ISO 3951-4:2011
(Main)Sampling procedures for inspection by variables — Part 4: Procedures for assessment of declared quality levels
Sampling procedures for inspection by variables — Part 4: Procedures for assessment of declared quality levels
ISO 3951-4:2011 establishes sampling plans and procedures by variables that can be used to assess whether the quality level of an entity (lot, process, etc.) conforms to a declared value. The sampling plans have been devised so that their operating characteristic curves match those of the corresponding attributes plans in ISO 2859-4 as closely as possible, so that the choice between using sampling by attributes and sampling by variables is not influenced by attempts to increase the chance of accepting an incorrectly declared quality level. In this part of ISO 3951, there is a risk of between 1,4 % and 8,2 % of contradicting a correct declared quality level. The risk is 10 % of failing to contradict an incorrect declared quality level which is related to the limiting quality ratio. Sampling plans are provided corresponding to three levels of discriminatory ability, and for the cases of unknown and known process standard deviation. In contrast to the procedures in the other parts of ISO 3951, the procedures in ISO 3951-4:2011 are not applicable to acceptance assessment of lots. Generally, the balancing of the risks of reaching incorrect conclusions in assessment procedures will differ from the balancing in the procedures for acceptance sampling. ISO 3951-4:2011 may be used for various forms of quality inspection in situations where objective evidence of conformity to some declared quality level is to be provided by means of inspection of a sample. The procedures are applicable to entities such as lots, process output, etc. that allow random samples of individual items to be taken from the entity. The sampling plans provided are applicable, but not limited, to inspection of a variety of products such as: end items; components and raw materials; operations; materials in process; supplies in storage; maintenance operations; data or records; administrative procedures. The procedures are intended to be used when the quality characteristics are measurable variables that are independent and normally distributed, and where the quantity of interest is the fraction of items that are nonconforming.
Règles d'échantillonnage pour les contrôles par mesures — Partie 4: Procédures pour l'évaluation des niveaux déclarés de qualité
General Information
Standards Content (Sample)
INTERNATIONAL ISO
STANDARD 3951-4
First edition
2011-08-15
Sampling procedures for inspection by
variables —
Part 4:
Procedures for assessment of declared
quality levels
Règles d'échantillonnage pour les contrôles par mesures —
Partie 4: Procédures pour l'évaluation des niveaux déclarés de qualité
Reference number
ISO 3951-4:2011(E)
©
ISO 2011
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ISO 3951-4:2011(E)
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ISO 3951-4:2011(E)
Contents Page
Foreword .iv
Introduction.v
1 Scope.1
2 Normative references.2
3 Terms, definitions, symbols and abbreviations.2
4 Principles.3
5 Declared quality level (DQL).4
6 Sampling plans .5
7 Operating a sampling plan .8
8 Further information .15
Annex A (informative) Method of matching variables plans to attributes plans .20
Annex B (informative) Examples of use of the procedures.21
Bibliography.25
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ISO 3951-4:2011(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 3951-4 was prepared by Technical Committee ISO/TC 69, Application of statistical methods,
Subcommittee SC 5, Acceptance sampling.
ISO 3951 consists of the following parts, under the general title Sampling procedures for inspection by
variables:
⎯ Part 1: Specification for single sampling plans indexed by acceptance quality limit (AQL) for lot-by-lot
inspection for a single quality characteristic and a single AQL
⎯ Part 2: General specification for single sampling plans indexed by acceptance quality limit (AQL) for
lot-by-lot inspection of independent quality characteristics
⎯ Part 3: Double sampling schemes indexed by acceptance quality limit (AQL) for lot-by-lot inspection
⎯ Part 4: Procedures for assessment of declared quality levels
⎯ Part 5: Sequential sampling plans indexed by acceptance quality limit (AQL) for inspection by variables
(known standard deviation)
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ISO 3951-4:2011(E)
Introduction
The procedures in this part of ISO 3951 differ in their scope from the procedures in ISO 3951 Parts 1, 2, 3 and
5. The acceptance sampling procedures that are specified in ISO 3951 Parts 1, 2, 3 and 5 are intended to be
used in bilateral agreements between two parties. Those acceptance sampling procedures are intended to be
used as simple, pragmatic rules for releasing product after inspection of only a limited sample of a
consignment, and therefore the procedures do not make reference (either explicitly or implicitly) to any
formally declared quality level.
Under acceptance sampling, there is no sharp borderline between quality levels that should be considered
acceptable and qualities that should be rejected by the procedure. For the procedures in ISO 3951 Parts 1, 2,
3 and 5, the two parties agree upon some limiting quality level (AQL) which is the worst tolerable process
average when a continuing series of lots is submitted. The switching rules and the sampling schemes in those
four standards are designed to encourage the suppliers to have process averages consistently better than the
AQL selected. In order to keep sample sizes moderate, the protection against accepting individual lots of
inferior quality may be less than that provided by sampling plans targeted for sentencing individual lots.
Procedures in ISO 3951 Parts 1, 2, 3 and 5 are well suited for acceptance sampling purposes, but they should
not be used in reviews, audits, etc. to verify a quality that has been declared for some entity. The main reason
is that the procedures have been indexed in terms of quality levels that are relevant solely for the pragmatic
purposes of acceptance sampling, and the various risks have been balanced accordingly.
The procedures in this part of ISO 3951 have been developed as a response to the growing need for sampling
procedures suitable for formal, systematic inspections such as reviews or audits. When performing such a
formal inspection, it is necessary for the authority to consider the risk of reaching an incorrect conclusion, and
to take this risk into account in planning and executing the review/audit/testing, etc.
This part of ISO 3951 provides guidance and rules to assist the user in taking this risk into account in an
informed manner.
The rules in this part of ISO 3951 have been devised such that there is only a small, limited risk of
contradicting the declared quality level when in fact the actual level conforms to the declared level.
If it were also desired that there should be a similarly small risk of not contradicting the declared quality level
when in fact the actual quality level does not conform to the declared quality level, then it would be necessary
to investigate a rather large sample. Therefore, in order to obtain the benefit of a moderate sample size, the
procedures in this part of ISO 3951 have been devised in such a way that they allow a somewhat higher risk
of failing to contradict the declared quality level when in fact the actual quality level does not conform to the
declared quality level.
The wording of the result of the assessment should reflect this unbalance between the risks of reaching
incorrect conclusions.
When the sample result contradicts the declared quality level, there is strong evidence of nonconformance to
the declared quality level.
When the sample result does not contradict the declared quality level, this should be understood as “we have
not, in this limited sample, found strong evidence of nonconformance to the declared quality level”.
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INTERNATIONAL STANDARD ISO 3951-4:2011(E)
Sampling procedures for inspection by variables —
Part 4:
Procedures for assessment of declared quality levels
1 Scope
This part of ISO 3951 establishes sampling plans and procedures by variables that can be used to assess
whether the quality level of an entity (lot, process, etc.) conforms to a declared value. The sampling plans
have been devised so that their operating characteristic curves match those of the corresponding attributes
plans in ISO 2859-4 as closely as possible, so that the choice between using sampling by attributes and
sampling by variables is not influenced by attempts to increase the chance of accepting an incorrectly
declared quality level. In this part of ISO 3951, there is a risk of between 1,4 % and 8,2 % of contradicting a
correct declared quality level. The risk is 10 % of failing to contradict an incorrect declared quality level which
is related to the limiting quality ratio (see Clause 4). Sampling plans are provided corresponding to three levels
of discriminatory ability, and for the cases of unknown and known process standard deviation.
In contrast to the procedures in the other parts of ISO 3951, the procedures in this part of ISO 3951 are not
applicable to acceptance assessment of lots. Generally, the balancing of the risks of reaching incorrect
conclusions in assessment procedures will differ from the balancing in the procedures for acceptance
sampling.
This part of ISO 3951 may be used for various forms of quality inspection in situations where objective
evidence of conformity to some declared quality level is to be provided by means of inspection of a sample.
The procedures are applicable to entities such as lots, process output, etc. that allow random samples of
individual items to be taken from the entity.
The sampling plans provided in this part of ISO 3951 are applicable, but not limited, to inspection of a variety
of products such as
⎯ end items,
⎯ components and raw materials,
⎯ operations,
⎯ materials in process,
⎯ supplies in storage,
⎯ maintenance operations,
⎯ data or records, and
⎯ administrative procedures.
The procedures are intended to be used when the quality characteristics are measurable variables that are
independent and normally distributed, and where the quantity of interest is the fraction of items that are
nonconforming.
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ISO 3951-4:2011(E)
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.
ISO 2859-4:2002, Sampling procedures for inspection by attributes — Part 4: Procedures for assessment of
declared quality levels
ISO 3534-1, Statistics — Vocabulary and symbols — Part 1: General statistical terms and terms used in
probability
ISO 3534-2, Statistics — Vocabulary and symbols — Part 2: Applied statistics
ISO 3951-2: 2006, Sampling procedures for inspection by variables — Part 2: General specification for single
sampling plans indexed by acceptance quality limit (AQL) for lot-by-lot inspection of independent quality
characteristics
ISO 9000, Quality management systems — Fundamentals and vocabulary
3 Terms, definitions, symbols and abbreviations
3.1 Terms and definitions
For the purposes of this part of ISO 3951, the terms and definitions given in ISO 3534-1, ISO 3534-2,
ISO 3951-2 and ISO 9000 and the following apply.
3.1.1
quality ratio
ratio of the actual to the declared quality level of the entity under investigation
3.1.2
limiting quality ratio
LQR
value of the quality ratio that is limited to a small risk (10 % in this part of ISO 3951) of failing to contradict an
incorrect declared quality level
3.2 Symbols and abbreviated terms
The symbols and abbreviated terms used in this part of ISO 3951 are as follows:
B (.) Distribution function of the symmetric beta distribution with both parameters equal to v
v
B(v,v) Beta function with both arguments equal to v, i.e. B(v,v)=Γ(v)Γ(v)/Γ(v) where Γ(v) is the gamma
function (see below)
D Declared quality level (as a symbol)
DQL Declared Quality Level (as an acronym)
k Form k acceptability constant under the “s” method, used when the sample standard deviation is
s
unknown
k Form k acceptability constant under the “σ” method, used when the process standard deviation is
σ
presumed to be known
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ISO 3951-4:2011(E)
L Lower specification limit (as a subscript, denotes the value at L)
LQR Limiting Quality Ratio (as an acronym)
m Number of quality characteristics, all assumed to be independent and normally distributed
n Sample size under the “s” method
s
n Sample size under the “σ” method
σ
OC Operating Characteristic
p Process fraction nonconforming in the entity
pˆ Estimate of the fraction nonconforming in the entity
pˆ Estimate of the combined fraction nonconforming at both specification limits, i.e. pˆˆ=+ppˆ
c c LU
p * Form p* acceptability constant (for both the “s” and “σ” methods)
Q Quality statistic (see 7.2.2 and 7.3.2)
s Sample standard deviation
U Upper specification limit (as a subscript, denotes the value at U)
x Sample mean
Φ(.) Standard normal distribution function
∞
v−1
Γ()v Gamma function, defined by Γ=()vt exp(−t)dt for ν > 0
∫
0
σ Process standard deviation
4 Principles
In any assessment procedure based on sampling, there will be an inherent uncertainty due to possible
sampling fluctuations. The procedures in this part of ISO 3951 have been conceived so as to lead to
contradiction of the declared quality level only when there is sufficient evidence to support a conclusion that
the actual quality is poorer than the declared quality level.
The plans have been devised in such a way that their operating characteristic curves match those of the
corresponding attributes plans ISO 2859-4 as closely as possible. Details of the matching method are given in
Annex A. The attributes plans of ISO 2859-4 were selected such that when the actual quality level is equal to
or better than the declared quality level, the risk is less than 5 % of contradicting the declared value. It follows
that when the actual quality level is worse than the declared quality level, there is a risk that the procedures
will fail to contradict an incorrect declared quality level. Owing to the fact that the match between
corresponding OC curves in ISO 2859-4 and ISO 3951-4 is imperfect, the corresponding risk in this part of
ISO 3951 varies around 5 %.
This risk depends on the value of the quality ratio, i.e. the ratio between the actual and the declared quality
level. The limiting quality ratio, LQR, is introduced to denote the highest quality ratio considered tolerable.
When the actual quality level is LQR times the declared quality level, the procedures in this part of ISO 3951
have a risk of 10 % of failing to contradict the declared quality level (corresponding to a 90 % probability of
contradicting the incorrect declared quality level).
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ISO 3951-4:2011(E)
Three LQR levels I, II and III are considered; details of the three LQR levels provided in this part of ISO 3951
are given in 6.1. Sampling plans are provided for both the case where the process standard deviation is
unknown (the “s” method) and the case where it is known (the “σ” method). (See ISO 3951-2 for details on the
implementation of sampling by variables plans.)
The sampling plans provided in this part of ISO 3951 are indexed by the limiting quality ratio (LQR) level and
the declared quality level (DQL) and are provided in Table 1.
5 Declared quality level (DQL)
The DQL together with the LQR level is used for indexing the sampling plans provided in this part of ISO 3951.
The values of DQL in the tables are known as preferred DQLs. The series of preferred DQL values
correspond to the series of preferred AQLs for inspection for nonconforming items given in ISO 3951-1.
There shall be a sound basis for the DQL used. The DQL shall not be deliberately overstated or understated.
When a DQL is designated for a certain type of nonconformity, it indicates that the supplier has good reason
to believe that the quality is not worse than this designated value.
CAUTION — When the DQL is estimated from a sample taken from the entity of interest, the
procedures in this International Standard shall not be used. Such a verification of an estimate from a
sample requires that the sample size and inspection result be taken into account in order to
incorporate the uncertainty associated with the estimate. This uncertainty affects the assessment of
the risks of making incorrect conclusions on the actual status of the entity of interest. Such
verification usually requires larger sample sizes than those used in the procedures described in this
part of ISO 3951.
Table 1 — Master table of sampling plans
DQL LQR Level I LQR Level II LQR Level III
in %
nonconforming n k n k 100 p * n k n k 100 p * n k n k 100 p *
s s σ σ s s σ σ s s σ σ
items
0,010 132 3,286 23 3,277 0,040 31 ← ←
0,015 117 3,156 21 3,143 0,064 05 ← ←
0,025 101 3,016 20 3,003 0,103 0 179 3,148 33 3,140 0,071 38 ←
0,040 86 2,879 19 2,867 0,161 4 158 3,012 31 3,003 0,113 6 258 3,187 46 3,181 0,065 03
0,065 73 2,728 17 2,710 0,260 4 132 2,867 29 2,858 0,181 7 223 3,051 44 3,045 0,103 5
0,10 60 2,573 16 2,556 0,415 6 112 2,723 27 2,712 0,285 4 189 2,912 40 2,905 0,163 2
0,15 50 2,412 15 2,393 0,662 1 93 2,565 25 2,553 0,458 7 160 2,762 37 2,754 0,261 8
0,25 40 2,237 13 2,211 1,070 76 2,400 23 2,387 0,732 7 134 2,614 34 2,604 0,410 3
0,40 31 2,061 12 2,033 1,685 61 2,230 20 2,212 1,162 110 2,449 31 2,438 0,659 8
0,65 24 1,863 11 1,830 2,747 48 2,043 18 2,021 1,876 89 2,279 28 2,266 1,052
1,0 18 1,659 9 1,611 4,376 37 1,853 16 1,827 2,962 70 2,101 26 2,087 1,667
1,5 13 1,426 8 1,367 7,199 27 1,636 14 1,604 4,802 54 1,904 23 1,886 2,688
2,5 9 1,189 7 1,114 11,44 20 1,411 12 1,370 7,626 41 1,702 20 1,680 4,238
4,0 6 0,887 6 0,786 19,45 13 1,195 8 1,127 11,42 30 1,471 17 1,442 6,857
6,5 4 0,536 3 0,379 32,13 9 0,869 8 0,801 19,60 21 1,227 14 1,190 10,85
10 3 0,044 2 0,021 48,79 6 0,497 4 0,402 32,11 14 0,935 9 0,877 17,61
The plans are indexed by the declared quality level (DQL) of nonconforming product and limiting quality ratio (LQR)
levels.
← Use the sampling plan to the left, which corresponds to a higher limiting quality ratio as no sampling plan exists for
this level of the limiting quality ratio.
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ISO 3951-4:2011(E)
6 Sampling plans
6.1 LQR (limiting quality ratio) levels
6.1.1 Level I
Level I may be used when a smaller sample size is desirable. For Level I sampling plans, the limiting quality
ratios range in value from 7,6 to 14,1. For example, if the declared quality level is 1,0 % nonconforming items,
and the actual quality level is 12,2 times this declared quality level, then the risk is 10 % of failing to contradict
the declared quality level (see Table 2).
Table 2 — Level I plans, limiting quality ratios (LQRs) and probabilities
of falsely contradicting correctly declared quality levels (DQLs)
“s” method “σ ” method
Probability of Probability of
DQL 100 p *
n k LQR falsely contradicting n k LQR falsely contradicting
s s σ σ
a correct DQL a correct DQL
in % noncon-
in % in %
forming items
0,010 132 3,286 13,6 2,5 23 3,277 13,1 1,7 0,040 31
0,015 117 3,156 14,1 2,1 21 3,143 14,0 1,5 0,064 05
0,025 101 3,016 13,5 2,4 20 3,003 13,2 1,6 0,103 0
0,040 86 2,879 13,2 2,6 19 2,867 12,6 1,7 0,161 4
0,065 73 2,728 12,9 2,7 17 2,710 12,6 1,8 0,260 4
0,10 60 2,573 13,3 2,7 16 2,556 12,7 1,6 0,415 6
0,15 50 2,412 13,7 2,3 15 2,393 13,1 1,3 0,662 1
0,25 40 2,237 13,1 2,7 13 2,211 12,7 1,6 1,070
0,40 31 2,061 12,7 3,1 12 2,033 12,0 1,6 1,685
0,65 24 1,863 12,2 3,2 11 1,830 11,5 1,5 2,747
1,0 18 1,659 12,2 3,2 9 1,611 11,8 1,6 4,376
1,5 13 1,426 12,5 2,9 8 1,367 12,0 1,2 7,199
2,5 9 1,189 11,1 3,6 7 1,114 10,6 1,3 11,44
4,0 6 0,887 10,3 3,4 6 0,786 9,9 0,91 19,45
6,5 4 0,536 8,9 3,1 3 0,379 9,9 2,5 32,13
10 3 0,044 7,6 1,6 2 0,021 8,1 3,7 48,79
EXAMPLE Suppose the “s” method plan n = 60, k = 2,573 is used, corresponding to a declared quality level (DQL)
s s
of 0,10 % nonconforming items. For this plan, there is a risk of 10,0 % of failing to contradict this DQL when the actual
quality level is 13,3 (LQR) times the declared quality level, i.e. if the actual quality level is 1,33 % nonconforming items.
If, on the contrary, the actual quality level had been the DQL, i.e. if the actual quality level is 0,10 % nonconforming items,
then there is a risk of 2,7 % of falsely contradicting this correct DQL.
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ISO 3951-4:2011(E)
6.1.2 Level II
Level II is the standard level that shall be used unless specific conditions warrant the use of another level. For
Level II sampling plans, the limiting quality ratios range in value from 5,34 to 7,48. For example, if the declared
quality level is 0,10 % nonconforming items, and the actual quality level is 7,05 times the declared quality level,
then the risk of failing to contradict the declared quality level under the “s” method is 10,0 % (see Table 3).
Table 3 — Level II plans, limiting quality ratios (LQRs) and probabilities
of falsely contradicting correctly declared quality levels (DQLs)
“s” method “σ ” method
Probability of Probability of
DQL 100 p *
n k LQR falsely contradicting n k LQR falsely contradicting
s s σ σ
a correct DQL a correct DQL
in %
nonconforming in % in %
items
0,025 179 3,148 7,22 3,4 33 3,1407,07 2,5 0,0713 8
0,040 158 3,012 7,06 3,4 31 3,0036,95 2,6 0,113 6
0,065 132 2,867 6,97 3,7 29 2,8586,76 2,7 0,181 7
0,10 112 2,723 7,05 3,6 27 2,7126,84 2,5 0,285 4
0,15 93 2,565 7,48 3,0 25 2,553 7,21 1,9 0,458 7
0,25 76 2,400 7,10 3,5 23 2,387 6,80 2,2 0,732 7
0,40 61 2,230 6,95 3,8 20 2,212 6,77 2,5 1,162
0,65 48 2,043 6,76 4,0 18 2,021 6,59 2,5 1,876
1,0 37 1,853 6,78 3,9 16 1,827 6,60 2,3 2,962
1,5 27 1,636 7,14 3,4 14 1,604 6,90 1,7 4,802
2,5 20 1,411 6,48 3,9 12 1,370 6,35 2,0 7,626
4,0 13 1,195 6,04 5,9 8 1,1276,25 3,9 11,42
6,5 9 0,869 5,66 4,6 8 0,801 5,60 2,2 19,60
10 6 0,497 5,34 3,2 4 0,402 5,94 3,9 32,11
EXAMPLE Suppose the “s” method plan n = 112, k = 2,723 is used, corresponding to a declared quality level (DQL)
s s
of 0,10 % nonconforming items. For this plan, there is a risk of 10,0 % of failing to contradict this DQL when the actual
quality level is 7,05 (LQR) times the declared quality level, i.e. if the actual quality level is 0,705 % nonconforming items.
If, on the contrary, the actual quality level is equal to the DQL, i.e. if the actual quality level is 0,10 % nonconforming items,
then there is a risk of 3,6 % of falsely contradicting this correct DQL.
6.1.3 Level III
Level III is for situations where a smaller LQR is desired, at the expense of a larger sample size. For Level III
sampling plans, the limiting quality ratios range in value from 4,72 to 5,97. For example, if the declared quality
level is 0,10 % nonconforming items and the actual quality level is 5,30 times this declared quality level, i.e.
0,530 %, then under the “σ” method there is a risk of 10 % of failing to contradict the declared quality level
(see Table 4).
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ISO 3951-4:2011(E)
Table 4 — Level III plans, limiting quality ratios (LQRs) and probabilities
of falsely contradicting correctly declared quality levels (DQLs)
“s” method “σ ” method
Probability of Probability of
DQL 100 p *
n k LQR falsely contradicting n k LQR falsely contradicting
s s σ σ
a correct DQL a correct DQL
in %
nonconforming in % in %
items
0,040 258 3,187 5,63 2,8 46 3,1815,54 2,1 0,065 03
0,065 223 3,051 5,57 2,9 44 3,0455,43 2,1 0,103 5
0,10 189 2,912 5,41 3,4 40 2,9055,30 2,5 0,163 2
0,15 160 2,762 5,61 2,9 37 2,7545,49 2,0 0,261 8
0,25 134 2,614 5,82 2,5 34 2,6045,71 1,7 0,410 3
0,40 110 2,449 5,57 3,0 31 2,4385,45 2,0 0,659 8
0,65 89 2,279 5,49 3,1 28 2,2665,37 2,1 1,052
1,0 70 2,101 5,30 3,6 26 2,0875,11 2,2 1,667
1,5 54 1,904 5,45 3,1 23 1,8865,27 1,7 2,688
2,5 41 1,702 5,61 2,7 20 1,6805,45 1,4 4,238
4,0 30 1,471 5,97 1,9 17 1,4425,86 0,9 6,857
6,5 21 1,227 5,01 3,3 14 1,1904,96 1,8 10,85
10 14 0,935 4,72 3,3 9 0,8775,02 2,8 17,61
EXAMPLE Suppose the “σ” method plan n = 40, k = 2,905 is chosen, corresponding to a declared quality level
σ σ
(DQL) of 0,10 % nonconforming items. For this plan, there is a 10 % risk of failing to contradict this DQL when the actual
quality level is 5, 30 (LQR) times the declared quality level, i.e. if the actual quality level is 0,530 % nonconforming items.
If, on the contrary, the actual quality level is equal to the DQL, i.e. if the actual quality level is 0,10 % nonconforming items,
then there is a risk of 2,5 % of falsely contradicting this correct DQL.
6.2 Selection of an “s” method sampling plan
Given the chosen DQL and LQR levels, use Table 1 to select an “s” method single sampling plan.
EXAMPLE For example, if the process standard deviation is unknown and LQR Level II is chosen with a DQL of
0,65 % nonconforming items, Table 1 yields an “s” method sampling plan with a sample size of 48, and a Form k
acceptability constant of 2,043 (or, equivalently, a Form p* acceptability constant of 0,018 76), which, from Table 3,
provides an LQR of 6,76.
If the declared quality level is not one of the tabulated values, then the next higher tabulated value of DQL shall be used to
select the plan.
NOTE This will result in a limiting quality ratio that is somewhat higher and a probability of falsely contradicting a
correct declared quality level that is somewhat lower than the values given in Tables 2 to 4 (see 8.2).
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ISO 3951-4:2011(E)
6.3 Selection of a “σ” method sampling plan
Given the chosen DQL and LQR levels, use Table 1 to select a “σ” method single sampling plan.
EXAMPLE For example, if the process standard deviation is presumed to be known and LQR Level II is chosen with
a DQL of 0,65 % nonconforming items, Table 1 yields a “σ” method sampling plan with sample size 18 and Form k
acceptability constant 2,021, which provides an LQR of 6,59 (see Table 3).
If the declared quality level is not one of the tabulated values, then the next higher tabulated value of DQL shall be used to
select the plan.
NOTE This will result in a limiting quality ratio that is somewhat higher and a probability of falsely contradicting a
correct declared quality level that is somewhat lower than the values given in Tables 2 to 4 (see 8.2).
7 Operating a sampling plan
7.1 Sample selection
The sample shall be selected by simple random sampling. If the sample size exceeds the size of the entity
under investigation, then all items of the entity shall be inspected.
7.2 Rules for contradicting a declared quality level: “s” method
7.2.1 General
*
Determine the applicable sampling plan (,nk), or equivalently (,np), from Table 1.
s s s
If the sample size equals or exceeds the size of the entity under investigation, then the DQL shall be verified
by comparing it to the actual quality level determined by inspecting all items in the entity.
Otherwise, select a random sample of size n . For each item in the sample, measure the value of the quality
s
characteristic x. Calculate the sample mean x and the sample standard deviation s.
7.2.2 Single specification limit
For a single upper specification limit U, calculate the quality statistic QU=−()x/s.
For a single lower specification limit L, calculate the quality statistic Qx=−()L/s.
If ,Qk≥ the declared quality level has not been contradicted. If Qk< , the declared quality level has been
s s
contradicted.
EXAMPLE A Level I DQL of 0,25 % is to be used, with an upper specification limit U = 11,5. The quality
characteristic is normally distributed with unknown process standard deviation. From Table 1, it is seen that a sample size
n = 40 is required and that the accompanying Form k acceptability constant k = 2,237. Suppose that the random sample
s s
of 40 items from the entity yields a sample mean x = 10,62
...
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