BS 6002-2:2007

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Sampling procedures for inspection by variables – General guide to single sampling plans indexed by acceptance quality limit (AQL) for lot-by-lot inspection of independent quality characteristics

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BSI	2007	96

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Categories: 03.120.30 - Application of statistical methods, BSI

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3	Contents Introduction 1 1 Scope 5 2 Normative references 6 3 Terms and definitions 7 4 Symbols 10 5 Acceptance quality limit (AQL) 12 6 Switching rules for normal, tightened and reduced inspection 13 7 Relationship to BS 6001-1 and BS 6002-1 14 8 Consumer protection 16 9 Planning 17 10 Choice between variables and attributes 18 11 Choice between the “s” and “s” methods 19 12 Choice of inspection level and AQL 19 13 Choice of sampling scheme 19 14 Preliminary operations 21 15 Standard univariate “s” method procedures 21 16 Standard multivariate “s” method procedures for independent quality characteristics 30 17 Standard univariate “s” method procedures 33 18 Standard multivariate “s” method procedures for independent quality characteristics 37 19 Standard multivariate combined “s” and “s” method procedures for independent quality characteristics 39 20 Procedure during continuing inspection 40 21 Normality and outliers 41 22 Records 41 23 Operation of switching rules 42 24 Discontinuation and resumption of inspection 43 25 Switching between the “s” and “s” methods 43 Annexes Annex A (normative) Table for determining the sample size code letter 44 Annex B (normative) Form k “s” method single sampling plans 45 Annex C (normative) Form k “s” method single sampling plans 48 Annex D (normative) Form p* “s” method single sampling plans 51 Annex E (normative) Form p* “s” method single sampling plans 54 Annex F (normative) Values of fs for maximum sample standard deviation (MSSD) 57 Annex G (normative) Values of fs for maximum process standard deviation (MPSD) 60 Annex H (normative) Estimating the process fraction nonconforming for sample size 3: “s” method 63 Annex I (normative) Values of cu for upper control limit on the sample standard deviation 66
4	Annex J (normative) Supplementary acceptability constants for qualifying towards reduced inspection 67 Annex K (normative) Procedures for obtaining s and s 68 Annex L (normative) Estimating the process fraction nonconforming 70 Annex M (informative) Consumer’s risk qualities 75 Annex N (informative) Producer’s risks 79 Annex O (informative) Operating characteristics for the “s” method 83 Bibliography 85 List of tables Table 1 – Example results 32 Table 2 – Example results with sample standard deviations redesignated as process standard deviations 38 Table 3 – Example results where some of the characteristics have known process standard deviations 40 Table A.1 – Sample size code letters and inspection levels 44 Table B.1 – Single sampling plans of Form k for normal inspection: “s” method 45 Table B.2 – Single sampling plans of Form k for tightened inspection: “s” method 46 Table B.3 – Single sampling plans of Form k for reduced inspection: “s” method 47 Table C.1 – Single sampling plans of Form k for normal inspection: “s” method 48 Table C.2 – Single sampling plans of Form k for tightened inspection: “s” method 49 Table C.3 – Single sampling plans of Form k for reduced inspection: “s” method 50 Table D.1 – Single sampling plans of Form p* for normal inspection: “s” method 51 Table D.2 – Single sampling plans of Form p* for tightened inspection: “s” method 52 Table D.3 – Single sampling plans of Form p* for reduced inspection: “s” method 53 Table E.1 – Single sampling plans of Form p* for normal inspection: “s” method 54 Table E.2 – Single sampling plans of Form p* for tightened inspection: “s” method 55 Table E.3 – Single sampling plans of Form p* for reduced inspection: “s” method 56 Table F.1 – Values of fs for maximum sample standard deviation for combined control of double specification limits: normal inspection, “s” method 57 Table F.2 – Values of fs for maximum sample standard deviation for combined control of double specification limits: tightened inspection, “s” method 58 Table F.3 – Values of fs for maximum sample standard deviation for combined control of double specification limits: reduced inspection, “s” method 59 Table G.1 – Values of fs for maximum process standard deviation for combined control of double specification limits: “s” method 60
5	Table G.2 – Values of fs for maximum process standard deviation (MPSD) for separate control of double specification limits: “s” method 61 Table G.3 – Values of fs for maximum process standard deviation (MPSD) for complex control of double specification limits: “s” method 62 Table H.1 – Estimated process fraction nonconforming, pˆ, as a function of the quality statistic Q 63 Table I.1 – Values of cu for upper control limit on the sample standard deviation 66 Table J.1 – Supplementary acceptability constants for qualifying towards reduced inspection 67 Table L.1 – Values of an for normal approximation to pˆ 72 Table M.1.s – Consumer’s risk quality (in percent) for normal inspection: “s” method 76 Table M.1.s – Consumer’s risk quality (in percent) for normal inspection: “s” method 76 Table M.2.s – Consumer’s risk quality (in percent) for tightened inspection: “s” method 77 Table M.2.s – Consumer’s risk quality (in percent) for tightened inspection: “s” method 77 Table M.3.s – Consumer’s risk quality (in percent) for reduced inspection: “s” method 78 Table M.3.s – Consumer’s risk quality (in percent) for reduced inspection: “s” method 78 Table N.1.s – Producer’s risk (in percent) for normal inspection: “s” method 80 Table N.1.s – Producer’s risk (in percent) for normal inspection: “s” method 80 Table N.2.s – Producer’s risk (in percent) for tightened inspection: “s” method 81 Table N.2.s – Producer’s risk (in percent) for tightened inspection: “s” method 81 Table N.3.s – Producer’s risk (in percent) for reduced inspection: “s” method 82 Table N.3.s – Producer’s risk (in percent) for reduced inspection: “s” method 82
7	Foreword
9	Introduction a) automatic protection to the consumer (by means of a switch to tightened inspection or discontinuation of sampling inspection) should a deterioration in quality be detected; b) an incentive (at the discretion of the responsible authority) to reduce inspection costs (by means of a switch to a smaller sample size) should consistently good quality be achieved.
11	Summary table
12	Summary table (concluded)
13	1 Scope NOTE It follows that, where there are different producers or production processes, this British Standard will need to be applied to each one separately. a) where the inspection procedure is to be applied to a continuing series of lots of discrete products all supplied by one producer using one production process; b) where the quality characteristics of the items of product are measurable on a continuous scale; c) where the measurement error is negligible (i.e. with a standard deviation no more than 10% of the corresponding process standard deviation); d) where production is stable (under statistical control) and the quality characteristics are distributed, at least to a close approximation, according to normal distributions; e) where, in the case of multiple quality characteristics, the characteristics are independent, or almost independent, of one another; f) where a contract or standard defines a lower specification limit U, an upper specification limit L, or both on each of the qu… 2) x u U (i.e. the upper specification limit is not violated); 3) x W L and x u U (i.e. neither the lower nor the upper specification limit is violated). 4) xii W Li; 5) xii u Ui; 6) xii W Li and xii u Ui.
14	2 Normative references
15	3 Terms and definitions 3.1 inspection by variables 3.2 sampling inspection 3.3 acceptance sampling inspection acceptance sampling 3.4 acceptance sampling inspection by variables 3.5 process fraction nonconforming 3.6 acceptance quality limit AQL NOTE See Clause 5. 3.7 quality level 3.8 consumer’s risk quality CRQ 3.9 producer’s risk PR
16	3.10 nonconformity 3.11 nonconforming unit 3.12 “s” method acceptance sampling plan NOTE See Clause 15. 3.13 “s” method acceptance sampling plan NOTE See Clause 16. 3.14 specification limit 3.15 lower specification limit L 3.16 upper specification limit U 3.17 combined control NOTE See 5.3, 15.3.2 and 17.3. 3.18 separate control NOTE See 5.3, 15.3.3 and 17.2. 3.19 complex control NOTE See 5.3, 15.3.4 and 17.3.
17	3.20 acceptability constant k, p* NOTE See 15.2 and 16.2. 3.21 quality statistic Q NOTE See 15.2 and 16.2. 3.22 lower quality statistic QL NOTE See Clause 4, 15.2 and 16.2. 3.23 upper quality statistic QU NOTE See Clause 4, 15.2 and 16.2. 3.24 maximum sample standard deviation MSSD smax NOTE See 15.3. 3.25 maximum process standard deviation MPSD smax NOTE See 17.2 and 17.3.
18	3.26 switching rule NOTE See Clause 23. 3.27 measurement 4 Symbols 4.1 Univariate symbols
19	4.2 Multivariate symbols
20	5 Acceptance quality limit (AQL) 5.1 Concept 5.2 Use 5.3 Specifying AQLs a) combined control of double specification limits, where nonconformity beyond both limits belongs to the same class, to which a single AQL applies; b) separate control, where nonconformity beyond both limits belongs to different classes, to which separate AQLs apply; c) complex control, where nonconformity beyond the limit that is of greater seriousness belongs to one class to which one AQL applies, and nonconformity beyond both limits combined belongs to another class to which a larger AQL applies.
21	5.4 Preferred AQLs 5.5 Caution 5.6 Limitation 6 Switching rules for normal, tightened and reduced inspection
22	7 Relationship to BS 6001-1 and BS 6002-1 7.1 Relationship to BS 6001-1 7.1.1 Similarities to BS 6001-1 a) This British Standard is complementary to BS 6001-1; the two documents share a common philosophy and, as far as possible, their procedures and vocabulary are the same. b) Both use the AQL to index the sampling plans and the preferred values used in this document are identical with those given for percent nonconforming in BS 6001-1 (i.e. from 0.01% to 10%). c) In both British Standards, lot size and inspection level (inspection level II in default of other instructions) determine a s… d) The switching rules are essentially equivalent. e) The classification of nonconformities by degree of seriousness into class A, class B, etc., remains unchanged.
23	7.1.2 Differences from BS 6001-1 a) Determination of acceptability. Acceptability for a BS 6001-1 attributes sampling plan for percent nonconforming is determine… b) Normality. In BS 6001-1, there is no requirement relating to the distribution of the characteristics. However, in this Britis… c) Independence. In BS 6001-1 there is no requirement relating to independence of multiple quality characteristics. However, in … d) Operating characteristic curves (OC curves). The OC curves of the variables plans in this standard are not identical to those… e) Producer’s risk. For process quality precisely at the AQL, the producer’s risk that a lot will not be accepted tends to decre… f) Sample sizes. The variables sample sizes for combinations of sample size code letter and AQL are usually smaller than the cor… g) Double sampling plans. Double sampling plans by variables are presented separately, in BS 6002-3.
24	h) Multiple sampling plans. No multiple sampling plans by variables are given in this British Standard. i) Average outgoing quality limit (AOQL). The AOQL concept is mainly of value when 100% inspection and rectification is feasible… 7.2 Relationship to BS 6002-1 7.2.1 Similarities to BS 6002-1 a) This British Standard is complementary to BS 6002-1 and the two documents both present single sampling procedures for inspection by variables. b) The procedures of BS 6002-1 are included in this British Standard but referred to as of “Form k”. 7.2.2 Differences from BS 6002-1 a) Part 2 is more general than Part 1, as it includes multivariate procedures for independent quality characteristics and also includes procedures for separate or combined control of double specification limits. b) Because Form k procedures may only be used for a single quality characteristic with a single AQL, Part 2 also includes the more general Form p* procedures. 8 Consumer protection 8.1 Use of individual plans
25	8.2 Consumer’s risk quality tables 8.3 Producer’s risk tables 8.4 Operating characteristic curves 9 Planning
26	10 Choice between variables and attributes a) In terms of economics, it is necessary to compare the total cost of the relatively simple inspection of a larger number of it… b) In terms of the knowledge gained, the advantage lies with inspection by variables, as the information obtained indicates more precisely how good the product is. Earlier warning will therefore be given if the quality is slipping. c) An attributes scheme can be more readily understood and accepted; for example, it may at first be difficult to accept that, w… d) A comparison of the size of the samples required for the same AQL from standard plans for inspection by attributes (i.e. from… e) Inspection by variables is particularly appropriate in conjunction with the use of control charts for variables. f) Variables sampling has a substantial advantage when the inspection process is expensive, for example in the case of destructive testing. g) A variables scheme becomes relatively more complicated to operate as the number of quality characteristics and the number of measurements to be taken on each item increases. h) The use of this British Standard is only applicable when there is reason to believe that the distribution of measurements of …
27	11 Choice between the “s” and “s” methods 12 Choice of inspection level and AQL 13 Choice of sampling scheme 13.1 Standard plans
28	13.2 Special plans
29	14 Preliminary operations a) check that production is considered to be continuing and that the distribution of the quality characteristics can be considered to be normal and independent; b) check for each quality characteristic separately whether the “s” method is to be used initially or whether the process standard deviation is stable and known, in which case the “s” method should be used; c) check that the inspection level to be used has been designated. If none has been given, inspection level II should be used; d) check, for every quality characteristic with double specification limits, whether the limits are under combined, separate or … e) check that an AQL has been designated for each class of nonconformity, and that it is one of the preferred AQLs for use with this British Standard. If it is not, then the tables are not applicable. 15 Standard univariate “s” method procedures 15.1 Obtaining a plan, sampling and preliminary calculations a) With the inspection level given (normally this will be II) and with the lot size, obtain the sample size code letter using Table A.1. b) For a single specification limit, enter Table B.1, B.2 or B.3 as appropriate with this code letter and the AQL, and obtain th…
30	c) Take a random sample of size n, measure the characteristic x in each item and then calculate the sample mean, and the estimat… 15.2 Form k acceptability criterion for the “s” method
32	15.3 Form p* acceptability criterion for the “s” method 15.3.1 Introduction 15.3.2 Combined control for the “s” method 15.3.2.1 General
33	15.3.2.2 Combined control for the “s” method with n = 3
34	15.3.2.3 Combined control for the “s” method with n = 4
36	15.3.2.4 Combined control for the “s” method with n W 5 – Exact method
37	15.3.2.5 Combined control for the “s” method with n W 5 – Approximative method
38	15.3.3 Separate control for the “s” method 15.3.4 Complex control for the “s” method 16 Standard multivariate “s” method procedures for independent quality characteristics 16.1 General methodology
39	16.2 Example a) by selecting two random samples from the lot, one of size 18 and one of size 24; or b) by randomly selecting a sub-sample of 18 items from the random sample of 24 items.
40	Table 1 Example results
41	17 Standard univariate “s” method procedures 17.1 Obtaining a plan, sampling and preliminary calculations a) With the inspection level given (normally this will be II) and with the lot size, obtain the sample size code letter using Table A.1. b) For a single specification limit, enter Table C.1, C.2 or C.3 as appropriate with this code letter and the AQL, and obtain th… c) Take a random sample of size n, measure the characteristic under inspection, x, for all items in the sample and calculate the… 17.2 Acceptability criterion for a single specification limit or for double specification limits with separate control
43	17.3 Acceptability criterion for double specification limits with combined or complex control a) Before sampling, determine the value of the factor fs by entering Table G.1 (for combined control) with the single AQL or by entering Table G.3 (for complex control) with both AQLs. b) Calculate the maximum allowable value of the process standard deviation, using the formula smax = (U p L)fs for the MPSD. c) Compare the value of the process standard deviation s with smax. If s exceeds smax, the process is unacceptable and sampling inspection is discontinued until it is demonstrated that the process variability has been adequately reduced. d) If s u smax, then use the lot size and given inspection level to determine the sample size code letter from Table A.1. e) From the sample size code letter, AQL and inspection severity (i.e. whether inspection is normal, tightened or reduced) determine the sample size, n, and acceptability constant, p, from Table E.1, E.2 or E.3. f) Select a random sample of size n from the lot and calculate the sample mean . g) Using the method given in L.2.2, calculate pˆU, pˆL and pˆ = pˆU + pˆL. h) If pˆ > p, the lot is not acceptable for either combined or complex control and no other calculations or comparisons are required. i) For combined control the lot is acceptable if pˆ u p. j) For complex control, determine from Table E.1, E.2 or E.3 the Form p acceptability constant for the single specification lim…
45	18 Standard multivariate “s” method procedures for independent quality characteristics 18.1 General methodology
46	18.2 Example Table 2 Example results with sample standard deviations redesignated as process standard deviations
47	19 Standard multivariate combined “s” and “s” method procedures for independent quality characteristics 19.1 General methodology 19.2 Example
48	Table 3 Example results where some of the characteristics have known process standard deviations 20 Procedure during continuing inspection a) the characteristic being inspected is normally distributed; b) records are kept; c) the switching rules are obeyed.
49	21 Normality and outliers 21.1 Normality 21.2 Outliers 22 Records 22.1 Control charts 22.2 Lots that are not accepted
50	23 Operation of switching rules a) these lots would have been acceptable if the AQL had been one step tighter; b) production is in statistical control; c) reduced inspection is considered desirable by the responsible authority. a) a lot is not accepted; b) production becomes irregular or delayed; c) reduced inspection is no longer considered desirable by the responsible authority.
51	24 Discontinuation and resumption of inspection 25 Switching between the “s” and “s” methods 25.1 Estimating the process standard deviation 25.2 State of statistical control 25.3 Switching from the “s” method to the “s” method 25.4 Switching from the “s” method to the “s” method
52	Annex A (normative) Table for determining the sample size code letter Table A.1 Sample size code letters and inspection levels
53	Annex B (normative) Form k “s” method single sampling plans Table B.1 Single sampling plans of Form k for normal inspection: “s” method
54	Table B.2 Single sampling plans of Form k for tightened inspection: “s” method
55	Table B.3 Single sampling plans of Form k for reduced inspection: “s” method
56	Annex C (normative) Form k “s” method single sampling plans Table C.1 Single sampling plans of Form k for normal inspection: “s” method
57	Table C.2 Single sampling plans of Form k for tightened inspection: “s” method
58	Table C.3 Single sampling plans of Form k for reduced inspection: “s” method
59	Annex D (normative) Form p* “s” method single sampling plans Table D.1 Single sampling plans of Form p* for normal inspection: “s” method
60	Table D.2 Single sampling plans of Form p* for tightened inspection: “s” method
61	Table D.3 Single sampling plans of Form p* for reduced inspection: “s” method
62	Annex E (normative) Form p* “s” method single sampling plans Table E.1 Single sampling plans of Form p* for normal inspection: “s” method
63	Table E.2 Single sampling plans of Form p* for tightened inspection: “s” method
64	Table E.3 Single sampling plans of Form p* for reduced inspection: “s” method
65	Annex F (normative) Values of fs for maximum sample standard deviation (MSSD) Table F.1 Values of fs for maximum sample standard deviation for combined control of double specification limits: normal inspection, “s” method
66	Table F.2 Values of fs for maximum sample standard deviation for combined control of double specification limits: tightened inspection, “s” method
67	Table F.3 Values of fs for maximum sample standard deviation for combined control of double specification limits: reduced inspection, “s” method
68	Annex G (normative) Values of fs for maximum process standard deviation (MPSD) Table G.1 Values of fs for maximum process standard deviation for combined control of double specification limits: “s” method
69	Table G.2 Values of fs for maximum process standard deviation (MPSD) for separate control of double specification limits: “s” method
70	Table G.3 Values of fs for maximum process standard deviation (MPSD) for complex control of double specification limits: “s” method
71	Annex H (normative) Estimating the process fraction nonconforming for sample size 3: “s” method Table H.1 Estimated process fraction nonconforming, pˆ, as a function of the quality statistic Q
72	Table H.1 Estimated process fraction nonconforming, pˆ, as a function of the quality statistic Q (continued)
73	Table H.1 Estimated process fraction nonconforming, pˆ, as a function of the quality statistic Q (concluded)
74	Annex I (normative) Values of cu for upper control limit on the sample standard deviation Table I.1 Values of cu for upper control limit on the sample standard deviation
75	Annex J (normative) Supplementary acceptability constants for qualifying towards reduced inspection Table J.1 Supplementary acceptability constants for qualifying towards reduced inspection
76	Annex K (normative) Procedures for obtaining s and s K.1 Procedure for obtaining s
77	K.2 Procedure for obtaining s
78	Annex L (normative) Estimating the process fraction nonconforming L.1 General L.2 Exact formulae L.2.1 The exact MVUE estimator of p for the “s” method
79	L.2.2 The exact MVUE estimator of p for the “s” method L.3 Approximative procedure for the “s” method with n W 5 a) Calculate Q = (U p )/s and/or ( p L)/s; b) Calculate ; c) If x u 0, then pˆ = 0, while if x W 1, then pˆ = 1 otherwise, continue with steps d) to g). d) Calculate where is given in Table L.1 below;
80	e) Calculate ; f) If w W 0, set , g) Look up pˆ = 9(t) Íin tables of the standard normal distribution function. Table L.1 Values of an for normal approximation to pˆ
81	L.4 Simplified exact formula for pˆ for the “s” method with n = 3
82	L.5 Simplified exact formula for for the “s” method with n = 4
83	Annex M (informative) Consumer’s risk qualities
84	Table M.1.s Consumer’s risk quality (in percent) for normal inspection: “s” method Table M.1.s Consumer’s risk quality (in percent) for normal inspection: “s” method
85	Table M.2.s Consumer’s risk quality (in percent) for tightened inspection: “s” method Table M.2.s Consumer’s risk quality (in percent) for tightened inspection: “s” method
86	Table M.3.s Consumer’s risk quality (in percent) for reduced inspection: “s” method Table M.3.s Consumer’s risk quality (in percent) for reduced inspection: “s” method
87	Annex N (informative) Producer’s risks
88	Table N.1.s Producer’s risk (in percent) for normal inspection: “s” method ‘ Table N.1.s Producer’s risk (in percent) for normal inspection: “s” method
89	Table N.2.s Producer’s risk (in percent) for tightened inspection: “s” method Table N.2.s Producer’s risk (in percent) for tightened inspection: “s” method
90	Table N.3.s Producer’s risk (in percent) for reduced inspection: “s” method Table N.3.s Producer’s risk (in percent) for reduced inspection: “s” method
91	Annex O (informative) Operating characteristics for the “s” method O.1 Formula for probability of acceptance O.2 Example O.3 Comparison with tabulated value for the “s” method
92	Chart A Sample size code letters of standard single sampling plans for specified qualities at 95% and 10% probabilities of acceptance
93	Bibliography

Additional information

Status	Withdrawn
Title	Sampling procedures for inspection by variables – General guide to single sampling plans indexed by acceptance quality limit (AQL) for lot-by-lot inspection of independent quality characteristics
Replaces	BS 6002-1:1993
Publisher	BSI
Committee	SS/5
Pages	96
Publication Date	2007-02-28
Withdrawn Date	2013-09-30
Replaced By	BS ISO 3951-2:2013
ISBN	978 0 580 50164 7
Standard Number	BS 6002-2:2007
Descriptors	Normal distribution, Inspection by variables, Statistical quality control, Power spectra, Inspection, Quality assurance, Standard deviation, Control charts, Quality control, Acceptance inspection, Mathematical calculations, Sampling tables, Sampling methods
ICS Codes	03.120.30 - Application of statistical methods

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