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 2500-2 (1965): Sampling inspection procedures, Part 2:Inspection by variables for percent defective [MSD 3:Statistical Methods for Quality and Reliability]

IS : 2500 ( Part II ) • 1965( Reaffirmed 2001 )

Indian Standard

SAMPLING INSPECTION PROCEDURESPART II INSPECTION BY VARIABLES FOR PERCENT DEFECTIVE

Seventh Reprint AUGUST 1998( lncorporating Amendment No.1)

UDC 519.271.3 : 620.111 (083)

C Copyright 1996

BUREAU OF INDIAN STANDARDSMANAK BHAVAN. 9 BAHADUR SHAH lAfAR. MARG

NEW DELHI 110002

Gr8 June 1966

IS : 2500 ( Part II ) • 1965

Indian Standard

SAMPLING INSPECTION PROCEDURESPART II INSPECTION BY VARIABLES FOR PERCENT DEFECTIVE

Indian Iron & Steel Co Ltd, Burnpur

Precious Metals Sectional Committee, SMDC 13,lSI

Directorate General of Inspection (Ministry orDefence)

Italab Private Ltd, Bombay

Light Metals and Their Alloys Sectional Committee,SMDC 10, lSI

Refractories Sectional Committee', SMDC 18,-ISIMinistry of Railways

SHRt A. GUUA

Copper and Copper Alloys Sectional Committee,SMOC 11. lSI

Cast Iron and Malleable Cast Iron SectionalCommittee, SMDC 9, lSI

SRRI PItEM RAJ GUPTA. Ferro Alloys Sectional Committee, S~IDC 8. lSISHaI P. C. HAZIlA Indian Bureau of Mines (Ministry of Steel and

Mines), NagpurSHIll N. K. MUKHERJEE (A/tnlUde)

Saa. P. L. JAIN Steel Tubes, Pipes and Fittings Sectional Com-mittee. SMDC 22, lSI .

SIDlI F. A. JAIDAHWALA Lead, Zinc, Tin, Antimony and Their AUoy.Sectional Oommiuce, SMOC 12, lSI

na N. ]AYARAMAN Essen at Co, BangaloreSHIll K. N. GuaUIlAJACHA.a. (Alternate)

SHU R.. M. KRISHNAN Foundry Sectional Committee, SMDC 17, lSISHU N. R.. KIUIHHASWAMY Directorate General of Ordnance Fat..tories

(~linistry or Defence), CalcuttaSHIll D. K. CHAkRAVARTY (Alurnall)SHU D. SaN (Al""'tI~)

SHRI ~f. M. GUPTA (.AlttmQ~)

SHat D. N. ELClIJDANASHar S. S. HONAVAR (A/tnnau)SIUtI J. P. PA'r&L (AltlmQu)

SHRI S. B. FIRKE

SHa1J. C. BANEIlJEESaRI A. K. BHA1'TACHARYA

SHRI M. N. BUIDE (.-fltmult,)Da U. N. BHIlANY

SHRI J. N. BURMAN (AlumtllI)DR M. K. BoSE

SHRI S. K. DunA

Methods of Sampling Sectional Committee, SMDC 4-

CIurinruua &jwlstnlin6Da A. V. SUItHATMB The Tata Iron & Steel Co'Ltd,Januhedpur

M""bn"Smu V. D. AOARWAL

(Conlintutl 0" PCl(2)

BUREAU OF INDIAN STANDARDSMANAK UI-IAVA'N, 9 BAIIADUR SHAH z.a.rAlt ~fARG

NEW Dl~LHl 1\OOU1

IS s 2500 (Part D) - 1965

(C,nti".4j'rtml /JG" I)

National Tnt House, CalcuttaDirectorate General of Supplies &: Di.posaJI

(Inspection Wln~); Wrought Steel ProductsSectional Committee, SMDC 5, lSI; SteelCastings Sectional Committee, SMDC 20,lSI; and Pig Iron Sectional Committee,SMDC 24, lSI

()r('s and Raw Matrrials Sectional Committee,SMDC 16, lSI

Steel Forgings Sectional Committee, SMDC 21, lSIMethods of Chemical Analysil Sectional Com..

rruttee, SMDC 2, 151Methods of Physical Tests Sectional Committee,

SMDC 3, lSIDirector, lSI (Ex-officro Mlmbtr)

&J1r's,,,'UafIndian Statistical Institute, Calcutta

SHRI B. N. SINGH,Assistant Director (Starisuc«)

SHat S. VISWANATHAN

SHR.A.PADMANAEHANSHRr A. SANOAMESWARA RAO

SlllU P. I. A. NARAVANAN

Indian Non-Ferrous Metal. Manufacturen' A..­dation, Calcutta

SHa. M. M. MOUDOILL (AltnlUJu)SKat N. T. MATHEW Anny Headquarters

DRM. N. BHATTAcHAavA (A/lima")SHal N. C. MITRA Government of India Mint (Ministry of Finance)

DR M. K. BOSE (Alllmall)SHaJ S. N. MUICERjlSHa. E. K. N. NAMBIAR

Mtm6nsSHIUD. B. LAHla.

Da A. MATTHAI (Alima.")SHItI N. MAJUMDAR

StcrdarySURI Y. K. BUAT

Extra Assistant Director (Statistics), lSI

Panel for Sampling Inspection Tables, SMDC 4 : P6

DR A. M~TTHAI Indian Stati.lical Institute, Calcutta

2

IS I 2500 (Part D). 1965

CONTENTS

PAGE

O. FOREWORD

I. SCOPE

2. TERMINOLOGY AND SYMBOLS

3. CONSIDERATIONS FOR THE CHOICE BETWEEN THE ATTRIBUTES AND

VARfADLES INSPECTION

4. PRELIMINARIF.S TO THE SELECTION OF VARIABLES INSPECTION PLANS4.1 Formation of Lots4.2 Types of Single Sampling Variables Plans4.3 Drawing of Samples

5. SELECTION OF SAMPLING PLANS5.1 Classification of Sampling Plans5.2 Inspection Level5.3 Lot Acceptability5.4 Normal, Reduced and Tightened Inspection

6. TABLES AND ILLUSTRATIVE EXAMPLES ..TABLE I SAMPLE SIZE CODE LETTERS BY INSPECTION LEVELS

AND SIZES OF LOTS

T ABLE 2 SINGLE SAMPLING VARIABLES PLANS FOR VARI­ABII.ITY KNOWN METHOD

TABLE 3 SINGLE SAMPLING VARIABLES PLANS FOR VARIABILITV

UNKNOWN - STANDARD DEVIATION METHODTABLE 4 SINGl.E SAMPLING VARIABLES PLANR FOR VARIABILITY

UNKNOWN - RANGE METHOD

STABLE 5 THE UPPER LIMIT FOR THE VALUE OF U_ L FOR V ARIA-

BILITY UNKNOWN - STANDARD DEVIATION l\'fF.THOD

R RTABLE 6 THE UPPER LIMIT FOR THE VALUE 01" U-L OR U-L

FOR VARIADILITY UNKNOWN - RANGE 1vlETHOD

EXAMPLt:S I, 2, 3 AND 4-ApPENDIX A SYMBOLS

ApPENDIX B FORMULJE AND TABLES FOR CONSTRUCTING SINGLE

SAMPLING AQL-L"fPD VARIABLES PLANS "'OR ONE-SIDED SPECI­

FICATION LIMITSB·O. Stipulations of the Plans (Values to be given)B-1. Variability Known Method .. . .B-2. Variability Unknown - Standard Deviation Method ..

TABLE 7 1'1-1£ VALUES OF t CORRESPONDING TO A GIVEN FRAC·

'CION VARYING FROM 0·00 TO 0·99B-3. Variability Unknown - Range Method ..

ApPENDIX C THE OPERATING CHARACTERISTIC CURVES

3

466

78899

101011121315

15

16

17

18

19

2021,22

24

25252525

262728

IS : 2500 ( Part II ) • 1965

Indian Standard

SAMPLING INSPECTION PROCEDURESPART II INSPECTION BY VARIABLES FOR PERCENT DEFECTIVE

o. FO R E W 0 R D

0.1 This Indian Standard was adopted by the Indian Standards Institutionon 1 October 1965, after the draft finalized by the Methods of SamplingSectional Committee had been approved by the Structural and MetalsDivision Council.

0.2 Part I of this standard dealing with inspection by attributes and by countof defects had been issued earlier with a view to facilitating the wide-spreaduse of sampling inspection in those situations where items can be classifiedas defectives or non-defectives, satisfactory or non-satisfactory. However,this type of inspection may require comparatively larger sample size and hencein certain situations like the determination of warp breaking strength ofcotton fabrics, it may become uneconomical due to the destructive nature orprohibitive cost of testing. In such cases, inspection by variables, whereinquality is measured on a continuous scale like tensile strength of steel wire,may be more useful and economical. This standard has been prepared tomeet the growing demand for the use of sampling plans for inspection byvariables.

0.3 Further economy in sampling inspection may be obtained if the units inthe lot are quite uniform in quality as a smaller sample may then be adequateto represent the lot. This uniformity may be achieved by controlling thequality at the production stage itself and helpful guidance may be obtainedin this respect from IS : 397·1952*.

0.4 Sometimes the quality characteristics can be inspected both by attributesand by variables, for example, the diameter of a shaft can be checked eitherby gauging or by actual measurement, In such cases, a decision has to bemade whether inspection should be by attributes or by variables. Some ofthe important considerations which provide the basis for a suitable choicehave been given in 4.2.3 of IS : 1548-1960t. However t a more detaileddiscussion of such considerations is given in 3 of this standard.

• Method for statistical quality control during production by the use or control chart(JWI mJintl).

tManual 011 buic principlet or lot sampling (nnu M1iJltI).

4

II I 2500 (Part D). 1965

0.5 This standard furnishes a collection of tables from which samplingplans can be selected for inspection by variables. Only single samplingplans have been given in this standard as the variables plans for double andmultiple sampling become rather complicated for practical usc. All theplans given in this standard require the lot quality to be specified in termsof percentage defective.

0.6 This standard is one ora series of Indian Standards relating to techniquesof statistical quality control. Other standards published so far in the seriesare:

* IS : 397-1952 Method for statistical quality control during productionby the use of control chart

• IS : 1548-1960 Manual on basic principia of lot samplingIS : 2500 (Part 1)-1963 Sampling inspection tables: Part r Inspec­

tion by attributes and by count of defects

0.7 In preparing this standard. considerable assistance has been derivedfrom the following publications:

MIL-STD-414 Military standard sampling procedures and tables forinspection by variables for percent defective. 1957. Departmentof Defence, USA.

Bowker (A H) and Goode (II P). Sampling inspection byvariables. 1952. Ed 1. McGra'.v-Hill Book Company, Inc, NewYork.

Dodge (1-1 F). A general procedure for sampling inspection byattributes based on the AQL concept. Technical Report No. 10.1959. Rutgers. The State University.

Ireson (W G). Sampling tables for inspection by variables. Teet­nical Report No.7. 1952. Applied Mathematics and StatisticsLaboratory. Stanford University.

Ireson (W G) and Resnikoff (G J). Sampling tables (or vari­ables inspection based on the range. Technical Report No. 11.1952. Applied Mathematics and Statistic» Laboratory. StanfordUniversity.

Rcsnikoff (0 J). A new two-sided ar ccptance region for samplingby variables. Technical Report No.8. 1952. Applied Mathematicsand Statistics Laboratory. Stanford University.

0.8 I n reporting the result of a test or analysis, if the final value, observedor calculated, il) 10 be rounded off, it shall be done in accordance withIS : 2-1960t .

·Slncr revised.tRulcs for rounding olf numerical values (rtl1iJed).

5

IS , 2500 (Part D) - 1965

1. SCOPE

1.1 This standard provides tables for single sampling plans for lot-by-lotinspection, when the inspection is done by variables, Besides, the Iormuheand the necessary tables for the construction of one's own single samplingplans are also given.

2. TERMINOLOGY AND SYMBOLS

2.0 For the purpose of this standard, the following definitions shall apply.The symbols used in this standard including those for some of the termsdefined below are explained in Appendix A.

2.1 SampliDglDspectioD - Inspection in which only a portion of a lot isinspected with a view to making a decision about accepting or rejectingthe lot.

2.2 SampliDg Plaa - A statement of the sampling procedure and the rulefor making decisions about the lot.

2.3 Item - Ultimate unit of product or material on which inspection willbe performed.

2.4 Lot - A collection of items from which a sample is drawn and inspectedto determine its acceptability.

2.5 Lot Size (N) - Number of items in a lot.

2.6 Sample - Collection of items selected for inspection from a lot.

2.7 Sam.ple Size (n) - Number of items in a sample.

2.8 Defective - An 'item' the quality of which does not meet the specifiedrequirements.

2.9 Percent Defective - Hundred times the ratio of the number ofdefectives to the total number of items.

2.10 Slagle SampHng Plaa -A type of sampling plan in which the decisionto accept or reject a lot is always reached after one sample from that lot hasbeen inspected.

2.11 Me.. (x) - The sum of the observations divided by the number ofobservations (S', also Appendix A).

2.12 Lot Staadard DeviatloD (a) - The square root of the mean of thesquares of the deviation of all the observations in a lot from their mean(s" als« Appendix A).

2.13 Sample Staadard DeviatiOD (.) - The square root of the quotientobtained by dividing the sum of squares of deviations of the observationsfrom their mean by one less than the number of observations in the sample(s" also Appendix A).

6

IS , 2500 (Part D). 1_

2.14 Ran.. (R) - The difference between the largest and the smallestobservations or test results in a sample (SI' also Appendix A).

NOTa I - lethe sample size is leu than JO, the r.nre shall be calculated for the .ampleas such. If. however. the sample size 11 10 or more (in multipln of 5), the observatioNshall be divided into subgroups of five each by takinl them consecutively in the aameorder u obtained. The range of each subgroup .haJJ then be separately determined.

2.15 Me.. Ra.a. (ft) - The mean of a set of ranges calculated for sub­groups of five observations in the sample (sel also Note 1 and Appendix A).

2.16 Proce•• Averale - The average percent defective of the productssubmitted by the producer for original inspection. (Original inspectionis the first inspection for the particular quantity of product as distinguishedfrom the inspection of products which have been re..submitted after priorrejection.]

2.17 A~eeptableQ.uality Level (AQ,L) - The maximum percent defectivethat, for the purpose of sampling inspection, can be considered as a satisfac­tory process average.

NOT~ 2 - When a consumer d~lgnatessome specific value of AQL, he indicatet to theproducer that his (the consumer's) acceptance sampling plan will acceft the l1'eat majorityofthe 10lS that the producer submits, provided the process average Ieve o(percmt defectiveIn lhr~e lots IS not greater than the designated value of AQL. ThUl, the AQL is adf"'lgnat,..d value of pt"rtcnt defective that the consumer indicates will be accepted mOltof the. lime (approximately 89 to 95 percent In this standard).

2.18 Lot Toler••ce PereeDt Defective (LTPD) - The percentage ofdcfecuves in a lot that can be tolerated in only a specified proportion of lots.

2.19 Producer'. Risk - The ri!)k (chance) of rejecting lots of qualityequal to the specified AQL. The risk of rejecting lots of quality better thanthe AQL will be smaller than the designated producer's risk.

2.20 CoDS1IDler'. Ri.k - The risk (chance) of accepting lots of qualityequal to the specified LTPD. The risk of accepting lots of quality worsethan the LTPD will be smaller than the designated consumer's risk.

3. CONSIDERAnONS FOR THE CHOICE BETWEEN THEATrRIBUTES AND VARIABLES INSPECTION

3.0 When a characteristic of an item is amenable to both attributes andvariables types of inspection, as is the case when a dimension can be eithergauged or measured, the following considerations would help in choosingthe appropriate type of inspection.

3.1 For any desired degree of protection, lesser number of items have to beinspected for variables inspection than for attributes inspection in orderto determine the acceptability (or otherwise) of a Jot. In other words,for the samples of the same size, inspection by variables gives a smaller riskof accepting Jots of unacceptable quality than inspection by attributes. Thevariables plans would he, therefore, ordinarily more useful and economicalin those cases where destructive or costly testing is involved.

7

IS I 2500 (Part D) .. 19&5

3.2 The measurement of an item under the variables inspection gives muchmore information about the quality of the item than the attributes inspection.Thus a container of copper naphthenate may show the copper content as 10·0percent against the speCification requirement of 8·5 percent minimum. Incase of attributes plans the container would be simply classified as satis­factory with respect to the copper content but the fact that the observedvalue ill 1-5 percent higher than the specified minimum value will not beutilized as such. The variables plans, on the other hand, use this additionalinformation in their acceptance criteria which are based on the mean andthe variation of the test results. The variables inspection would, therefore,yield more information about the quality of the lot than attributesinspection.

3.3 Inspection by attributes may to some extent be subjective in the sensethat an item classified as defective by one inspector may be classified asnon-defective by another or by the same at a later time. This is particularly10 in the case of visual inspection and items of borderline quality. Inspectionby variables, on the other: hand, would require actual measurement of theitem quality and is, therefore, more objective and minimizes the possibilitiesof inspection bias and error.

3.4 The attributes inspection may be generally performed either visually orby gauging and as such the cost of inspection per item is low; on the otherhand. inspection by measurement involves more time, labour, skill and morecomplicated tools, thereby making the inspection costlier.

3.5 Variabl~ inspection may require more record keeping and calculationby way of computation of mean, standard deviation, etc.

3.6 Variables plans are based on the assumption that the distribution of thequality characteristic is normal. Hence it is important that these plans arcnot used indiscriminately. In case the assumption of normality is in doubt,it is advisable to obtain the guidance of a competent statistician to ascertainthe feasibility of application of these plans.

4. PRELlMlNAIUES TO THE SELEcrlON OF VARIABLESINSPECTION PLANS

4.1 I'ormadoa of Lot. - A lot (see 2.4) should, as far as possible, consistof items of • single type. grade, class, size, etc, produced under relativelyuniform conditions of manufacture by a si~le firm so that the items in thelot are of uniform quality. In such a case, the size of the sample to betested in the Jot for a given protection would be small and inspection wouldbe economical. Furthermore, a lot should, consistent with the conditionsof homogeneity, be as large as possible so that the incidence of cost of inspec­tion per item is minimized. A lot can be a 'stationary lot' or a 'moving lot'subject to the convenience of stacking, access to each item in the lot andidentification. Each Jot shall be properly identifiable and each 'stationarylot' shall be presented as far as possible, in such a way that the inspector

8

IS I 2500 (Part D). 1965

will have easy access to all parts of the lot to select at random the itemsmaking up the sample.

4.2 Type. of Sialle SampU.K Variable. PIaD. - There are threetypes of single sampling plans available in this standard for inspection byvariables. These are applicable under the following three situations:

a) Variabiliry Known - When the variability in the lot, that is, thevalue of the lot standard deviation (a) is known beforehand eitherfrom the past experience, control chart data or any other means.

b) Variability Unknown - Standard Deviation Method - When the vari­ability in the lot is not known and is estimated from the samplestandard deviation (r).

c) Variahility Unknoum - Rang, M,thod - When the variability inthe lot is not known and is estimated from the sample range (R)or sample mean range (R).

Each of the above sampling plans has been treated separately for thefollowing two categories:

I) When one-sided specification limit, that is, either an upper speci­fication limit (U) or a lower specification limit (L) is given.

2) When two-sided specification limits, that is, both the upper and thelower specification limits are given.

4.2.1 For the same degree of protection, the sample size is minimumfor the variability known method and maximum for the range method and,therefore, the former is 'most economical. Moreover, from the administra..tive and computational points of view the variability known plans areeasiest to operate as they require the calculation of mean alone, furthercomputations remaining the same for all the three types of plans. Therange plans come next as they require the calculation of both the meanand the range and lastly the standard deviation plans which require thecalculations of both the mean and standard deviation, the latter being morecumbersome to calculate than the range. Though the variability knownplans are the simplest to operate, the requirement of a priori knowledgeof variability is a stringent one.

4.3 Drawing of Samples - The sampling inspection plans includedin the standard assume that the items constituting a sample are selectedfrom the lot at random. Technically, a sample is said to have been selectedat random if the method of selection gives the same chance to every itemin the lot for being included in the sample. Various methods availablefor collecting a random sample including the approximation methodslike systematic sampling with random start have been discussed in 5.3.2of IS : 1548.1960·.

• Manual on basic principles of lot sampling (since revis,d).

9

· IS r 2500 (Part D). 1965

5. SELEcrION OF SAMPLING PLANS

5.0 Selection of a sampling plan should be guided by the cost of inspectionthat will be incurred and the protection desired by the producer and theconsumer. The protection provided by a sampling plan to the producerand the consumer is described completely by its Operating CharacteristicCurve (00 Curve) which gives the probabilities of accepting (or rejecting)lots with 'varying percentage of defectives. The steepness of the DC Curvereflects the power of the sampling- plan to discriminate between good andbad lots; the steeper the OC Curve, the better is its power of discrimination.While the operating characteristic provides a complete picture of the pro­tection afforded by the sampling plan (see 8.3.3.1 of IS : l548-1960·)in the form of a function or a curve, it docs not give a single value of themeasure of the protection. In view of this, the choice of a sampling planis generally made with reference to certain specified points on the DCCurve, as for example, those associated with the Acceptable Quality Level(AQL), Lot Tolerance Percent Defective (LTPD), etc. The AQL, LTPDor such other values may be chosen on the basis of the previous data availableand by an agreement between the parties concerned. A comprehensiveset of sampling plans classified in terms of the AQL is given in this standard.

NOTE 3 - Each of the plan! selected from the tables given in this standard has its ownOC Curve. The plans based on the variability known, variability unknown - standarddeviation method and variability unknown - range methods corresponding to a particularcombination of a sample size code letter and the AQL value have approximately the sameDC Curve. For the variability unknown - standard deviation method, the OC Curvesare given in Appendix C. These curves would also approximately hold good for thecorresponding plans based on known variability and range methods. They can also beutilized for reading LTPD or any other desired value associated with the OC Curve.

5.1 C1as.ificatioD of S~lDpling Plans5.1.1 A QL Plans - The comprehensive set of sampling plans classified

in terms of AQL provided in the standard includes the following:a) Variability known plans (set Table 2 on P 16);b) Variability unknown plans, standard deviation method (.reI Table 3

on P 17) ; andc) Variability unknown plans, range method (se, Table 4 on P 18).

5.1.1.1 The choice ora sampling plan for a particular product requiresthe decision on the quality requirements; if AQL plans are being used,it is necessary to choose an appropriate AQL. In selecting an AQL valuea compromise may have to be struck between the quality desired and thequality attainable. If the AQL is superior to the quality that cau beattained under usual production conditions, an excessive amount of productwill be rejected; on the other hand, if AQL is not exacting enough, anexcessive amount of inferior products may be accepted. The value ofAQL may be specified by agreement between the parties concerned, giving

.Manual on basic principles of lot sampling (.sin" reuised),

10

IS I 2500 (Part D) - 1965

due weightage, whenever possible, to the past performance of the supplierof the product. Sampling plans for certain nominal values of AQL rangingfrom 0·10 to 10·0 percent are provided for in the Tables 2 to 4. Whenthe specified AQL is a particular value, other than those for which samplingplans have been furnished a suitable value close to it and available in thetables may be chosen subject to the agreement between the parties concerned.

5.1.2 One's Own Plans - To help the construction of one's own plans,tables for values of factors and formulee have been given, w hen stipulationsarc made in terms of AQL -md LTPD values. These are the following:

a) Variability known plans;b) Variability unknown plans, standard deviation method; and

c) Variability unknown plans, range method.

These plans have been explained in Appendix B.

5.2 1a8pectioll Level - In order to determine which of the samplingplans given in the standard are to be used in a particular case, havingdecided upon the quality requirement, it is necessary to decide upon the'Inspection Level'. The term 'Inspection Lever is used to designate therelative amount of inspection one is required to do. A higher inspectionlevel means relatively more inspection and consequently lesser risk of accept­ing lots of quality worse than the chosen AQL. That is to say, the higherthe inspection level, the greater is the protection against acceptance of lov­quality lots; but then the cost of inspection goes up. It may, therefore,be necessary to strike a compromise between a large sample which givesa reliable estimate of the lot quality and a small sample which reduces theinspection cost. Such a compromise can be arrived at by proper selectionof the inspection level.

5.2.1 Table 1 (set P 15), which is auxiliary to Tables 2, 3 and 4 givesfive inspection levels with the sample size in code letters (see 5.2.2)Inspection level I calls for the smallest sample size, thereby minimizingthe cost of inspection. Inspection level V gives relatively the largestsample size, thereby lessening both the risk of accepting bad quality Iotsand rejecting good quality lots but increasing the cost of inspection, Formajority of products under normal conditions of acceptance inspection,a reasonable compromise between the high inspection costs and tl,e ri-kinvolved may be achieved by taking the sample size corresponding to theinspection level IV.

5.2.2 Table 1 give! code letters B, 0, D, •. . .. to indicate the samplesizes under different inspection levels for varying lot sizes. It is in termsof a code letter selected from Table 1 that a plan will be selected from Tables2 to 4. These rode letters are known as 'sample size code letters' dMilJna!inzas they do a particular sample size. For example, J denotes a i~)npJc

11

II. 2500 (Pan D) -1965

of 25 for standard deviation method (s. Table 3) and 30 for ra.DF method(s" Table 4).

5.3 Lot AceeptablUty - The acceptability of a lot of materials submittedfor inspection shall be determined by using one of the sampling plans ass0­ciated with a specified value of AQL. Depending upon the specificationof one-sided or two-sided limits, the acceptability criteria for the threetypes of sampling plans would be as given in 5.3.1, 5.3.2 and 5.3.3.

5.3.1 YariabiliV' Know" M,thod5.3.1.1 For one-sided _(peci,ficati?n limits - On the basis of the AQL and

the sample size code letter chosen, the value of the sample size (n) and asuitable factor (kH

) shall be obtained from Table 2. The mean of the II testresults shall then be calculated and the lot shall be declared as acceptable if:

a) the value of the expression (x+k"a) <: V, when the upper speci­fication limit (U) is given;

ORb) the value of the expression (x-k·o»L, when the lower speci­

fication limit (l~) is given.

5.3.1.2 For two-sidld s/JIcification limits - The lot shall be declaredas acceptable if:

a) the value of the expression U~L <; the maximum value specified

below for the chosen AQL:

AQL (%) 0·10 0·15 0·25 0·40 0·65 1·00 1·50 2·50 4·00 6·50 10·00

Upper Linlit~

f _0_ 0·1520·1580-1650·1750·184 0·194 0·206 0·223 0·243 0·270 0·304o U-L

b) the value of the expression (x+k· (I) <. U, and

c) the value of the expression (x-}: o)"'>L.

5.3.2 Variability Unknown - Standard Demation M,t1uHl

5.3.2.1 Fur one-sided sp.,ijication limits - On the basis of the AQL .andthe sample size code letter chosen, the value of the sample size (n) and asuitable factor (k') shall be obtained from Table 3. The mean (x) and thestandard deviation (s) shall then be calculated from the test results andthe lot shall be declared as acceptable if:

a) the value of the expression (x+k's)< U, when the upper speci­fication limit (U) is given;

ORb) the value of the expression (x-k's»L, when the lower speci­

fication limit (L) is given.

12

II I 2500 (put D) • 1_

5.3.2.2 Ftw IuJtl-sitIMI sl*ijietUitJlI limits - The lot shall be declaredas acceptable if:

a) the value oCthe expression U~L<the maximum value specified

in Table 5 (Sit P 19) for the particular combination of the chosenAQL and the sample size code leuer,

b) the value of the expression (J+k's)<U. andc) the value of the expression (x-k's»L.

5.3.3 YaritJ1Jili~ Unknown - Rang, Mellaotl

5.3.3.1 For otll-sitktl sPecific4lio1l limits - On the basis or the AQL andthe sample size code letter chosen, the value of the sample size (n) and asuitable factor (k) shall be obtained from Table 4. The mean (.f) and therange (R) if the sample size is less than 10, or the mean range (R) if thesample size is 10 or more, shall then be calculated from the test results.The lot shan be declared as acceptable if:

a) the value of the expression (.f+kR) or (R+kR) <: U, when theupper specification limit (U) is given;

ORb) the ~cAlue of the expression (x-kR) or (R-k'R) ;> L, when the

lc ver specification limit (L) is given.

5.3.3.2 For lwo-sidMI sJueifieali01l limits - The lot shall be declared asacceptable if:

a) the value oCthe expression U~L or UR. L <: the maximum value

specified in Table 6 (s" P 20) for the particular combination ofthe chosen AQL and the sample siz~ code letter,

b) the value of the expression (R+kR) or (z+kR) <: U. andc) the value of the expression (J-lcR) or (i-ill) > L.

S.t Normal, .ed.aM1 ... Tlpt..... Iupecdoa

5.4.0 When the quality or the submitted loti shows significant I~it is desirable to make appropriate changes in the sampling plans. If thequality deteriorates, it is necessary to tighten the inspection; if the qu~lity

improves, it may be desirable to relax the inspection.

5....1 Nomtlll l1Uf*1itna - Ina~tion under a sampling plan that is inCoree for • particular product and producer is called cNormal Inspection'.It may be continued &I long as the quality of the product submitted it betterthan or equal to the chOlell AQ;L. The coDSiltency in maintaining this

13

IS I 2500 (Part D) • 1965

level of quality by the producer can be ascertained either from a continuousrecord of inspection data which can be used to estimate the process averageof the producer or from a knowledge of the proportion of the lots that arenot accepted. In case the quality becomes consistently better than thestipulated AQL, 'Reduced Inspection' may be undertaken. If, however,the quality becomes consistently worse than the chosen AQL, 'TightenedInspection' is to be resorted to.

5.4.2 Tightmetl InsjJ«tio" - Inspection shall be tightened either byraising the inspection level, that is, by selecting a sample size code letterhigher than the one adopted for normal inspection or by employing a smallerAQL. Since the former approach leads to an increased amount of inspec­tion, tightening is done by using a sampling plan with an AQL smallerthan that used previously.

5.4.2.1 The following criteria shall be applied for changing from normalto tightened inspection and vice versa:

a) If 2 out of the 5 (or less) consecutive lots have been rejected whileon normal inspection, change over to tightened inspection,

b) If, while on tightened inspection, 5 consecutive lots have beenaccepted, change over to normal inspection.

5.4.2.2 From the tables given in this standard the choice of a plan fortightened inspection shall be made in the following manner:

Retain the same sample size code letter as before but refer to an AQLa step lower than the AQL used for normal inspection. For example,if the AQL used for nonnal inspection is 4·0 percent for the code letterK (in Tables 2, 3 or 4), the AQL to be used for tightened inspection shallbe 2·5 percent for the same code letter.

5.4.2.3 In certain cases such tightened insrection may lead to anincrease in the size of the sample. For example, i an AQL of 0·65 percenthas been used for normal inspection for the sample size code letter D, theAQL and the code letter to be used for tightened inspection shall be 0·40and E respectively.

5.4.3 IUduretl I~,'ion - If the quality of the submitted lots is consistentlybetter than the AQL chosen, reduced inspection may be resorted to eitherby selecting lower sample size code letter than the one used for normalinspection or by relaxing the AQL. Since the former approach leadsto economies in inspection it is preferable, unless there is an agreementto the contrary, to reduce inspection by changing over to a plan with alower sample size code letter than the one adopted for normal inspection.

5.4.3.1 The following criteria may be applied for changing over fromnormal to reduced inspection and vice versa:

a) If none out of 10 consecutive lots has been rejected while on normalinspection. change over to reduced inspection.

14

IS s 2500 (Part D) • 1965

b) Ifa lot is rejected, and ifat the same time the rejected lot is precededby less than 10 lots accepted on reduced inspection, change overto normal inspection.

5.4.3.2 From the tables given in this standard, the choice of a planfor reduced inspection may be made in the following manner:

Retain the same AQL as before but refer to the sample size codeletter one step lower than that used for normal inspection. For example,if the code letter used for normal inspection is J with an AQL of 2·5percent (in Tables 2, 3 or 4), then the code letter to be wed for reducedinspection may be H for the same AQL.

5.4.3.3 In certain cases suitable plans for reduced inspection givenin 5.4.3.2 may not be available. Reduced inspection may then beresorted to by choosing the sample size code letter one step lower and theAQ.L one step higher than that used for 'normal inspection. For example,if the code letter D and the AQL of 0·65 percen t have been used for normalinspection, then the code letter and AQL to be used for reduced inspectionmay be C and 1·0 percent respectively.

6. TABLES AND ILLUSTRATIVE EXAMPLES

TABLE 1 SAMPLE SIZE CODE LET'l'ERS BY INSPECnON LEVELS ANDSIZES 01' LOTS

LoT 81Z&

2 to 89 u 15

16 " 2526 u 5051 It 100

101 u 150151" 300SOl" 500561 Jt I 000

1001 u 5000! OOJ tI 10000

10 001 and above

(CldWI .5.2.1 muJ 5.2.2, and E~ampl,s I, 2, 3 GIld.)

INSPECTION LaVELlA-

Il III IV v(S a m pI e Size Code Letters)

B B B B CB B B B DB B B C EB B B D FB B C E GB B D F HB C E G JB D F H KC E G J LD F H K ME G J L MF H K M N

15

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843

IS I 25GO (Pan D) • .115

~l:

The specified minimum yield point for certain steel castinp is 40·0kg/mm l . Suppose lots containing 400 items are submitted for inspectioo.A single sampling variables plan with inspection level III and aD AQ,Lof 2·5 percent is adopted for the purpose of inspection. Suppose the vari­ability (a) is known to be 1·0 kg/mml •

Reference to Table I then gives the sample size code letter F correspondingto which Table 2 shows the sample size 5 and factor i· equal to 1'39.

From the lot, 5 castings shall then be selected at random and their yieldpoints determined. Suppose the yield points for the 5 ca,tings are 42~lt

39·9. 40·7. 43·2 and 42'6.

The mean (x) is then found to be 41·7.

Also, the value of the expression (.f-k·o) cernes out to be 41·7 -) ·39 X 1·0=40·51. Since this value is greater than the lower specification limit

of4O'0, the lot shall be accepted.

The LTPD value associated with the above plan as read from Fig. 8(.. P 36) is obtained as 22 percent (with the probability of acceptancebeing 10 percent).

&.",pl, 2 :

i) The specified maximum resistance of a certain electrical componentis 660 ohms. Suppose lots containing 150 components are submitted forinspection. Ifit is agreed to use inspection level IV, an AQL of'one percent ..and single sampling variables plans Cor variability unknown (standarddeviation method), then reference to Tables I and 3 gives the sample size10 and the factor k' equal to 1·72.

Suppose the resistances (in ohms) of the 10 components selected at randomfrom the lot are 639, 640,650, 647, 662, 637. 652. 643. 657 and 649.

Then .i=647·6, and

s = J(639--647'6)1+(640-647~6jt+ + (649-647'6) I .

= .y65·38=8·09

The value·of the expression (.i+A's)=647·6+1·72x8·09=66I·~. Sincethis value is greater than the maximum specification limit of 660 ohms,the lot shall be rejected.

ii) In the above example, if it is intend-d to use the single samplingvariables plan for variability unknown (range method), then reference toTables 1 and 4 gives the sample size as 10 and factor k equal to 0·703.

21

IS I 2500(Part D) • 1965

Taking the above values of sample results, it is found that:.i=647·6

Range of tint 5 results=-662-639=~3

Range of Jast 5 results=657-637=20

Hence mean range (R) 23;-20 =21'5

The value of expression (x+k1() =647'6+0·703 X 21·5 =662·7. Sincethis value is greater than the maximum specification limit of 660 ohms.the lot shall be rejected.

'The LTPD value associated with the above plans can be read from Fig, 6(see P 34). Thus lots containing approximately 19'5 percent defectiveswould be accepted only 5 percent of the times by the above plans.

Exampl, 3:

The specified hardness range for certain types of spannen is 40 to 50liRe. Lots containing 2 000 spanners are submitted for inspection. Asingle sampling variables plan with inspection level V and an AQL of 4·0percent is adopted. Suppose the variability (a) is known to be 2-5 HRC.

Reference to Tables 1 and 2 gives the sample size as 20 and the factor

r equal to 1-38. The upper limit for U~ L is obtained as 0 243 from the

values under 53.1.2.

Applying the first condition given in 5.3.1.2, we have the following:

U:'L 50~~ =0·25 which is more than the permissible upper limit

of 0·243. Hence the lot is rejected straightway without drawing the 20items for sample inspection.

Extmf/Jle 4:i) The specified melting range for a certain type or wax is 60° to 70°C.

Lots containing 200 containers are submitted for inspection. A singlesampling variables ~Ian with variability unknown (standard deviationmethod) with inspection level IV and an AQL of 2'5 percent is adopted.

References to Tables I, 3 and 5 give the sample size 15, the factor k1 equal

to ....71lnd the upper limit for U~L equal to 0·284.

Suppose the melting points for the material in 15 containers selectedat random are:

6:4,5, 66·0w sse, 68-5, 69,5, 66-5, 67,0, 62-5, 66-0, 67-5, 64-0, 69,0, 70-0,66'0,66-5

22

IS I 2500 (Part D) - 1965

997·5Then compute mean (.i) =1"5"=6605

and s=-· /(63-5-66-5)1+ 0 •••• 0 +(66-5-66·5)'V 14

J67-00= 14=V4 '786 = 2-19

Applying the first condition given in 5.3.2.2, U~L = 2~~9 =0-219 which

is less than the permissible limit of 0-284.

Applying the second and third conditions, (~+1's)=66'5+1-47x2-1q=69-7 is less than the .naximum specification limit,

and (~-k's)=66-5-1'47x2-19=63-3is more than theminimumspeci­fication limit.

Hence the lot shall be accepted.

ii) In the above example, if it is intended to use the single samplingvariables plan for variability unknown (range method), then reference toTables 1, 4 and 6 gives the sample size 15, the factor Ie. equal to 0·610 and

upper limit for U~L equal to 0688.

Taking the above values of sample results, we obtain,

Mean (x) = 66'5Range of first 5 test results =69-5 -63-5 =6-0

Range of next 5 test results=67-S-62-5=5'0Range of last 5 test results=70-0-64-0=6-0

- 6-0+5'0+6-0Hence mean range (R) = 3 =5-7

Applying the first condition given in 5.3.3.2, U~L = ~~ =0·57 which is

less than the permissible limit of 0·688.

Applying the second and third conditions,

i+kR~66'5+0-610X5-7=70'O is equal to the maximum specificationlimit,

and ~-kR=66·5-0-610x5-7=63-0 is greater than the minimumspecification limit.

Hence the lot shall be accepted.

IS I 2500 (Part D) • 1965

APPENDIX A

(Clauses 2.1"1, 2.12, 2.13, 2.14 and 2.15)SYMBOLS

~ mean; if XI' KI,' ...• .x.. are the II measurements of the item in a

I th - XI+K1·••· +x..samp e, en x- n

a lot standard deviation; if KI. KI" •••••• •X)I are the .N measurementsof items in a lot and X is the corresponding mean, then:

I sample standard deviation; ifx.. X., • . • • • • • • , x.. are the II measure­ments of items in the sample, then:

s= /(Xl-X)I+ +(X..-X)I /("11+ +x,,')-M'V (n-I) V n-l

R range; if XI. Xl' ••••••• •X" are the II measurements of items in asample, arranged in the ascending order of magnitude, then:

R=x.-xi

R mean range; if RI • R., R,. are the ranges of 171 lUb.groupsof five observations each (so that sample size n=5 171), then:

R= RI+RI····· ..• R.m

k coefficient of R or R in the arceptance/reJection criterion for singlesampling plans by variables with variability unknown (rangemethod).

k' coefficient ofs in the acceptance/rejection criterion for single samplingplans by variables with variability unknown (standard deviationmethod).

k" coefficient of a in the acceptance/rejection criterion for linglesampling plans by variables WIth variability known.

U upper specification limit.L lower specification limit.<: less than or equal to.:> greatel' than or equal to.

24

IS I ~(PutD).l_

APPENDIX B

(Clause 5.1.2)

I'ORMUlJ£ AND T ABLES FOR CONSTRUCTING SINGLESAMPLING AQ.L-LTPD VAlUABLES PLANS FOR

ONE-SIDED SPECIFICATION LIMITS

B-O. STIPULATIONS OP THE PLANS (VALUES TO BE GIVEN)

III =- acceptable quality level (AQL) expressed as fraction defective/I'. = lot tolerance percent defective (LTPD) as fraction defective

" == producer', risk (fraction)Il =: consumer', risk (fraction)

L or U=lower or upper specification limit for item quality

I t is desired to construct single sampling variables plans such that lotswith P'l fraction defective would be accepted (1 - <) times, and lots withP'I fraction defective will be rejected (1- fJ) times. From P'I' /l.) I(. and fJread the corresponding' values, namely:

I , " " and I from Table 1/J1 P'. I(, ~

where t. is the normal deviate exceeded with probability *.8-1. VA1UABILITY KNOWN :METHOD

8-1.1 The acceptance criterion for this method is given as before by(!+i-u)< U or (R-Ya);> L Where the sample I~e (tI) and f(' have tobe found out. The values of ,. and k" which determine the desired samplingplan uniquely are given by the following fonnulz :

8=(,::~~")II

k-"'A' - ~r 1 vi"

8-2. VARIABILITY UNKNOWN-STANDARD DBVlATlONMETHOD

8-2.1 The acce~tance criterion for this method is given .. before by(z+!'s)<.U or (.i-i',,) is >L where the umple size <II> and I' have to be

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28

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6

IS , 2500 (Pan D) • 1965

found out. The values of Ie' and II which determine the desired samplingplan uniquely are given by the following formulz :

i' '0(. 'p'.+'fJ'P'.

~"+'p

_( 'f(,+IIJ)I 1'1+211- x--

Ip'1 - tp', 2

B-3. VARIABILITY UNKNOWN ... RANGE ME'J HOD

8-3.1 The acceptance criterion for this method is given as before by(.i+1R) or (.i+kR)<U OR (.i-kR) or (x-1R);>L [R is to be calculatedif the sample size IS less than 10. In case the sample size is 10 or more,R, based on the sub-~oupaof five samples each» is to be calculated]. Thevalues of" and II which determine the desired sampling plan uniquely areobtained as given in 8-3.1.1.

B-3.1.1 The values of II and Ie' are first obtained by using the formulzgiven under B-2. In case n comes out to be leu than or equal to 9~ k isdetermined from k=DIe' where the values of D are given below for thevarious sample sizes :

stnn/JU Sit" n Ya/IUS of D

2 0·886 2

3 0·590 8

4 0·...857

5 0·429 9

6 0-3946

7 0-3698

8 0-351 2

9 0·3367

I~ however, the value of" exceeds 9 and is not a multiple or 5, it shaUbe raised to the first higher number which is a multiple of 5 and then tis determined from k ==0·4299 k',

27

IS l 2500 (Pan D) • 19&5

APPENDIX C(No" 3 undn Claus, 5.0)

TIlE OPERATING CHARACTERISTIC CURVES

Q.l. Figures 1 to 11 (s. P 29-39) give the complete set of operating charac­teristic curves for the sampling plana based on variability unknown-«standard deviation method as given in this .tandard. These curves areobtained by plotting the cpen:ent of 10m expected to be accepted' against-percent defectives in submitted lots'. The curves given would also approxi­mately hold good for the plans pertaining to the variability known and rangemethods with the same sample size code letter.

c.2. For any specified value of the acceptable quality level (sel 2.17),all the OC Curves obtained for the different sample size code letten aregiven in the same ~re. Thus, there are 11 figures corresponding to the11 values of AQL from 0·10 to 10·00 percent specified in the standard.The sample size code letter corresponding to any plan bas been indicatedon the relevant OC Curves in each of the figures gtven.

c.3. For any percent defective, the percentage of loti expected to be acceptedmay be read from the OC Curve corresponding to the chosen samplingplan. A£ an example, for • plan with the sample size code letter Land AQL o£2·5 percent, the relevant OC Curve is given in Fig. 8. It maythen be aeen &om thia OC Cwve that 8 percent defective lobi would beaccepted in about 27 percent of the cases for the plan under consideration.

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