:
.3.:1.2 Exhausrsve Evenr - The coJlect:u:>n of' all possible outco~es .an any "trIal is knovvlI a.sexhaust-ave event'
( .Page S, clause 3.1.4 ) - Subs"t.l"tu:te the Collovv.lDg Cor the existJng cla.use.
·3....4 r~.~·~I.'r~,bl. E ....e'.' - TI'Io IiCt. or ou~co.-.-acs r~~vour&..bJc lo u.~ cvcn~ .n 0 '\r.lul 18 "the sct~.r I'll o,.t~u... 'IC. ""I••el. on\ull \.J~o J,uppcnlns 01'" \.1'10 c::venl.-
( J~Clge S. clCluse 3.:1.6 ) - Subst.l\utc:: '\he 1""0110vv11"18 ror t11c cXlut.ng cluusc:
·3.~.G Lllcrel"e"..renl .z::venl3 - ~cvcral events arc said to be .Independent .Ie t.he chance oroccurrclIcc ( or non-occurrence) of" an cvcn~ .s no~ a.ffect.ed by lhc supplen-1ent.ary knovvledgeconcerning the occurrence ( or nou-oCCUrrCI"lCe ) or any of'" tJ"lC remaIning events'
( pezge 6, c~Quse 3.3.1 ) - ShIC~ "the mattcr of" 3.4 at "the cod of" thIS clause.
( Page 7, clause 3.3.2 ) - ShIC~ the Inatt:er oC 3.5 at the end of'thIs clau.se.
( Page 7, crouse 3.4 ) - I:>elct.e c.3.4'
( Page 8, cLause 3.5 ) - Dele"te c3.5'_
(Page 11, c'ause 5.1.1, last lJ.ne ) _ Subst.tute C( ~) o n-.p.· for C(_";) qn-. p.'.
( .Pezge IS, cLause 5.6 ) - Substl"tu.te the £ollovvIog Cor t:he eXisting equation
~~~2-*-y/np < 1 - P )
J .N (0.. 1 ) ax'X".-'IP- i~".r" c I - JI ,-
< 1'>~,ge J6. cLause 6.3 ) - ~nsert "the rolJovvang at "the end 01' t.hls clause:
CCo, nrlancc - T1.e covariance be~vvccn the Zl.b and thc JU& possiblc outcomes IS obt.alned a.s:
COlo" ( X'h X'J) -= ~np, PJ'
< Page 22, cLause 8.2, f6ne 3) - SubstItute "t.he £olIov.."'J.ng Cor "the eXisting .corID.ula Corstandard devJat.lon:
~,/v(r) =[ ~ ( 1-~) (Z='; )J1'
( Page 22. cIause 8.2.1 .. Lzne 5 ) - Su..bst.Itu"te "the £ollovvJ.ng Cor the eXlstang equation: _
.A. / (16) ( 16) 200-25'~Standard devJat.lon=, V 2S 200 1 -200 >< 200 -
( Page 29. rOlV 21, coL 10) - Substltut.e <O 7361' for cO 7461'
< ""...~ 1-> 3 )
1- ... 111 l c..d A-.llll_oll 1-... 1111018_ A.llg.. rl~ .. In
IS I 9300 (Part I) • 1919
Indian StandardSTATISTICAL MODELS FORINDUSTRIAL APPLICATIONS
PART I DISCRETE MODELS
o. FOREWORD
0.1 This Indian Standard ( Part I ) was adopted by the Indian StandardsInstitution on 12 September 1979, after the draft finalized by the QualityControl and Industrial Statistics Sectional Committee had been approvedby the Executive Committee.
0.2 Statistical models have been found to be of immense help in differentindustries. The behaviour of various characteristics in industrial processes may often be well explained by many statistical distributions to whichthey can reasonably be approximated. For example, the number ofdefectives in the case of a manufactured item may follow the binomialdistribution, whereas the number of defects per unit in a finished productmay follow Poisson distribution. The theoretical distributions like binomial, Poisson and exponential may readily be used in many cases forstudying the behaviour of various characteristics of manufacturingprocesses in the industry.
0.3 To facilitate easy application, this standard on statistical models isbeing published in two parts. Whereas the discrete models are coveredin this part, the continuous models would be considered in Part II.
0.4 This standard is one of a series of Indian Standards pertaining tostatistical quality control and acceptance sampling, Other standardspublished so far in the series arc given on fourth cover page.0.5 In reporting the result of a test or analysis made in accordance withthis standard, if the final value, observed or calculated, is to be foundedoff, it shall be done in accordance with IS : 2-1960·.
1. SCOPE
1.1 This standard describes certain discrete statistical models, theirpotentiality and application in industries with suitable illustrations.
• Rules for rounding off numerical values (rt:l';std).
3
IS I 9300 ( Part I ) .1979
1.1.1 The models covered in this standard are those of binomial,multinomial, Poisson and hypergeometric,
2. TERMINOLOGY
2.0 For the purpose of this standard, the following definitions shall apply.
2.1 Item - Ultimate unit of product or material on which inspection willbe performed.
2.2 Population - The totality of items under consideration.
2.3 Characteristic - A property which helps to differentiate between theitems of a given population. The differentiation may be either quantitative ( by variables) or qualitative ( by attributes).
2.4 Relative Frequency - The ratio of the number oftirnes a particularvalue ( or a value falling within a given class interval) is observed to thetotal number of observations.
2.5 Frequency Distribution - The relationship between the values ofa characteristic and their absolute or relative frequencies, The distribution is often presented as a table with special groupings ( class intervals)if the values are measured on a continuous scale.
2.6 Arithmetic Mean ( Average) - Sum of the values of the ObSCl vations divided by the number of observations.
2.7 Variance - 1\ measure of dispersion based on the mean sq11arcdeviation from the arithmetic mean.
2.8 Sample - Group of items drawn from a lot for inspection.
2.9 Sample Size -Number of items in the sample.
2.10 Defective - An C item t the quality of which does not meet thespecified requirements.
2.11 Fraction Defective - The number of defective items divided by thetotal number of items.
2.12 Percent Defectfve - The fraction defective multiplied by 100.
2.13 Degrees of Freedom - The number of independent componentvalues which are necessary to determine a statistic.
2.14 Goodness of Fit -A measure of the agreement between an observed distribution and a theoretical distribution, specified a priori or fittedto the observations.
2.15 Observed Vallie - The value of a characteristic determined as aresult of an observation or test.
4
IS I 9300 ( Part I ) • 1979
2.16 Null Hypothesis - The hypothesis (or assumption) of the equivalence ( or no difference) between the effects of rnethod (s) so that thesample(s) emanates From the same lot.
2.17 Level of ~igDificaDce- The probability of rejecting the nullhypothesis when it is true. Conventionally it is taken to be 5 percent or1 percent.
2.18 Statistic - A function of observed values derived from the sample,
~GENERALCONCEPTS
3.1 The concept of probability is essential in understanding the statisticaldistributions or models. Probability is effectively the mathematicalstatement of chance. It may be either:
a) 'a priori' consideration, or
b) 'frequency consideration • or c empirical definition'.
But before defining probability, it is necessary to become familiarwith various terms used in the definition.
3.1.1 Trial and Event - If an experiment repeated under identicalconditions does not give unique results but different outcomes, then theexperiment is known as a trial and the outcomes are known as events.
3.1.2 Exhaustive Events - The total number of possible outcomes in anytrial is known as exhaustive events.
3.1.3 Mutually Exclusitu Events - If an occurrence of anyone of theevents precludes the occurrence of all others, then the events are said tobe mutually exclusive.
3.1.4 Fauourable Event - The number of cases favourable to an eventin a trial is the number of outcomes which entail the happening of theevent.
3.1.5 Equally l.Jikely Eomts - The outcomes of a trial are said to beequally likely, if taking into consideration all the relevant evidence, thereis no reason to expect one in preference to other.
3.1.6 Independent Ecents - Several events are said to be independentif the happening ( or non-happening) of an event is not affected by thesupplementary knowledge concerning the occurrence of an)' number ofremain ing events.
s
18 I 9300 ( Part I ) • 1979
3.2 De&aitioas of Probability
3.2.1 C A Priori' DtJinition - C A priori' definition is based on the priorinformation. For example, without tossing a coin it is known that thereare two possible outcomes of head or tail if the coin is tossed. Both theseoutcomes are mutually exclusive and exhaustive and equally likely. Onlyone case is favourable to appearance of either head or tail. Thus, if atrial results in n exhaustive, mutually exclusive and equally likely casesand out of which m cases are favourable to an event E, then the probability of happening of E denoted by P{E) or p where,
mP(E) =p=-
n
3.2.2 Frequency or Empirical Definition - Suppose a trial is conducted
N times, out of which nl times an event E has occured. Therefore!!}
is the proportion of times that the event has occurred. By repeating thisexperiment with large and larger values of N, this proportion will stabilize to a certain value and this stabilized value is the probability of theevent.
If P(E) = ~ !otal nu~ber~f~ccurences.of the eventfatal number of trials
then the limit of peE) as N-+oo is the probability cp' of the event,
obviously P(E) lies between 0 and 1.P{E) == 1 means that the event is a certainity and peE) = 0 means theevent is impossible.
3.3 Laws of Probability - The two laws of probability are:
a) additive ( or total probability) law,b) multiplicative ( or compound probability) law.
3.3.1 Additive Law - If an event E occurs when anyone of the set ofmutually exclusive events E1, E., Ek happens then the probabi-lity of P(E) is given by:
P(EJ = P(E1) + P(E1) + + peEk)
This is known as the additive or total probability law.
3.3.2 Multiplicative Law - This law governs the joint occurrence ofevents. Let us take one event E to be the joint occurrence of events E1E2• ••• ••• ••• EK • Now joint occurrence of these events can be either J
a) independent, or
6
IS I 9300 ( Part I ) • 1979
b) otherwise; as E1 occurs, E. occurs depending on the occurrenceof El , E. occurs depending on the occurrences of E1 and E1 .••••
lastly EK. occurs depending on the occurrences of E1, E•.•••••EK- 1•
Thus mathematically, this law states that:
i) peE) = peEl) peE,) .•.••. P(EK ) for independent events, and
ii) P(E) - P(E1) P(E.IE) P(EsIEtE,.) ...••• P(EKIE1E• .. ••• EK - 1)Where P(E2/E1) means the conditional probability E I after E1has occurred and so on.
3.4 Example - Let the supplies of 5 000 and 3 000 components receivedfrom two firms X and r have the following break-up of quality:
Having a major defectHaving a minor defectHaving no defects
Total
Number ofCompontntl~-----__A- ~
Supplied By Supplied ByX 1"
140 60400 150
4 460 2 790
5 000 3 000
Total
200550
7250
8000
The probability of getting a component with a major defect from the
entire supply = 82g~ .. 0'025.
Knowing that the component is supplied by X, the conditional probabi
lity of getting a component with major defect == 5 1~00_ 0= 0'028.
Knowing that the component is supplied by r, the conditional probabi
lity of getting a component with major defect - 3~OOO = 0'020.
Hence the probability of getting a component with major defect from theentire supply - (probability of getting a component from X) X( conditional probability of it having a major defect) + ( probability ofgetting a component from r) X (conditional probability of it having amajor defect)
5 000 3 000- _._-- 4 X 0'028 +-- --- X 0'020
8000 8 000
7
IS : 9300 (Part I ) • 1979
II:=: 0'625 X 0-028 + 0'375 X 0'020c= 0'017 5 + 0-007 5
= 0'025
3.5 Esample - Three Jots of manufactured items contain 2, 5 and 3 percent of defectives respectively_One item is selected from each of the lot,What is the probability that selected items show 2 defectives and 1 nondefective?
The above example demonstrates the application of both the lawsof probability. Denoting defectives by B ( bad item) and non-defectivesby G ( good item ), it can be seen that 2 defectives and one non-defectivecan come from:
Lot 1Lot 2Lot 3
Case I
BBG
Case II
BGB
Case III
GBB
The probability for case I is the probability for joint occurrence of B,from Lot I, B from Lot 2 and G from Lot 3 (multiplicative law) andsimilarly for cases II and Ill.
Now the probability of obtaining a defective B from Lot 1 isP1(B) = 0-02
and probability of non-defective from Lot 1 isp}(G) == 0'98
Similarly P2(B) = 0'05, p.(G) == 0'95 for Lot 2
and Pa(B) -= 0'03, Pa(G) = 0'97 for Lot 3
Thus probability for cases I. II and III are:
Case I - P1(B) P.(B) Pa{G) == 0'02 X 0'05 X 0'97 = 0'00097Case II - Pl(B) P,,(G) Pa(B) == 0'02 X 0'95 X 0·03 == 0'000 57Case III - p](G) Ps(B) Ps(B) = 0'98 X 0-05 X 0'03 == 0-00147
Since the event of appearance of 2 defectives and 1 non-defectiveis possible when anyone of the three cases happens, the probability ofthe event will be the sum of the probabilities for cases I, II and III.
Thus the probability of obtaining 2 defectives and 1 non-defectiveis:
0-00097 + 0'00057 + 0'001 47 == 0-003 01
8
IS I 9300 ( Part I ) ·1979
This probability is quite small. Hence, occurrence of such an eventis very rare. Even from common sense it is evident that the possibility ofgetting 2 defectives out of 3 samples drawn from 3 different lots, eachhaving very few defectives is quite remote.
4. PROBABILITY DISTRIBUTIONS
4.1 Suppose a sample of n items is drawn from a lot of manufactureditems and the number of defectives in the sample are noted. The defectives in the sample can be from 0 to n. If N sets of such samples aretaken then we get frequency of samples having 0, 1, 2•••••••••n defectives.Let i denote the number of defectives in the sample and 11 the frequencyof the samples having i defectives. It is then possible to prepare thefollowing table:
TABLE 1 THE FREQ,UENCY OF DEFECTIVES
TOTAL
(1)
o12
n
RF.r.ATIVE I'-1tEQUENCY
11 fi IN(2) (3)
10 loIN11 fl/Ni2 rc»
lu fn/"V
PROBABILITY
(4)
PoPIPI
pn
4.2 The third column gives the relative frequency of samples having 0,1, 2 n defectives. By the frequency definition of probability,these relative frequencies can be taken as the probabilities Po, Ph I II •••P,.of Xl taking 0, 1, 2 n values.
4.2.1 Thus, in general if x takes values Xl, x, ..• •..... X.. with probabilities PhPt. .•.•••••. p,., the assembly of xi's with probabilities PI constitute theprobability distribution of the variable x, Since x takes only discretevalues, the probability distr ibution is a discrete one. Any variable whichtakes different values with different probabilities is called a random orstochastic variable.
IS I 9300 ( Part I ) • 1979
4.2.2 Since x takes all possible values with one or other probability PIobviously IpJ == 1.
4.3 Mean and Variance or Probability Distribution - Just like thefrequency distribution we can calculate the mean and variance of aprobability distribution. The mean of the probability distribution iscalled the expected value of the variable x and denoted by E(x). Thevariance of the probability distribution is denoted by V(x).
4.3.1 If x takes values Xl' Xs, ••••••••• x" with probabilities PI, Pl Pnthen:
nE(x) Ie: :E Pi Xl
i == 1
and V(x) = ~ P1 [%1 - E(x) JSi == 1
= i P1 Xl1 - [E(x) JIi = 1
5. BINOMIAL MODEL
ProbabilitySampler-------~------l
1st item 2nd item 3rd item
5.1 Consider the case of a production line inspection system in whichitems are inspected. Each inspection can have only two outcomes,whether the item is defective ( B ) or non-defective (G). The probability of getting a defective item is assumed to be constant, say JJ. Naturallythe probability of a non-defective item is q = ( 1 - p) by the law of probability. Let us now take a sample of 3 items for inspection. The numberof defectives say r in the sample can be 0, 1, 2 or 3. They can occur inseveral ways in the sample and are shown below along with their probabilities:
No. ofDefectives, r
0 G G G q8
1 G G B 1G B G ~ 3qlpB G G I
J2 G B B 1
B G B ~ 3qplB B G J
3 B B B p8
10
IS I 9300 ( Part I ) . 1919
5.1.1 Obviously total probability ( qS + 3q2p + 3qp" + ps ) = (q +P)3= 1 and the probabilities for r = 0, 1, 2 and 3 are the terms of thebinomial expansion of ( q + P r'. Accordingly for sample of size n, theprobabilities for r = 0, 1, 2..•. n arc the terms of the expansion ( q +P )n
that is q", ( ~ ) qn-l p, ( ~ ) q..-2p2, ...... ( : ) r respectively.
5.1.2 In general, the probability of, defectives and ( n - r) nan-
def 0 o(n) n!etectives IS r P' qfl-r = '( _ ) ,- p'qn-r, 0 n , .
and i (:) rr: ==1,=0
Thus when n independent trials are conducted, each trial resulting in onlytwo outcomes and the probability of success in each trial remaining constant, the probability distribution of nurn ber of success is said to followbinomial law. In the example considered above the distribution of r ina sample of n pieces which can be taken as equivalent to n independenttrials follows binomial law,
5.2 Cumulative Probability Cor Binomial Model
5.2.1 In binomial model, the probability of r defectives and ( n - , )non-defectives is given by:
Hence, the cumulative probability of having 0, 1, 2'.0 . up to xdefectives is given by the expression:
i ( : )prqn-rr=O
Varying x from 0 to n, one can get different cumulative probabilities.
For obtaining the individual and cumulative probabilities of binomial distribution a reference is invited to Appendix A and B respectively.
5.2.2 Example - A welded pipeline has a mean of five welds every100 metres, The probability of a weld being defective is 0'05. \Vhat arethe probabilities that:
a) no welds ( per 100 m ) are defective,
b) less than two ( per 100 m ) are defective, and
c) at least four (per 100 m ) will pass inspection.
11
IS I 9300 ( Part I ) • 1979
Here the probability of a defective weld = 0·05
and probability of a non-defective weld = 0'95a) Probability that no welds are defective = ( 0'95 )5
= 0·77b) Probability of less than 2 defectives I::: probability ( no defective)
+ probability ( 1 defective)
== ( 0 95 )5 + 5( 0·95 )' (0'05) c: 0'98c) Probability ( at least 4 passes) = probability ( no defective)
+ probability ( 1 defective)
= same as (b) = 0'98
5.2.3 Example - The probability of a single article being defective ina production process is 0'02. A sample of 10 is taken every time forinspection. At most how many defectives should be allowed if it isdesired that 98 percent of the samples should be passed by inspection?
Here p = 0'02 and n -= 10
Assuming that at most x defectives may be allowed, the probabilityof Ie defectives should be equal to 0'98, that is,
i( 10 ) ( 0'02 )' ( 0·98 )10-' = 0'98r = 0 r
where r is the number of defectives.
From the Appendix B we get the value of x = 1 corresponding top ==' 0·02 and n -- 10. Thus at most 1 defective can be allowed.
5.3 MeaD aad Variance
nMean II=: E(r) == ~ r ( n )p'qn-, = np
, = 0 rn
Variance -=- E [ r - E(r) ]1 = I (,-np )1 ( n )p'q,.-rr -== 0 r
== npqStandard deviation == V npq
5.3.1 Example - From a process producing 10 percent defectives, 30components are selected at random. What is the expected number ofdefectives and the standard deviation?
10Expected number of defectives = 30 X 100 = 3
12
Standard deviation = ,j 30 X
-,j-;O- X
-V 2·7= 1·64
IS : 9300 ( Part I) • 1919
10-(-10)100 1 - IOU
10 X 90100 X 100
50
5.4 Fitting of Binomial Model - It has been mentioned in 4.1 thatrelative frequencies for various values of the variate from the observeddata can be taken aCJ the probabilities of the variate taking these values.Thus for fitting a binomial model to a set of data, it is necessary tocalculate the theoretical frequencies corresponding to the observedfrequencies. The theoretical frequencies are calculated on the basis of thebinomial distribution fitted to the data. The following example willclearly show the different steps in fitting the binomial model.
5.4.1 Example - The distribution of 50 samples of 100 items eachaccording to number of defective items observed is given below:
No. nf Defectio« Items No. of Samples
( , ) ( frequency f )
o 81 122 133 94 75 06 17 0
50
Fit a binomial model to the above data. From the data, meannumber of defectives per sample
(0 X 8 + 1 x 12 + 2 X 13 + 7 X 0 )==c= 1'98
and therefore proportion of defectives in sample::: 1·98/100 = 0'0198 =:0·02 as the sample size is 100.
13
IS I 9300 ( Part I ). 1979
Taking this proportion of '02 as the probability of a defective thebinomial model can be set up. Thus the probability of T defectives and( n - r ) nondefectives in a sample of 100 is given by:
( l~ ) (0.02)' (0·98 )100-' = P(r)
Putting, = 0, 1, 2..•.•• the probabilites P(O), P( 1), P(2) ••••••.. , that is, theprobabilities of 0, 1, 2..,. ..• ..• defectives in the sample are obtained.Multiplying these probabilities by 50 we obtain the expected number ofsamples having 0, 1, 2 ••.•.•.••••. defectives as follows:
TABLE 2 OBSERVED AND EXPECTED FREQ.UENCIES OF SAMPLES
NUMBER OF DRFECTIVEITEMS ( r )
TOTAl...
(1)
o1234567
NUMBER OF SAMPIIES
~----------~----------1obser vcd Expected(2) (3)
8 6'612 13'513 13'79 s-t7 1 4'5 Io ~ 8 1'8 ~ 7'11 , 0'6 Io J 0'2 J
-----------_.-------50 50
5.5 Goodness or Fit From X" - Test - After calculating the expectedfrequencies for different number of defectives, their closeness to the observed frequencies are tested with the help of
( observed - expected )1 . .XI = I ---- -- -- ------ after pooling the adjacent classes
expected '
so as to make the expected frequency of each class at least 5. For theexample given in 5.4.1 the calculated value of X" which is 0·61 is muchless than 7'82, the tabulated value of X2* for 3 -= ( 5 - 1 - 1 ) degreesof freedom, at the 50;0 level of significance. Hence it is concluded thatbinomial is a good fit to the data.
5.6 The binomial model discussed earlier may be approximated by asuitable normal model]. If nand p are parameters of the binomial
*St't'Appendix A of IS : 6200 ( Part II )-1977 'Statistical test of significance, Part IIxl - test (first revision) ",
tSte Clause 9 of IS: i200 ( Part I )..1974 I Presentation of statistical data: Part ITabulation and summarization'.
14
IS : 9300 ( Part I ) • 1919
model, this can be approximated by a normal model with mean = np and
standard deviation e= V rlp-(-l --Pl- and the probability of getting Xl
successes in n trails is approximated by
Xl + l _Xl_-~I!_+ !Xt- ,I JV[np,'; np (l-P)] dx = (VI1Plf-=-P)
Xl - np - 1. .N ( 0, 1 )dx
v' np-(1=.pdNOTE - For Xl = 0, the integration is from - 00 to 1 whereas for Xi = n
the integration is from n- i to 00.
6. MULTINOMIAL MODEL
ProbabilityFrequency ofOccurrences
6.1 The multinomial model is a generalisation of binomial model. Thismodel refers to n independent trials of an identical random process inwhich there are k possible outcomes, unlike binomial model where onlytwo outcomes are possible.
k6.2 Let PIbe the probability of ith outcome. Thus }: PI = 1. Also
i-=llet Xl be the numbers of trials in which ith outcome has occurred.Obviously ~ Xl = n, The multinomial law gives the probability of aparticular set of values of the XJ regardless of the order in which thevarious outcomes occur. Thus the model can be represented in the following tabular Iorrn:
PossibleOutcomes
1
2
k
Total n
Pk----
I
The probability function for this law is thus defined as:
n !== l' , P1 Xl .•••••••• Pk XkXl XI ••••••••••• Xk •
IS
IS I 9300 ( Part I ) • 1919
The probabilities of different combinations of outcomes from nindependent trials are given by the various terms in the expansion ofthe multinomial (PI + Pl +Pk)D
6.2.1 This distribution is useful in industry when different grades of aproduct have to be judged.
6.3 Mean and Variance - The mean and variance of the ith possibleoutcome are obtained as:
Standard deviation
E (Xl) -= lIPl
V ( Xl) = nPlql
= Vnplql
6.4 Example - The item produced by a production process is classifiedas overweight, acceptable and underweight, the probabilities for which are0'2, 0'7, and 0'1 respectively. In a sequence of 10 independent trialscalculate the probability of 1 overweight item, 6 acceptable items and3 under weight items.
By applying the multinomial law we get the required probability as
ITI~!! 3! (0'2)1 (0'7)8 (0'1 )8
= 0-02
7. POISSON MODEL
7.1 Calculations based on binomial distribution are sometimes very laborious specially when the number of terms involved are many and numberof items, n is very large. A simple approximation may be obtained toany term of binomial and this approximation is called Poisson exponentialbinomial limit or briefly Poisson law or Poisson model. The larger thevalue of n and smaller the value of p, the more close is the Poissonapproximation. Thus when n tends to infinity ( very large) and p is verysmall but np is finite and constant, one can use the Poisson model.
The probability function of Poisson model can be written as:e-mmr
,1 ,0 ..... <;00
where m is the mean or expected number of successes (defectives)and r is the number of successes. The probabilities of 0 success, 1 success,2 successes ........• etc are given by:
m1e- mr», me-tn, -2-1-
16
IS : 9300 ( Part I ) • 1919
Again
co~
,==0
m'-,-=1
T •
1.1.1 Cumulative Probability for Poisson Model - In 7.1 it has beenshown that the probability of having r successes for Poisson model is
,-m m~. The cumulative probability of having 0, I, 2 up to x, .successes ( clefectives ) is therefore:
x e- m mr
,T
x varying from 0 to 00, one can get different cumulative probabilities.For individual terms of Poisson and cumulative terms a reference is invitedto Appendices C and D respectively.
7.1.2 Example - In a certain factory turning out razor blades, there isa small chance' of 0-002 for any blade to be defective. The blades aresupplied in packets of 10. Use Poisson model to calculate the approximate number of packets containing no defective, one defective, two defectives and at least two defectives respectively in a consicnment of 10 000packets.
The mean number of defective blades per packet:
= 10 X 0'002 = 0·02
Poisson model for r defectives in this case will be:e-· 02 ( ·02)'
r !---
With the help of Appendix C, it is obtained that
,-0 02( o-os )0probability of no defective blade = - - O!... = 0·080 2
t-O 02( 0 02 )1probability of one defective blade = -- - I! -- - = 0·019 6
e-O 02( 0·02 )1probability of 2 defective blades = ----2 ! --- ;-:; 0 000 2
17
IS : 9300 ( Part I ) • 1979
probability of at least 2 defective blades = Probability of 2 defective+ Probability of
3 defectives + .= 1 - Probability of no
defective -Probability of 1 defective
= 1 - 0'9802-0'0196== 1 - 0'999 8 = 0'000 2
Thus number of packets having:i) no defective c= 0'980 2 X 10000 == 9 1302ii) one defective = 0'019 6 X 10000 === 196
iii) two defectives == 0'000 2 X 10 000 = 2
iv) at least two defectives = 0'000 2 X 10 000 ~ 2
7.1.3 Example - Find the probabilty that at most 5 defective fuses willbe found in a box of 200 fuses if experience shows that 2 percent of suchfuses are defective.
Here mean m = 200 X 0·02 = 45 4r
e-4 --The required probabiJty = ~ , !r ==0
= 0·785
7.2 Mean and Variance - It has been shown ( see 7.1 ) that m in thePoisson model is the mean of the distribution.
Thus meanco m'~ r e-'" -- = m
r = 0 r l
00 m' ,and variance = ~ (, - m )1 e-fJI
--,- = m., = 0 r :
Standard deviation == Vmfor Poisson model, the mean and variance are the same.
7.3 Fitting a p.,isson Model - It has already been shown ( se« 7.2 )that the property of Poisson distribution is that its mean and variance aresame. Thus it is necessary to examine the mean and variance of theexperimental data before fitting a Poisson model to it. I f they varywidely Poisson model will not be a proper fit. In case the differencebetween mean and variance is of small order, this model should be triedand a proper goodness of fit test be carried out.
18
IS I 9300 ( Part I ) • 1919
1.3.1 Example - The following data gives the number of defective
screws in a box of 100 screws for ·100 boxes:
No. of defectivescrews per boxof 100 screwsfr) 0 2 3 1- 5 6 7 8 9 10 Total
No. of boxes 103 143 ~}8 42 8 1 2 0 0 0 0 400
Fit a Poisson model and carry out the goodness of fit.
Firstly calculate:
Mean = 1'32
and Variance = 1·30
Since lucan and variance are approximately same, a Poisson modelcan be titted, The expected frequencies are calculated from theexpression:
t-l·3~ ( 1'32 )'·~OO --- - -- --
T !by putting r = 0, 1, 2, ...........••.. 10. These are
107, 141, 93, 41, 13, '1, 1, 0, 0, 0, 0The various steps in the goodness of fit test arc shown in Table 3.
TABLE 3 COMPUTATION OF GOODNESS OF FIT - x2 TEST
No. OF DEFKCTIVJt1 () IlR nnvxn EXPJ~t'TED O-E (O-E)I (O-E)ljESCH,EWS P};U Box \') F'IU~QUIr.NCY«)) :FHEQUENCy(E)-----
(1) (2) (3) (4) (5) (6)
0 103 107 -4- Hi 0'151 143 141 2 4 0'032 98 93 5 25 0'273 42 41 1 1 0'021- B 13 5 25 1·925 4 1 4 1{; 2 ,
1 I7 0
,0
,8 0 >6 0 >5 0'20
I I9 0 I gJ10 o J
"rutal 400 400 2·59
Since there are 6 classes after pooling and two constraints, one forthe total frequency and the other for the mean, the available degrees of
19
IS I 9300 ( Part I ) · 1979
freedom ( df) == 4. The tabular value. of X" for 4 df at 5 percent level =9'49. Since the value of 1.2, calculated as 2·59 is less than the tabulatedvalue, the fit is taken as a good one.
8. HYPERGEOMETRIC MODEL
8.1 In the binomial model, the probability of getting a defective piecep is assumed to be constant. The model is still valid even if p is fairlyconstant. Thus while selecting 100 items from a lot of say 800 items, theprobability from one selection to the next is only approximately constantunless the sampling with replacement is done, that is, each item afterselection is put back in the lot before the next item is selected. If thereare 16 defectives in the Jot then the probability of a good item on the firstselection is 784/800 = 0·98. If the first one is good, the probability of agood item on the second drawing is 783/799 or if the first item is defective,the probability of getting a good item in the second draw is 784/799.Both of them are quite close to 0·98. Thus working on the assumption ofbinomial model is fairly justified. In fact binomial model is truly correctwhen the lot size is infinite but it still gives useful approximations when thelot size is 8 to 10 times larger than the sample size. But when the lot sizeis small, it is more justified to use the hypergeometric model which statesthat if there are D defectives in a Jot of N items then the probability ofgetting exactly' defectives in a sample of size n is given by:
P(T) ==
8.1.1 Actually hypergecmetric model is the case of sampling from finitepopulation whereas binomial is the case of sampling from infinite population. Hypergeometric models are useful in the calculation of exactprobabilities w hen lot sizes are small and the sample size forms a substantial proportion of the lot size. It is also needed when sampling is donewithout replacement. Direct calculation of the above probability is verylaborious. If the lot size does not exceed lOU, the tables of logarithms offactorials given in Appendix E may be utilized.
8.1.2 Example - In a lot of size 80 with 7 defectives a sample of size20 is drawn. Calculate the probability that the sample contains
a) no defective,b) one defective, and
·S" Appendix A of IS : 6200 ( Part II )-1977' Stitlstical test or significance: Part IIxl - test (fir,' "visic,. ). '
20
IS I 9300 ( Part I ) • 1979
c) two defectives.Here N = 80, D == 7, n = 20
Thus P(O) ( the probability of getting 0 defectives) is given by:
( '230) (07)73! 20! 60!
P(O) = ( ~~ ) = 20! 53! 80!
log P(O) II::: - 0-915 1P(O) = 0·1216
Similarly P(I) = ( I~) (I )._ = 0'3152
(;g)
( I~) (~) .and P(2) = ( :Z ) = 0'326 7
8.1.3 A shipment of 100 television picture tubes contains fivedefectives. If three tubes are randomly selected and given to a customer,find the probability that he will get (a) exactly one defective tube, (b)two defective tubes, and (c) all the three defective tubes.
Here N .. 100, n == 3 and D == 5.
The probability of exactly one defective is:
(95, (5)P(l) _ 2 ) 1
( l~)
:::z: 0·138 07
Similarly probability of getting exactly 2 defective tubes is:
( 9~) (~)P(2) = ( 100 '\
3 )
21
- 0·00588
IS : 9300 ( Part I) .1979
and the probability of getting all the 3 defective tubes is:
== 0·000 06
8.2 Mea. and Variance
Mean E(r) - ~D = np where p = ~ and fJ .. ( 1 - ~)
V . V nD ( D) N - n N - narrance (r) == N 1- N N _ 1 = "pq N - f
and Standard deviation = \/,1'(r) _ [ n nD ( 1_~) N - n._Jl/2eN N N- 1
V N-n- npg· ---N-l
8.2.1 Example - From a lot of 200 components containing 16 defectives,a sample of size 25 is chosen. Find the expected number of defectives inthe sample as also the standard deviation.
Expected No. of defectives = i~ X 16 = 2
Standard deviation = A / 25 (_~) ( 1 _ -.!~) X 200-=-25"V 200 200 200 - 1
= A I 25 X 16 X 184 X 175V 200 X 200 X 199
== ~r:6r81
= 1-27
22
IS :9300 (Part I).1979
APPENDIX A(Clause 521 )
BINOMIAL DISTRIBUTION INDIVIDUAL TERMS
lor n:: 2to 20 1nd p- 001( 001) 10, (005) 50
( :) PI (1- P)1-'
n V001 002 003 004 005 006 007 008 009 010 015 0'20 025 030 035 040 045 050
09801 09604 09409 09216 090~5 088]6 08619 0816~ 08281 08100 o722J 06400 05625 04100 04225 03600 o~025 0250000198 00392 0058Z 00768 009JO 01118 01302 01412 01638 01300 02550 03200 03750 o{200 04550 04800 o4QJO 0500000001 00004 00009 00016 00025 00036 000!9 00064 00081 00100 o0!~5 00100 00615 oOJOO 012~5 01600 olOb 02500
09703 09412 09127 08847 08574 08306 080H 07787 07536 0n90 06141 05120 04219 0H~O 02746 02160 0166~ 01250olJ294 00576 00847 01106 0135~ 01590 01816 02031 02236 0Hl0 03151 03840 04219 0HIO 04436 0H20 04QQt 0375000003 00012 00026 00016 00071 OOlOl 00137 00177 00~21 00170 0vSH 00960 01~6 01890 02389 02BBO 033H 03750
00001 00002 00002 00003 0000) 00007 00010 00034 00080 00156 OODO 00~29 005iO 00911 01250
09606 09224 08853 08493 08US 07807 01481 0716~ 06817 06561 05?20 04096 0316t 0HOI 01185 01296 00915 0062500388 00753 01O~5 01416 01715 01993 02252 02492 02713 02916 () J68J 04096 04219 0411~ 0l845 03456 0ll)9J 0250000000 00013 00051 00088 00115 001Ql 00254 OOn5 00i02 00486 OOQ75 015J6 02109 016~6 03105 03456 03675 03750
00001 00002 00005 00008 00013 00019 00027 000J6 00115 00256 00t69 00126 01115 01536 02005 0250000001 00001 00005 00016 00039 OOOBI 00150 00256 00410 00625
o 095100903908587081540771807339069570659106240 Oj90j OH11 03217 0237301681011600011800503003121 00480 00922 01328 01699 02036 02j4~ 02618 02866 03086 03280 03915 040)6 03955 03602 03124 02592 02059 015622 00010 00038 00082 00142 00214 00299 00~94 00198 00610 00729 01382 020~8 02637 0JOB7 0336~ 03456 03%9 031253 00001 00003 00006 00011 00019 00030 00043 OOObO 00081 00244 00512 00879 0131J 01811 0230~ 02757 031254 00001 00001 00002 00003 0OOO~ 00022 00064 0014ii 00)84 00488 00768 01128 011625 00001 00003 00010 00024 00053 00102 00185 00312
o9H5 08858 08330 07828 07351 06399 06410 06064 05679 05314 03171 02621 01780 01176 00754 00467 00217 0015600511 0108i 01546 01957 02321 o26H 02922 03164 0~310 03;43 03993 03932 03560 oJ01J 02437 01866 01359 Olmsa0014 00055 00120 0m04 00105 o0422 00550 00688 00833 00984 01762 02458 02966 0JHI 03280 03110 02180 02~44
o0002 00005 00011 00021 o00~6 0005j 00080 00110 00116 00415 00819 01318 0185~ 02~55 02165 03032 0312500001 o0002 00003 00005 00008 00012 00015 00154 00330 OOJQ1 00951 o13B2 01861 02141
00001 00004 00015 00044 001O~ 0alOS 00369 00609 00938o0001 00002 00007 00018 00041 o008~ 00156
(Contlnlltd)
23
IS 19300 (Part I).1979
BINOMIAL DISTRIBUTION INDIVIDUAL TERMS - rontd
II~ 001002 003 004 005 006 007 008 009 010 015 020 025 030 035 040 045 050
09321 08681 08080 01514 06983 o64a5 06017 05578 05168 04783 03206 o20g7 o13~j 00821 oOt90 00280 00112 0007800659 01240 01149 02192 02573 02897 03170 o339b 03518 03720 03960 0%70 03115 ll~m OlBlU 0130& 00372 0014700020 00076 00161 OO174 00106 00555 00716 00886 01061 01240 02097 o27~J 03111 Ojll1 o293J 02613 02140 01641
00003 00008 00019 00036 00059 00090 00128 00175 00230 00617 01147 01730 OUIlY 02619 02903 02918 0273400001 00002 00004 00007 00011 00017 00026 00109 00287 00577 00972 01442 01935 02300 o273~
00001 00001 00002 00012 00043 00115 00250 oOt6b 00174 01112 0164100001 00004 00013 00036 oOOal 00172 00320 00547
00001 00002 0000& 00016 00037 00078
09227 08508 078~7 07214 066H 06096 05596 05132 04701 04305 02725 01678 01001 00576 00319 00168 00084 0003900746 o1389 01939 02405 02793 03113 03370 03570 03721 oj826 03841 o3j55 () 2610 01977 01373 oOB9b 00548 00312oOOl6 00099 00210 00351 OOj15 001195 00888 01087 oilsa 01488 02370 o29Jb 03111 02965 oL)B7 olOGO 01564 0109100001 00004 00013 00029 0'0054 00089 00134 00189 oO~55 00331 00839 I)1468 o~07b 02)11 02786 02787 o2J68 o~188
00001 OOool 0OOO~ 00007 00013 00021 00031 00046 00185 o04SQ 00865 o13M 01815 02Jl2 o2G~1 02/34o0001 0'0001 00002 00004 oOOZG o00Q2 oO~H OOth7 00808 oIlJ9 01719 02188
00002 o001l 00018 00100 00217 00413 00703 o109~
00001 00004 00012 oOU~3 00019 00164 0031200001 00002 00007 00017 00039
09135 083)7 07602 06925 06302 o5730 05204 0472l 04279 03874 02316 01142 00111 00104 00207 00101 oDOW oOO~O
00830 01531 02116 02597 02985 o3192 03525 0369j 03809 o3R74 03b79 03020 o22J3 01516 () lOtH 00605 00339 0017600034 00125 00262 o0~33 00629 00840 01061 01285 01107 o1712 0l597 o30~0 03003 o hhH oJlh l oIhl2 01110 0070300001 00006 00019 OOOU 00077 00125 OOl8b 00261 00348 00446 01009 o170l o2JJG 02688 o~716 o210B omq oIb41
00001 00003 00006 00012 0'0021 00034 00052 00074 00283 00661 I) 11GB 01711 o~19 ~ 02108 02bOO o2iblo0001 00002 00003 00005 00008 00050 00161 o03H9 00735 01\81 01672 02128 OHbl
00001 00006 00028 o00U7 oOZIO OOlH 00143 01160 016410000] 00012 00039 00098 oOll2 o0~07 00703
00001 oOOOi 00013 oOO~5 00083 0017600001 00003 00008 00020
10 0 090# 08171 07374 06648 05987 05386 04840 04344 03894 03487 01969 o1014 00563 o0~82 o01~5 00060 o002j 000101 00914 01667 02281 02710 03151 03438 03643 03777 03851 0'3874 03414 02684 01877 oI~II 00725 oOtO] 00207 OOO1l82 00042 00153 00317 00519 00746 00988 01234 01478 01114 01937 o27~9 03020 02816 oH~5 (11157 o\lO9 o07b3 004391 00001 00008 00026 00058 00105 00168 OOHa 00143 00452 00)74 o1298 02013 02503 () 2668 o25~2 o21JO oIh6) 01l7Z4 00001 00004 00010 00019 00033 00052 00078 00112 00401 oOR81 o14GO 02001 0)117 02508 o2i84 o~O51
5 00001 o0001 00003 00005 00009 00015 00085 00264 oOJ8~ 01029 01536 02007 02HO 024616 00001 00001 00012 oOOj5 o01b2 003b8 00689 01115 o159b 020517 00001 00008 00031 00090 00212 oOl25 00146 011728 00001 oOOO~ 00014 oDOt3 00106 00229 004399 00001 oOOOi 00016 00042 00098
J10 00001 00003 oDOlO
(Contl"Utd)
24
IS: 9300 (Part I).1979
BINOMIAL DISTRIBUTION: INDIVIDUAL TERMS - Contd
_N~ 001 002 003 004 005 006 007 008 009 010 oIj 020 025 030 035 040 045 oJO,
11 0 08%~ oB007 o71j3 06382 05688 05063 04501 03996 03544 03138 01673 00859 o0~22 OOlQ8 00088 00036 oOOI~ 00005I 00)95 oINa o241l oZ9Z5 03293 o~5J5 03727 03823 03855 03835 o324U 02362 oIM9 009J2 oOJIB o02r6 o012J OOOH2 00010 o01R3 00376 00609 00867 oIlh o140J 01662 0190b 02131 o2Hbb 02953 02581 01998 Ol39J 00887 OOJI3 002693 00002 00011 oonn 00076 00137 00217 00317 00434 00566 007\0 01511 o221~ 02581 o~Jbtl 02254 01771 0/259 o080J4 00002 00006 00014 00028 oOO~D 00075 00112 00158 o05~6 01107 oOUI 02201 02428 02H5 0201)0 010115 00001 00002 00005 OOOOq 00015 00025 00132 oOlBa 00803 01321 01830 0207 023611 021566 00001 00002 00003 00021 OOO97 00268 00566 00985 01471 01931 02)167 00003 00017 00064 00173 00379 0001 01128 016118 00002 00011 00037 00102 00134 00462 008069 00001 00005 00018 00052 o0116 00269
10 00002 00007 00021 0005411 oOU02 00005
12 0 08864 07847 o6q~R 06127 05404 04759 04186 03671 03225 02824 01427 00687 00317 00138 00057 00022 00008 000021 o10H 01922 02175 03064 o341~ 03641 03781 o~8~7 038 7 o~7b6 03012 02062 01267 00112 o03G8 00174 00075 000292 00060 00216 00418 00702 onq88 o1230 01565 oIA35 02082 02301 02lH o2815 02323 01178 01088 00639 00339 001613 oOool 00015 00045 00098 00173 00272 0'O39~ 00>12 00686 001152 01720 02%2 02581 0239, 01454 oIH9 o092i o05174 00001 00003 oOOOq 00021 00039 00067 o010~ 00153 00213 00683 01329 o1936 o2H1 02367 02128 01700 012085 00001 00002 00004 00008 00014 00024 00038 00193 00532 o1012 01585 oiON 02170 o2llJ 01434fi 00001 00001 00003 00005 00040 00155 00401 00192 01281 o1766 021H 022167 00001 00006 00033 00115 00291 00591 o100J 01489 019348 00001 00005 00024 00018 00199 00420 007b2 o12089 00001 00004 00015 00048 00125 00277 00537
10 oOOOl 00008 OOOh oGOb8 0011511 00001 00003 01010 0002912 00001 00002
13 0 08775 07690 06730 05882 05133 04474 03SQ1 03383 0291, o?i42 01209 00550 00238 00097 00037 00013 00004 00001I 01152 02040 o270b 03186 03512 03712 o380Q o3HU 01773 0%72 02774 01787 01029 oOJ40 DONI 00113 00045 000162 00070 00210 oOJ02 0071)7 01109 o1~22 01720 01995 o223!} 02+48 02937 02680 o10SQ oI~88 0083b 001,3 oOl20 o009~3 00003 00019 00057 00122 00214 00333 oOP5 00636 00812 00997 01900 02157 02517 011B1 011111 o1107 oOb60 o03~94 00001 0000+ oOOI{ 00028 00051 00089 00138 00201 00277 00838 o1535 02097 02337 02222 o1fl45 o13JO oOR725 00001 00003 00006 00012 00022 oOOl6 0005' 00266 00691 01258 01803 0215+ o)21 ~ 01989 01)716 00001 00001 00003 00005 00008 00063 00230 00559 01030 01516 019(8 02169 020957 00001 00011 00058 00186 OOWI oOU3l 01312 o1775 020958 00001 00011 00047 00142 oO~36 00056 o1009 o1~719 00001 00009 00034 00101 00243 00495 oOB73
10 00001 00006 00022 oOOn5 00162 0034911 00001 00003 00012 00036 0009512 00001 oOU05 0001613 00001
14 0 08687 07536 06528 05647 04877 04205 03620 03112 02670 02288 01028 00440 00178 00068 00024 on008 00002 00001I o1219 02153 02827 03294 03593 03758 03815 03788 03698 03559 02539 01539 00832 00407 00181 00073 00027 oOOOq2 00081 00286 00568 00892 01229 01559 01867 02141 02377 02570 02912 02501 o1802 01134 0063+ 00317 00141 000563 00003 00023 00070 00149 o02i9 oOSQ8 00562 00745 00940 01142 02056 02501 o2W2 01943 01366 00845 00462 002224 00001 00006 00017 00037 00070 00116 00178 o025b 00149 00998 01720 02202 02290 02022 01549 o10~0 006115 00001 00004 00009 00018 00031 00051 00078 00352 00860 01468 01963 02178 02066 01701 012226 00001 00002 00004 00008 00013 00093 00322 00734 o12h2 o1759 o20b6 02088 018137 00001 00002 00019 00092 00280 00618 01082 01574 01952 020458 00003 00020 00082 00232 00510 00918 o13qB oIBl39 00003 00018 00066 oOIR3 00408 o07b2 01222
10 00003 00014 00049 00136 00312 oOb1111 00002 00010 00013 00093 0022212 00001 00005 oOOlq 0005613 00001 00002 oOooq14 00001
( Contrrr/lfd)
2S
IS I 9300 (Part I).1979
BINOMIAL D1STRmUTION I INDIVIDUAL TERMS - CDntd
II~ 001 002 003 oO~ 005 006 007 008 009 010 015 020 025 030 035 040 0'45 050
15 0 08601 07386 06333 05421 04633 03953 03367 02863 02430 020,9 00874 00352 00134 00047 00016 00005 00001 -I 01303 o2?61 02938 ol368 036)8 03785 03801 03734 03605 03H2 02312 oJJI9 00668 00305 00/26 00047 00016 000052 000)2 00323 00636 00988 o/348 o16YI 01003 o2173 02496 02669 02856 02309 01559 oOQl6 00476 o(P19 00090 000323 00004 00029 00085 0018 00307 00468 00653 00857 01070 0128, o218t 02501 02252 01700 01110 0034 00318 001394 00002 00008 00(122 00049 00090 00148 00223 oOJ17 o0~28 01156 01876 02252 02186 01792 01268 00780 00417, 00001 00002 oOOOb 00013 00024 00043 00069 00105 00449 oI03l oIb51 02061 02123 01859 01404 oOQIG
6 00001 00003 00006 00011 00019 00132 00430 00917 01472 01906 o2OG6 01914 015277 00001 00001 00003 00030 00138 00393 OOBI1 o1319 01711 02013 o196~
8 00005 00035 00131 00348 00710 oIIBI 01647 oIQ649 00001 00007 00034 00116 00298 0061l 01046 o1517
10 00001 00007 00030 00096 00245 00515 0091611 00001 00006 o002t 00074 00191 0041712 00001 oOOO~ 00016 00052 0013913 00001 00003 00010 0003214 00001 0000515
16 0 08515 07236 06143 05201 OHOI 03716 03131 o263~ 02211 01853 00743 00261 00100 00033 00010 00003 00001 -1 01376 02363 o30~0 o14h9 o3i06 0379, 03771 0366) 03499 03294 02097 01126 o05i, 00)28 00087 00030 00009 000022 00104 00362 o070J 01014 01463 o1817 02129 o23JO 02596 02745 02775 02111 01336 00n2 oOi53 00150 00056 000183 oOOO~ 00034 00102 00211 00319 o05~1 00748 00970 01198 o1423 02285 02461 02079 01461 oORR8 oOl8 00215 000854 00002 00010 00029 aOOnl 00112 o0lR3 00274 00385 a0514 01311 02001 a2252 020W 01553 oloa o057l o01785 00001 00003 ooaog 00017 o003J 000)7 oOOql ooln 00555 01201 01802 02099 a2008 01623 01123 oOb676 00001 00002 00005 00009 00017 00028 00180 00550 01101 01649 01982 o19R3 o16B4 oIll27 00001 00002 00004 00045 00197 00124 01010 015'4 o186~ 01969 017468 00001 00009 00055 o01Q7 o04R7 oOQ23 o1417 01812 019649 00001 00012 00058 a0185 00442 oOU40 01318 01746
10 00002 00011 00056 00161 oON2 00755 0121211 00002 ooon 00049 00142 00337 0061>712 00002 00011 00040 00115 0027813 00002 00008 00029 0008514 00001 00005 0001815 00001 0{XX}216 -
17 0 o842q 07093 05958 04996 04181 03493 02912 02423 02012 01668 00631 00225 00075 00023 00007 OOOO? - -1 01447 024(;\ 03133 o35~9 03741 03740 03726 03582 03383 03\50 01893 00957 o042b 00169 00060 011019 0000, 000012 00117 o04<J2 00775 01180 01575 01915 02244 02492 02677 02800 02673 oIql4 01136 00581 o0'1i0 00102 000\5 000103 00006 00041 00120 00246 0041; 00618 00844 01083 01324 01550 02359 01593 01893 01145 00701 00141 00144 000524 00003 00013 OOOlb 00076 00138 00222 oOl30 o0~58 00605 01457 02093 02209 01868 o1320 o07tl6 oO~II 001825 00001 00004 00010 o00?3 00044 00075 00118 00175 00668 o13bl 01914 02081 o1849 01379 001\75 oO~72
6 00001 00003 00007 00013 00023 oOOgq 00236 00680 01276 01784 01991 01839 01432 009H7 00001 00002 00004 00001 00065 00267 o06€8 01201 01685 01927 01841 014848 00001 00014 00084 00279 00644 01134 o16n6 o1883 018559 00003 00021 00093 00276 00611 o1070 01540 0181;
10 00004 00075 00095 00263 00571 01008 o14r+11 00001 00005 00026 00090 00242 00525 OOllH12 00001 00006 00024 00081 o0?11 0047213 00001 00005 00021 o00r8 0018214 00001 oOOOt 00016 0005215 00001 00003 0001016 00001
I17 -
(Conhn",d)J
26
BINOMIAL DISTRIBUTION I INDIVIDUAL TERMS - ContdIS I 9300 (Part I)-1979
"I~ 001 002 003 004 005 006 0'07 008 009 010 015 020 025 030 o3J 0'40 045 050
18 0 08345 06951 05780 04796 03972 03283 o270P o292q 01831 01501 005% oOIBO o005u oOOIG 0000+ orool - -1 01517 02554 03217 03597 o37b3 03772 03bb4 o3~a9 o32bO 03002 o170f 00811 OOJm oO!2b OOOi2 oUOl2 00001 000012 00130 004+3 00846 o12H oh8l 02017 02318 02579 0274\ o2B11 025511 o172i 0095U oOIJH 00190 ooaw oOOll ol)(J()b3 00007 00048 00140 00283 00473 oOb91 oU942 01196 o144b o16BO 02100 02297 01704 o IU ~o 00117 o02~b 00015 000314 00004 o001h 00044 U009J o01b7 002bh 00390 00136 00700 01592 02153 02130 I) 118\ I) 1101 o()bl1 00291 001175 00001 00005 0001+ 00030 000% 00095 00118 00218 00787 01:107 01983 oJll7 o Ibb4 0111h nObbb oOJZ7b 00002 00004 o(lOUl 00010 00012 00052 aOmI 00810 0113b o IHi3 01441 o 11151 oIIBI oU,087 00001 00003 ooon5 o(l010 00091 oOi,O 00820 013"6 01792 o 1892 () IUJ7 oW48 u0001 00002 00022 oO\2(l oOllh 0(1\\1 oH21 01711 OWh4 010099 oooo~ oOOH (l01H oOlRb o071~ o128i oI{)J~ 0185510 00001 00008 00042 OUUg oOl85 00771 01218 01&69II 00001 OUOIO OOOlh 00151 00374 007H a 121~11 00002 ~ 0)12 00047 0014-5 0035l o070B13 00002 00012 oOOi5 OOIH 0032714 00002 OUOII 00039 0011715 00002 00009 0003110 00001 0000617 0000118 -
19 0 08262 o6BI2 05606 o.w04 03774 o30B6 o251q 02051 o1666 013;1 00456 00144 00042 00011 00003 00001 - -1 01586 02642 o12q4 o3b45 03771 03143 03102 03389 03131 02852 01529 oOhS1 o02b8 000113 00029 onOOB 00002 -2 00144 o048J 00917 01367 01787 02150 02440 o2ln2 02787 02852 02428 01;40 00803 oO~5H 00138 Daotb oOU13 000033 00008 o005b 00161 00323 o053~ 00778 01041 01307 01562 o1796 02428 02182 01117 001169 00422 o017S 000112 000184 00005 00020 00054 00112 00199 00113 00455 00618 00798 01714 02182 02023 (lUlll 00909 o0lb7 00201 OOOH5 00002 00007 00018 00038 00071 00119 00183 00266 oOQ07 011>36 02023 0/91h O'11{)8 o041{ oOtlJ7 002226 00001 00002 00006 00012 00024 00042 o006Q 00374 o0'}55 o1571 01916 01814 oItll o014Q 005187 00001 00002 00004 oOOOB 00014 00122 o014~ oOQ74 01525 01011 Ol7Q7 o1143 004b18 oUOOI 00001 00002 00032 oOllb 00487 oOlJ81 oHn9 01797 01771 o14H9 oOU07 00051 00148 00114 oOQ80 o 14b~ 01771 0176210 00001 00013 oOOh6 00220 00528 ()(m6 01149 Ol7lJl11 00003 00018 00077 00233 00;3) 00970 oIHl12 00004 00022 00083 00237 005)9 0016113 00001 00005 00024 00085 00233 0051814 00001 00006 oOOH 00082 o022115 00001 00005 00022 oOOH16 00001 00005 000181700001 0000318 -19 -
20 0 08179 06676 05438 04420 03585 02901 02342 01887 01516 01216 00188 00115 00032 00008 00002 - - -1 01652 027L5 03364 03683 03774 03703 03526 o~282 03000 02702 o13fi8 00576 00211 (lOObS oDOlO 00001 00001 -2 oOlSq 00528 00988 01458 01887 02246 02521 02711 02818 02852 02293 013&9 00&69 o02i3 00100 oOOB 00008 oOOO~3 00010 OOOh5 00183 00364 00596 oOBbO 01139 01414 01672 o 1901 ol428 02054 a133q o07lG 00323 oom () OOHl 000114 00006 00024 00065 00133 00233 o03b4 00523 00703 ooaqa o 1821 02182 01897 o!JOt o07~8 003 )1) 00139 000405 - 00002 00009 00022 00048 00088 00145 00222 o0~19 01098 o1746 02023 01739 01272 o074b o01bJ oOllQ6 00001 00003 00008 0001' 00032 o00j5 00089 00454 01091 o168G o141b 01712 01244 o074h 003707 00001 00002 00005 00011 00020 oOIGO 00545 01124 o164l OIRH OW,lJ 01221 007398 00001 00002 00004 00046 00222 00609 01144 oIbH 01797 oIb)3 01201I)00001 00011 00074 00271 00654 01158 O11()7 01771 oIb0210 00002 00020 00099 00308 00686 01171 o15qj 0176211 00005 00030 00120 00336 00710 01185 o160l12 00001 00008 00039 00136 00355 00727 0120113
00002 (j0010 00045 00146 00366 007391400002 00012 00049 00150 0037015
0'0003 00013 00049 00148Ib00003 00013 0004617
00002 0001118 - 0000219 -20 -(CDntlnu,d)
27
IS 19300 (Part I).1979
APPENDIX 1B(Clauses 52.1 and 5.2 3)
BINOMIAL DISTRIBUTION: PARTIAL SUMSfor n=2to 20 and p=001( 001)0 1Q( 005)050,
~ (: )pq 1- P)N-IX=O
I I'r\ 001 002 003 OOl 005 006 007 008 009 010 015 020 025 030 035 040 045 050
-----------09801 09604 09W9 09216 09025 o88J6 08649 o846i 08281 08100 07225 06400 05625 04QOO 04225 03600 03025 0'500o9999 09496 09991 o9<lSl 09975 o9Jh4 o4{}51 09936 09919 09900 09775 o9boo 09371 oqlOO 08771 08100 07475 0750010000 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 10lOO 10000 10000 1UUOO
09703 0q412 09127 088H 08571 08306 o804l 07787 07531> 01290 06141 03120 01219 03430 02746 02160 01664 01250o99J7 09988 09974 01953 04928 o98Yh 09LLO 09818 09772 09720 09392 089(0 08438 07840 07182 0(4kO 01718 05COOI0000 1OUOO 01000 09999 09949 09498 09497 09J95 OQQ9J o99~0 o99b6 09920 oYU44 09730 09171 09GO o{0l9 oJi50
10000 10000 10000 I tOOo 10000 10LOO 100(0 10000 10000 10000 10000 10000 10000 10000 10000
09606 0QZ24 08853 08493 08145 o7C07 07481 07164 o68J7 06561 05220 04096 03164 02401 0\785 0129h 00915 00621o9~9~ 04977 09948 09904 0981)0 a~1!01 097B 0915& 01170 09H7 08905 081Ql 07383 06517 05630 Of52 03{1O OJllS10000 10000 0Y999 o99ga 09J95 09992 Og987 oJgBI o9Y7l 099b3 09880 o11728 09492 091(3 08735 oHl08 o 585 OW7:>
10000 10000 10000 10000 10000 10000 o919q 09499 09993 09984 09%1 oqll}y o~l/O OQ744 Ol5lJO 037510000 10000 1UOOO 10000 10000 10000 If 000 1COOO 10000 ILOOO
o9ilO 0q039 08587 08154 07738 o733q Om)7 06m OlJ'to o590j 04437 03277 02373 01681 ollhO 00778 00J03 00312a99QO 09962 09915 o985L oJ771 09681 OQJ75 09451 0lJ32b 09185 o81J2 OH73 o6~2B 05282 01281 03370 02Jb2 018h10000 09999 09997 09994 09988 oQ980 09969 OQ9JJ o(N37 (9)14 09734 oQ421 08435 O'83W 07648 ob82h OSQ31 oJoon
10000 10000 10000 10000 09999 aqQ99 09998 09QQ7 09995 09978 09913 oQ844 o9192 091(0 09130 08(88 0812510000 10000 10000 10JOO 10000 09999 09Q97 04990 o947b 0°947 oC8r8 OCRIJ OlJL88
lOUOO 10000 10000 10000 10000 10000 10000 10000
09415 08858 08130 078 18 07351 06899 06+70 06064 0)679 05314 03771 o2b21 01780 01176 00754 00167 00277 OOl5b09985 09943 09875 0978l 09672 o9Al o~)Q2 o~217 09148 08857 07765 065,4 o53~9 04202 031Ql o23~3 o1~36 0109410000 09998 o999J 09988 09978 o9911 09942 09915 OQ882 0981! 09527 08011 o830b o7U3 06471 05441 oill) 01138
10000 10000 10000 o9ql9 o949H 09997 09Q95 OQQ92 09987 o99H OY830 0%24 09295 o832b oBlOB 07447 ob5bL10000 10000 10000 10000 10000 OQ999 0999& 09984 o9Q,4 oqaql 09777 09590 oQ108 oSqOh
10000 10000 09999 09998 09Q93 09Q82 09959 OQlJI7 OQllH10000 10000 10000 10000 10000 10000 10000
09321 08681 a80BO 07114 o698J 06181 OGOl7 o157R oJI6q 01185 03206 o20Q7 01331 00824 00490 o(llro 00152 0007309980 09921 09829 o970lJ 091J6 o9l~2 09107 o897~ oB74J 0050) 07166 o57h7 04449 09194 o23JU o151 It o101~ 00('510000 0Q997 09991 09900 09962 09937 o1I~()3 09BGO o9a07 I)9/43 09'62 03520 07564 06471 0531)3 04199 031U (llbb
10000 10000 09999 o999J 09Q9h 09993 09988 o9~o2 o9~7j 09879 o96G7 o92Q4 08740 oHOOl 07102 o(on ~ o loon10000 10000 10000 10000 o99911 09999 oq998 o498H 09951 09871 09712 09444 011037 08471 oi71 ~
10000 10000 100ClO o999q 09996 09987 09962 OqQIO 09RI2 09613 OQ3510000 10000 09999 09998 a9Q9i o99H4 09Qf3 0lQ22
10000 10000 10000 10000 10000 10000
(Conltnld)
28
IS I 9300 (Part I).1979
BINOMIAL DISTRI,UTION I PARTIAL SUMS - CDntd
II~ 001 002 003 004 005 006 007 008 009 010 015 020 025 030 035 040 04, 050
0 o9227 08508 07837 07214 06634 06096 05596 05132 04703 04305 02725 01678 01001 00576 00319 00\f8 00084 000391 09973 0l897 09777 0(619 09428 092(8 08965 08702 08423 08131 ob572 05133 03671 02553 oIt91 01014 OOu32 003522 o9999 09q96 09987 o9t6q 09942 09904 o~853 09789 09711 o9tl9 08948 o79b9 06785 05518 04278 03154 02201 014413 10000 10000 09999 09998 oG996 09993 09987 09978 09966 o995G 09786 09437 08832 08059 07064 05941 04170 o363i4 10000 10000 10000 10000 09999 09999 09997 09996 oqq71 09896 oq727 09420 08939 082t 3 073% 0I %75 10000 10000 10000 10000 09998 09988 09958 o~887 oq747 09)02 0ql 11 08~55
6 10000 09999 09996 o9Y87 09964 09915 09BIQ 096487 10000 10000 09999 o9998 09993 09983 0996I8 10000 10000 I0000 I0000 I0000
09135 08337 07602 06925 06302 05730 05204 04722 04279 03874 02316 01342 00751 00404 00207 00101 00046 0002009966 09869 09718 09522 09288 09022 08729 08417 o8C88 07748 05995 04362 03003 01960 o1211 00705 00385 00195o9999 09994 09980 099'5 09916 09862 09791 09702 0995 0940 08591 07382 06007 04628 03373 02318 01495 0089810000 10000 0q99q 09997 09994 09987 09977 09963 09943 09917 09661 09144 o8i43 07297 06089 04826 03614 02~39
10000 10000 10000 09999 o999~ 09997 09995 09991 09944 09804 09511 09012 08283 0i334 06214 050001LOOO 10000 10000 10000 09999 oq9q4 09969 o99CO 09747 o9414 0Q006 08~4l 074hl
10000 10000 09997 09987 09Q57 09888 097JO 09502 OqlOl10000 09999 09996 o9986 09962 0Q909 09R05
10000 10000 o9999 09997 09992 09980I0000 I0000 I0000 10000
10 0 09014 08171 07374 06648 05987 05386 04840 04344 03894 03487 01969 01074 00563 00282 00135 00060 00025 00010I 09957 09818 09655 09418 09139 08824 08483 08121 07746 07461 05443 03758 02440 01493 008&0 0046t 002~1 00107~ 09999 09991 09972 09938 09885 09812 09717 09599 09460 08893 08202 06778 05256 03828 02616 01673 00996 005473 10000 10000 09999 o999b 09990 09980 099M 09942 09912 o987l 09500 08791 07759 o649b 05138 03823 02660 017194 10000 10000 09999 09998 09997 09994 09990 09984 09901 09672 09219 11 84q7 07515 063~1 05044 037105 10000 10000 10000 10000 o999q 09999 09986 o993b (19803 09527 09051 08338 07384 062306 10000 10000 09999 09991 09965 09894 oQ740 0Q45l 089RO 0R28\7 10000 09999 09996 09984 o9952 09877 09726 094~3
8 10000 10000 09999 o9995 09983 0995j 098939 10000 10000 09999 09997 09990
10 10000 10000 I0000
11 0 o8953 0'8007 07153 06382 05688 05063 04501 03996 03544 03138 01673 00859 00422 00198 00088 OOO~6 00014 000051 oQ948 0qa05 09587 09308 08981 08618 08228 07819 07399 06974 04922 03221 01971 01130 00606 00302 00139 00OJ92 o9qqa 09988 09963 09917 09848 09752 09630 09481 09305 09104 07788 06174 04552 03111 02001 01189 00652 003273 10000 I0000 099qa oQgqg 09984 09970 09947 09915 09871 09815 09306 08389 07133 o569b 04256 o29b3 01911 0113J4 10000 10000 ~q999 o9q97 09995 o99QO 09983 09972 09841 09496 08854 01SQ7 06b83 053'8 03971 027445 10000 10000 10000 09999 09998 09997 09973 09883 09657 09218 08513 07535 06331 05000fi 10000 10000 10000 09997 09980 09924 oq784 oq499 0900h 0R26l 0725h7 10000 09998 09988 09QS7 o9q78 09707 09390 0B8678 10000 o9J99 o999~ 09980 09941 09852 096739 10000 10000 oq948 09993 09978 099B
10 I0000 10000 09998 0999511 10000 10000
(Contlnutd)
29
IS 19300 (Part I) .1979
BINOMIAL DISTRIBUTION I PARTIAL SUMS - Contd
ftl~ 001 002 003 004 005 006 007 008 0'09 010 015 020 025 030 035 040 045 050
12 0 08864 07847 06938 06127 05404 04759 04186 03677 03225 02824 01422 00687 00317 00138 000)7 00022 00008 000021 09938 09769 09514 09191 08816 03405 01967 07513 o70Jl 06590 o44~5 oZ749 01584 00350 00124 00196 00083 00032l o99ga 09985 09952 09893 09801 09681 09532 09348 09134 08891 073)8 05583 03)07 02128 01513 008H o0411 001933 10000 o9JY9 09997 09940 olJ978 099)7 09925 o93UO 09020 09744 09078 07946 06488 049)5 03167 02253 01H5 007304 10000 10000 o9J99 09993 olJ996 oqq91 09984 09973 09957 0°761 09274 08421 072H 05833 04382 03Da 019385 10000 10000 10000 oYlll 09998 09997 09995 o9~)4 09806 09450 o8n22 07873 Obb52 05269 038726 10000 10000 10000 09999 09993 09961 09857 09614 09154 08H8 07393 061287 10000 09999 09994 09972 09905 09745 09H7 08883 080628 10000 09999 09996 09983 09944 09847 09644 092709 10000 10000 09998 09992 09912 09921 09807
10 10000 09999 09997 09989 0996811 10000 10000 09999 0999812 10000 10000
13
0 08775 07690 06730 05882 o51~3 04474 o389~ 03383 02935 02542 01209 oOJ50 00238 00097 00037 00013 00004 000011 oq928 09730 o943b 090h8 08646 OB18b 07102 07206 06707 o6113 03983 02336 01267 00b37 o029b 00126 00049 000172 09997 09980 09938 0986) oq715 o910U I)9t2l 09201 08946 08661 06920 05017 03326 02/))) 01132 00579 00269 001123 10000 09999 09991 09986 09909 oQ9iO oqH97 09837 09738 09658 08820 01473 05843 o+20b 02783 () Ib8b 00929 004614 10000 10000 09999 09997 o99q1 09987 09976 09959 0993) 09658 o910J 07940 06511 oJOOJ 03130 o2179 013345 10000 10000 oq999 09999 09997 09991 09991 099)) 09700 09198 o8jl6 07159 057H 04208 029056 10000 10000 10000 09999 09999 09987 09930 09757 09376 08705 07712 06437 050007 10000 10000 09999 09988 09944 09818 09)38 09023 08212 070958 10000 09498 09990 o99bO 098N 09079 09302 086669 10000 09999 o9q93 09975 omz 09797 09539
10 10000 09999 09997 09987 099)9 09888II 10000 10000 09999 09995 0998312 10000 10000 0999913 10000
140 08687 07536 06528 o56~7 04877 04205 03620 03112 02670 02288 01028 00440 00178 00063 00024 00008 00002 000011 09916 09690 09355 o89~1 o8~70 o79b1 OH36 06900 06368 01846 o3Sb7 01979 01010 00475 oOl05 00081 00029 000092 09997 09975 o9913 09833 o9199 o9S2l 09302 o904l 08745 08416 06479 04481 02811 01608 00839 00398 00170 000653 10000 09399 o99J4 o99BI 09911 o99lO o9Hn4 09186 oQ685 o9~59 o85~5 o69R2 0,213 o~~5l 02205 01243 o063Z 002874 10000 10000 09998 09996 09990 09980 o99b5 09941 09908 09533 oU702 07415 0)842 042'7 02793 oIb72 008985 10000 10000 09999 09993 09996 0999 ' o99H5 09881 09161 08883 07805 o6i05 04859 03173 ol1206 10000 10000 10000 09999 o999B 09978 09884 09617 090b7 08164 0692) 05461 039537 10000 10000 09997 09976 09697 09685 09247 o8~99 07414 060478 10000 09996 09978 09917 09757 09417 08811 078809 10000 09997 09983 09940 09825 09574 09102
10 10000 09998 09989 o99bl 09886 0971311 10000 09999 oq994 04978 0993512 10000 o9q99 09997 0999113 10000 10000 0999914 10000
(Continued)
seq
~O
BINOMIAL OISTllBUTION I PARtiAL SUMS - Conld
I ~ 001 002 003 004 005 006 007 0108 009 010 015 0'20 025 oJO 035 040 0145 050
15 0 08601 07386 06333 o5~21 04633 03953 03367 02863 02430 02059 00874 00332 o013~ 00047 00016 00005 00001 -I 09904 091>47 09270 08809 08290 oi138 07168 ob597 o603j o5t90 o31Hb /) Ib7l 00302 00153 00142 00052 00017 oOOOj2 o99Q6 09970 09906 09791 09638 09429 09171 o8B70 08531 oUl59 oh042 o~9HO 02301 o121 J oObl7 00271 00107 000373 10000 09998 09992 09976 09945 09896 09821 09727 o9bOl OYiH 08227 01148) {I 1613 {I 919 o1727 00901 00424 001764 10000 09999 09998 o999~ o998h 09972 o99JO 09918 09873 o93JJ ocl358 oh86J o~115 03519 02171 o120~ 001925 10000 10000 09999 09999 09997 09993 09987 09978 QIIBi! 09189 oB)16 OW6 05643 04032 o20GB 015096 10000 10000 10000 09999 09998 09997 oqq6~ () 9819 09434 oShag 07543 ob09~ 04522 030367 10000 10000 10000 0999+ oq918 09827 o(bOO 08868 07809 06)35 050008 09999 09992 09958 o9fJI8 09578 09050 o818! 069649 10000 o4gqq 09992 09963 o9g76 09662 09231 08491
10 10000 o99q9 oq993 09972 09907 09145 01)40811 10000 oy(j99 09995 09981 09937 0982412 10000 09999 09997 09989 0996313 10000 10000 09999 0999514 10000 1000015 10000
16 0 08515 07238 06143 05204 04401 03716 03131 02634 02211 018)3 00'43 00281 00100 00033 oO~lO 00003 00001 -1 oq891 09601 09182 08673 0'8108 ohll oC902 06299 05711 01147 o)839 oI~07 00635 00211 00098 00033 00010 000032 019995 09963 09887 09758 09571 09327 09031 08689 o8J06 07392 OJIJH 113JI8 o1971 ormt 004)1 00183 00066 000213 10000 09998 oq989 09968 09930 o9a68 oq779 o90J8 o9)Ot o9l1b II"Ull I 0) 181 04050 o t 9 01339 oOb51 00281 001064 10000 09999 09997 09991 oq981 olJ962 09932 oQU89 091130 o~209 fl7llP oh~02 o1199 02892 o16h6 00851 oOJ845 10000 10000 o999J 09998 09995 oQ990 oq9HI of/9b7 097/5 () (113~ 01103 01)48 04900 03)83 01976 oIOJ16 10000 10000 09999 09999 09997 oIJ991 o9)H o1733 o9l0~ o1147 ohaUl 015272 01660 o227l7 100UO 10000 10000 04999 o9YIl9 o~9~0 09729 oQ)Jfi 08406 07161 05629 040188 10000 09198 o1J9115 oqlJ25 o1743 oq3 9 08577 07441 019829 10000 () Y9Q8 o9Q8~ oQ19 09771 o9H7 08159 07728
10 10000 oq997 o(llUi oY918 o980Q 09514 oR91911 10000 091117 09987 09951 09851 0961612 10000 09998 09991 09965 oq89413 10000 Ol9YGG 09994 0997914 10000 o9q99 0999715 10000 1000016 -
(ContInued)
31
t& 19~OO (Part t)•19'9
IINOMJAL DtsfllBttrlON I ~A111A1 suMs - Conld
~I~ 001002 003 OOi 0'05 006 007 008 009 010 015 020 025 030 035 040 045 050
17 0 08429 01093 05958 04996 04181 03493 02912 02423 02012 01668 00631 00225 00015 00023 00001 000021 09877 09554 09091 08535 07922 07283 06638 06005 05396 04818 o2J25 01182 00501 00193 00067 00021 00006 000012 oq994 09956 09866 09714 09497 09218 08882 08497 08073 07618 05198 03096 oIb37 00774 00327 00123 00041 000123 10000 09997 09986 09QOO 09912 09836 09721 09581 09397 09174 07556 05489 03530 oLOl9 01028 o04b~ 00184 oOOb4
4 10000 09999 09996 09988 09974 09949 09911 09855 09719 09013 07582 05739 03887 02348 o12bO 00596 002455 10000 10000 09999 09997 09993 09985 09973 09953 09681 08943 07653 o5Y68 04197 02639 01471 007176 10000 10000 09999 09998 09996 09992 09917 09623 08929 07752 06188 04478 02902 016627 10000 10000 10000 09999 09983 09891 o9J98 08954 07872 00405 04743 031458 10000 09997 09974 0'9876 09597 oQOO6 08011 06626 050009 10000 09995 09969 09873 09617 09081 o816b 06855
10 09999 09994 09968 09880 09652 09174 0833811 10000 09999 09993 09970 09894 o96G9 0928312 10000 09999 09994 09975 09914 0975513 10000 09999 09995 09981 o99~6
14 10000 09999 09997 0998815 10000 10000 0999916 1000017
0 o83~5 06951 o5780 04796 03972 03283 02708 02229 01831 01501 00536 00180 00056 00016 00004 0000118 1 o98n2 oQS05 o8997 08393 07735 07055 06373 05719 05091 o4J03 02241 00991 00395 00142 00046 00013 00003 00001
2 o99Qg 09948 o9843 09h,7 09419 09102 08725 08298 07832 07338 04797 02713 01353 00600 001% 00082 00025 000073 10000 09996 oq982 0Q950 09891 09799 o9ry67 o9194 09277 09018 07202 05010 03057 01646 00783 o0318 o0110 000384 10000 o9998 09994 09q85 09906 09933 09884 09814 09718 08794 o716i 05187 03321 o1886 on9~2 00411 001545 10000 09999 09998 09995 09990 09919 09962 o99~6 09581 08671 o71n 05144 01550 o20H8 01077 00481b 10000 10000 10000 09999 09997 09994 09988 09882 09487 08610 07217 05191 03743 02258 011897 10000 10000 09999 09998 a9q73 o9B37 09431 08593 07283 05634 03915 o2oW3
8 10000 10000 09995 09917 oGS07 09404 08609 o73h8 05778 040739 09999 09Q91 09946 oQ7QO o9~O3 08613 07473 05927
10 10000 09998 09988 09939 09788 o9~24 08720 0759711 10000 09998 09986 09938 oq797 09463 0881112 10000 09997 09986 o99~2 09817 0951913 10000 oQq97 09987 09951 0984014 10000 09998 09990 0996215 10000 09999 0999316 10000 0999917 1000018
(Coatlftlud)
32
BINOMIAL DlsTRmunoN I PARTIAL SUMS - Conld
IS I 9300 (Part I)•1979
n ,JODI 002 o03 004 005 006 007 008 0'09 010 015 o~O 025 030 035 040 045 050
19 0 08262 06812 05606 04604 03774 03086 02519 02051 01666 01351 00456 00144 00042 00011 00003 00001 - -1 09M7 OQ454 08900 08249 07547 068~9 06121 05HO 04798 04203 01985 00829 00310 00104 00031 00008 00002 -2 09991 09939 09817 o9b16 09335 08979 08561 08092 07185 07054 OH13 02369 o]l13 00162 00170 00055 00015 000043 10000 o999J 09978 09939 09868 09757 09602 09398 09147 08850 06841 04551 02631 01132 00691 o02~O 00077 000224 10000 09998 09993 09980 09956 09915 09853 09765 o96~8 08556 06733 04654 o28~) 01500 00696 00280 000965 10000 09999 09998 09994 09986 09971 09949 o991~ o9~63 08369 06678 04119 02968 916~9 00777 003186 10000 10000 09999 09998 09996 09991 09983 09837 09324 08251 06655 04812 03081 01727 008357 10000 10000 09999 09999 09997 09959 09767 09225 08180 06656 04878 03169 017968 110000 10000 10000 09992 o9q13 09713 o91bl 08145 06675 04910 032389 09999 09984 09911 09bH 09125 08139 06710 05000
10 10000 09997 09977 09891 09653 09115 08159 0676211 10000 09991 09472 09886 09648 09129 o820~12 09999 o9q9~ 09969 09884 09658 o91b513 10000 o99q9 09993 09969 09891 0968214 10000 09999 o999~ 091172 oqClO415 10000 09999 ogqg1 0997816 10000 09999 oq99617 10000 1000018 -19 -
20 0 08179 06676 05438 04420 03585 02901 02342 01887 o1516 01216 00388 00115 00032 00008 00002 - - -1 09831 09401 08802 08103 07358 06605 05869 05169 04516 03917 o1756 00692 00243 000,6 00021 00005 00001 -2 09990 09929 09790 09561 09245 08850 o8~90 07819 01334 06769 o.wi9 02061 00913 0011) 00121 00036 00009 000023 10000 09994 09973 09926 09841 09710 09529 o929i 09007 08670 06477 04114 02252 01071 004H oOl60 00049 000134 10000 09997 09990 oqq74 oq94t 09893 09817 09710 09568 o82Q8 06296 04148 o~~71 01182 00510 00189 000595 10000 09999 09997 OQ991 o99BI 09962 09932 09887 09127 08042 06172 o416i o24r4 o1~56 o05~3 002076 10000 10000 09999 09997 09994 o11987 o997b 09781 09133 07858 oGOaD 04166 02100 01299 005777 10000 10000 09999 09998 09996 o9q41 oq679 oR982 on 1 06010 04159 02520 013168 10000 10000 09999 09987 o9QOO 09591 oHBbl o7fi24 05956 04143 025179 10000 09998 09974 09861 oQJ~O 08782 07553 05914 04119
10 10000 09994 091161 o(IB?q 09468 08725 01107 0508111 09999 0999\ oqqlq 09804 09415 08692 0748312 10000 09998 o~)R7 09940 09790 019420 0868413 10000 09m 09985 09935 09786 019423
14 10000 09997 09984 09936 0919315 10000 09997 09985 0994116 10000 09997 0'998717 10000 0999818 1000019 -20 -
IS 193~O (Part I) •U7~
APPENDIX C(Clause 1.l.1 )
POISSON DISTRIBUTION: INDIVIDUAL TERMS
For m=~ ~I (O 01 )0W, (005 )100 (0 1)~ (~I~ ) 5
nrr""-,-1
, m~ ~ ~I ~ ~~ ~ ~l 0O~ ~ OJ ~ ~o 001 0O~ 0O~ 0W 0IJ 0~O 0lJ 0JO ~ J5 040 045 050 055 060 0oj 010 01j 0~O 0ijj 090
J -- _o~~OO ~ ~~Ol ~ ~104 0OOo~ 0qjl~ ~ 941~ ~ ~n4 0nJI 0m~ 0~04~ 0~601 0~W1 0m~ 0140~ 010~1 061~J 06J16 0W6j 0j1hq 01m 0jnO 04155 0m4 0mJ 0m4 0~OODo009~ 0Ol~o ~ I)l~1 0~J~4 00~16 0056j 006jJ 0on~ 0om 0O~Oj 01~91 016J1 019H 0~m 0~466 0~681 0~8u9 0JO~J 0~11J 0Jm 0ll9l 01416 01J4J 0lm 0J6B 0J6J9
QOO~~ 000J4 OOOO~ OOOI~ OOOl1 OOO~J 00010 OOOJ1 00045 00091 00161 OOUJ oom Dom OO~J6 O~646 D015~ Dom 009~~ 111101 olm 0IJ1~ ~ UJij 015H D1641oOCH)I 00001 ~ 0001 0OOO~ 00005 00011 0oo~O OOOJJ 00050 ooon 001)91 0Ol~b ~ ~Ibl) 00198 0om 0~l8~ 0om 0OJ~J 0om 00494
0000100001 OOOOJ 00004000010001100016 OUOU OOOJO 000J9 000,0 0006! oom 0009l 001l!00001 00001 0tool 00011! 0OOO~ 0000) 0001)1 00009 OOOll OOOlb 0OO~O
oOOUI OOJOI 0(0)1 0OOO~ OOOOl 00001
~O~i 100 11111311151611181910!! 1I 16 !8 303131 363840 I! " 4648 iO
o 0J8G1 0%19 0m9 0JOI~ 0~m 0H66 0~m 0~019 0ISH 016~J 01496 0IJ5J 01I0~ 00901 0014J 0060~ 004~~ 0O'O~ 0mH 0om 0om OOIBJ 0mo 0om 0010\ 0008~ 00061I 0jo14 0J619 0J66~ 0Jbl4 0mJ 0~m 0~M1 0mo 0~IOb 0~~1j 0~~4~ 0~101 0m~ a~111 0I~~I 0110J 01~94 0I~04 0mj 00~84 00830 0om OObJO OOjlO 0046~ 0OJ~) 0om~ 011H 018~~ 0~014 0~16~ 0~~O~ 0~411 0~JIO 0h~4 0l640 0~61~ 0~iOO 0~101 0~6~1 0~blJ 0~jlO 0~~~4 0~~40 01081 0'19!9 01111 016lj 01i6} olm 0118~ 0106~ 009W 0OM~J OOJH 0Obl~ 0~m 00~61 0O~~ 0m8 01m 01li8 01496 01601 01110 0IBO~ 0196b 0~090 0l!1b 0~~~j 0mo 0Ul6 0~186 0Wj 0m6 019jt 018)~ 0I11J 0IfiJI 01m 0H044 0om 0OIJ~ 0O~OJ 00~60 0om 00~9j 00411 005jl OOUJU 0om 0om 0090~ 0108~ 01~'4 01414 olm 01680 0\181 0IBJ8 0191~ 01914 01951 019H 0\911 Ol81j 0IB~O 011jjj OOO~j 00011 00041 OOOo~ 00084 00111 00141 00116 00~16 00~60 00~09 00J61 00416 OObO~ 001~5 oom 0100801140 o11M olm oun OIJ6~ 016J~ 0lbS1 om5 Dim olm6 00004 OOOOj 00008 oom 00018 OOO~6 oom 00041 00061 0001~ 0009~ OOI~O 00114 00~41 oom 00401 00504 00008 0011b OO~16 009~6 01011 OIH 01l~1 0IJ~~ 0 J98 OI46l1 00001 00001 00001 0OOO~ 0OOO~ 00005 001108 00011 OOOlj 0oo~O 0oo~1 0OO~~ OOOj5 0008~ 0om 0016J 00~16 0om 0OJ~8 004V 00508 0om 00686 00118 001169 00%9 0\ll~4
~ 0000100001 00001 oooo~ OOOO~ 00005 00006 00009 00015 OOO~j OOO~ij 00051 OOO~I 001110014800191 OOUI OO~98 OO~60 oom 00500 oo,n 006JJ9 00001 00001 00001 00001 00004 00001 0001\ 0001~ 00011 00040 00056 00016 0010~ 0OIJ~ 00168 00109 0om OOJOI 00~6~
10 00001 0OOO~ 0OOO~ 00005 00008 0001~ 00019 00018 OOOJ9 OOOjJ 00011 0009~ 0Oll~ 00141 0'0181II ijijOijl ij0001 0OOo~ 00004 00006 00009 0OOI~ 00019 000~1 OOOJ1 0004~ 00064 00082I~ ~ 0001 ijroll 0ooo~ 0OOO~ 00004 00006 00009 0lXm 00019 00016 000'4I~ 00001 00001 0OOO~ 0ooo~ 00005 00001 00009 0OO\~14 00001 00001 00001 ooon ocoo~ O~IOj15 oCOOl 0000101:002
----------------.....- .....------------_.._-----,-
APPENDIX D(Clause 1,1.1 )
POISSON DISTRIBUTION IPARTIAL SUMS
For m=001 (0 01 )0W(0OJ) I00 (0I)~ (0~ )j
KOOIOOlOOIOOIOOlOOO00100000001001502002501ll03501001501001500006501001500008J090
o~~OO 09~0~ 09,O~ 0~OO~ 09JI~ 0~1I~ 09~H 09~~1 091~9 09048 08001 08181 0m~ 0140~ O,OH 0610J 06j,o 0bOb) 0JIU9 0Jms 0mo 04%6 0~m 0HJJ 0W4 0iOu610000 09Q98 0~9~u 0999~ 099~8 09~~~ 09m 09~10 0996l o99JJ 09~98 0qs~) 09m 091lJI 09m 09J8~ 09Hu 00098 08QlJ 0818\ 08~14 08141 08~( 6 0BOn6 01901 01nJ
10000 10000 10000 10000 10000 09999 09999 09Q99 0~998 o999j 09989 0991~ 09904 099~j o9911 098~1 098~b o~m 09M O~lI1 09n19 o939J 09J b 09tJI 09mI0000 10000 10000 10000 I0000 09999 09999 09991 099~J 09Jn 09933 099~~ 099h 09900 o99J6 09J4l 099n 09909 U9889 09~bJ
1~1~lml.O.O~O.O~O~ O~O~ O~O.O~ O~10000 10000 10000 10000 10000 09999 09999 099~q 09998 09~9, Oml
10000 10000 100001000010000 10000
K095 10 II II II 14 15 16 11 18 19 20 II 1'4 26 18 10 II II I~ 18 40 II II 16 18 50
o 0~S61 0~619 0~~~ 0!Ol~ 0~m 0~~~6 0~~~I 0~019 01m 016j~ 0H% olm 01108 00~1 0om 00608 0'0~98 00~8 0O~!~ 0om 0om 0018~ 0mo 0Ol~~ 0mOl 0008~ 00061I 01~41 0n18 06990 06b~6 06~68 0J918 0JJ18 0m9 049j~ 0%~8 0~m 0~060 Oj46 0~084 0~614 0WI 01991 0111~ 01%8 0IlJI 01014 00916 00180 OOb6j OOJ6J 0om 00404l 09~81 09191 09004 0819J 08m 08m 08088 018~4 0m~ 0nOG 010~1 0Uf01 0ml 0Jb91 0JI84 0~9J 0m~ 0~199 0m, 0jOli 0~o89 0nBl 0~IOl 018Jl 016~6 0Hlj 01141~ 098j9 09810 0914j 09b6~ 0~)b9 0~~6j 09jH 09~1~ 0~Oo8 ~ ~m 0~141 0~j11 0819~ 01181 0mo 0u919 0om 0OO~j 0jiSt 0m~ 0ms 04Jl5 0~Q54 0j59~ 0~m 0)94~ 0l6S04 0~911 0996J 09946 099~j 0989J 09851 09814 0916J 09104 090J6 095J9 09m 09m 090B 0m4 08111 0815J 01800 014H 01~ol 066/8 06~M 05898 05m 0JI~~ 041b~ 0H055 09995 0999~ 09990 OQ9BJ 0)9,8 09%8 099J5 099~ 099~0 OQ896 09868 oq8~4 09m 090lJ 09jl0 09H9 09161 089~ 0810) O~HI 081j6 01851 oml 011~9 0~858 06J1O OblOO6 09999 09999 09999 099~1 09996 0'9994 09991 09981 099~1 09914 09966 o99j5 099~5 09884 098~8 09156 09665 09J54 094~1 0~lb1 09091 0889j 0861J 08~~u 08180 01908 016~~1 10000 1'0000 10000 10000 09~99 09999 09998 0'9991 09990 09994 0~99~ 09989 09980 0996, 0q941 09919 0q881 09m 09169 0Q6~1 09599 09489 09%1 0gm 090~9 08801 086608 10000 10000 10000 10000 09999 099~9 09998 0~998 09995 09991 09985 09916 0996~ 0994~ 0~11 0988J 0q8~ 09186 09111 096~~ 09519 09H~ 09m9 10000 10000 10000 10000 09Q99 09998 09996 0999J 09Y89 0998~ 099iJ 099bO 099tl 09919 098~9 09851 0Q80j 091~9 0llO8lW 1~I~O~O.O.o.oEomO~O~O.OHO~O.O.II 1'0000 10000 09999 09999 09998 099~6 09994 09991 09900 09980 09911 09960 09945I~ I0000 IOCOO 0m9 09999 09~98 09991 09996 09~9~ 09990 09986 09980I~ 10000 10000 10000 09999 09999 09998 09991 09995 0r99J~ 1~I.O.O.O.o~~ I~I.I~OMm O.v I~
IS I 9300 ( Part I) • 1979
APPENDIX E(See Clause 8.1.1 )
~OGARITHMS OF FACTORIALS OF INTEGERS FROM 1 TO 100.........•"AMPLE SIZE LOG10 nl SAMPLE SIZE I..oo 10 n 1 SAMPLE SIZE 1..00 10 n 1
II n II
(I) (2) (1) (2) (1) (2)
:; 1 0·0000000 41 49·5244289 81 120·763 21272 0·301 0300 42 51-14767B2 82 122-677 02663 0·778 1513 43 52·781 1467 83 124·596 10474- 1·380 2112 44 [>4·424 5993 84- )26· 520 38405 2·079 1812 45 56·077 8119 85 128·449 80296 2·857 3325 46 57·740 5697 86 130·38430137 3·702 4305 47 59·412 6676 87 132·323800fi8 4-605 5205 48 61·093 9088 88 134"268 30339 5·559 7630 49 62"78·10 1049 89 136'217 6933
10 6·559 7630 50 64:·483 0749 90 138"171 935811 7·601 1557 51 66'1906450 91 140"130 977212 8'680 3370 52 67·9066484 92 142-094 765013 9·794 2803 53 69"630 981)3 93 14t'Ob314 10·9404084 54 71-363 3180 94- 146·0 ~6 3758]5 12"116 4996 55 73"10l b807 95 148 014099416 13·320 6196 56 74·851 8(j87 96 149·996 370717 14·551 0685 57 76·607 7416 97 151·983 1414-18 15'806 3410 58 78·371 1716 9B 1r;3-974- :~6B5
19 17·085 094·6 59 80-1420236 99 15S·970 003720 18·386 1246 60 81-920 1148 100 157 970 003721 19·708 3439 61 83-70.5 50t722 21·050 7666 62 85"497 8964-23 22·412 4944 63 87'297 236924- 23"792 7057 64- 89-103 416925 25·190 6457 65 90·916 330326 26·605 6190 66 92'735 874227 28·0369828 67 94·561949028 29-486 1408 68 96·394457929 30·946 5388 69 98·233 307030 32-423 6601 70 100·078 405031 33·915 0218 71 101-929663432 35·420 1717 72 103·786 995933 36·9396857 73 105·650 318734 38'490 1646 74 107·519 550535 40·0142326 75 109·394 611736 41-570 5351 76 111·275 425337 43"138 7369 77 113·161 916038 44·718 5205 78 115·054 010639 46'309 5851 79 116·951 637740 47"911 6451 80 118·854 7277
37
INDIAN STANDARDS
ON
QUALITY CONTROL, SAMPLING AND OTHER STATISTICAL TECHNlQ,UES
IS:997 { Part I )-1972 Method for statistical quality control during production: Part I
Control charts for variable (first revision)397 ( Part II )-1975 Method for statistical quality control during production:
Part II Control charts for attributes and count of defects (first "vision)1548-1969 Manual on basic principlea of lot sampling2500 ( Part I )-1973 Sampling inspection tables: Part I Inspection by attributes and
by count of defects (first "vision).2500 ( Part II )-1965 Sampling inspection tables: Part II Inspection by variables for
percent defective4905-1968 Methods for random sampling ..~5002-1969 Methods for determination of sample siZe to estimate the average quality
of a lot or process5420 ( Part I )-1969 Guide on precision of test methods: Part Pr-inciples and
applications5420 ( Part II )..1973 Guide on precision of test methods: Part II Inter-laboratory
testing6200 {Part I )-1977 Statistical tests of significance: Part I I.t t, normal and F tests
(first "vision)6200 ( Part II )..1977 Statistical tests of significance: Part II (X2·test (first revision )7200 ( Part I )-1974 Presentation of statistical data: Part I 'fabulation and sum
marization7200 ( Part II )-1975 Presentation of statistical data: Part II Diagrammatic Repro ..
sentation of data ~
7300 1974 Methods of regression and correlation7600·1975 Anal ysis of variance7920 ( Part I )-1976 Statistical vocaq,ulary and symbols: Part I General statistical
terms 'i7920 ( Part II )-1976 Statistical vocabliary and symbols: Part II Terms used in
sampling and process contro'",
J
J
~.
•~ .~
.¥ ,, '.i
.\\
\
INTERNATIONAL SYSTEM OF UNITS (51 UNITS)
B••• Unit.
QUlntlty Un" Symbol
Length metre mMasa kilogram kg
Time I.cond •Electric current ampere AThermodynamic kelvin K
temperatureLumlnou8 Intensity candela cdAmount of lubstance mol. mol
8upplementary Unit.
QuanUt, Unit Symbol
Plane angl. radian ladSolid angle Iteradlan .,Derived Unit'
Qu,nl/I, Unl' Symbol aeflnltlon
'orc. newton N t N - 1 kg.m/a-Energy Joule J 1 J - 1 N.mPowe, watt W ,
W - 1 J/.flux web., Wb , Wb - , V••Flux denalt, teala T t , - 1 Wb/ml
Frequency hertz Hz 1 Hz - 1 ell (a-I)Electrtc conductanc. IIemen. S 1 S - 1 A/VElectromotive force volt V t V - 1 W/AP,elsure, .t,e.1 pascal Pa 1 ... - 1 HIm.
INDIAN STANDaIlD. INSTITUTION".n•• Bbl••n•• eahadut Shah Z.f., U.rg, NEW DELHI 11DIOI
Telephon•• I H N.t, 11 .t It Tele..,••• I .an.u.n.tIt.
R.,lon,' Offlc••1 '.I."honeWe.tern I Novelt, Chamber•• Qr.nt 80.' BOUBA Y 400007 17 97 IIE••t.rn ,I Chowrlngh•••PPloaob CALCUTTA 700071 27 SO 90South.rn I C. I. Y. C.mDua, Ad,., MADSAS 100120 .1 244.B,anch Ollle••1
'PUlhPlk' t.Nu''''OlNlmed Shit"" "Ilg. ICM",., AHMADABAD a80001 I 03 ,t., • Block, unit, Blda. N.r••lmhl'.'. Square BANGALORE 580001 I 7&4.C;.ngotrl Complex, Bh.dbhlda Road, T. T. Nagl' BHOPAL 482003 t 2711liE Kalpana Area BHUBANESHWAR 711014 I a817Ahlm•• Bldo, sea 82-83. Sector 11C CHANOfGARH 180017 2 II 2D1·I-SeC L. N.Supt. Mara HYDERASAD 500001 II t\1.3R14 Yudhtste, Mera, C Scheme JAIPUR 302001 ,"alt17/4188 S.ryodaya NI,.' KANPUR 208001 112< 7.J ..'.tJlput,. Indultrla' Eltat. PATHA 100011 0.01Hanle. Blda .'nd '1001 a.a,, Station Roa. "IIYANDIlUIl ....., ".1
P,I_t., •• Na. r.,I. Pt..."" Pt••• tebo,•••t-,- ~of