ISO 3951-1:2005

INTERNATIONAL ISO
STANDARD 3951-1
First edition
2005-04-01
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
Règles d'échantillonnage pour les contrôles par mesures —
Partie 1: Spécifications pour les plans d'échantillonnage simples
indexés d'après le niveau de qualité acceptable (NQA) pour le contrôle
lot par lot pour une caractéristique de qualité unique et un NQA unique

Reference number
©
ISO 2005
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ii © ISO 2005 – All rights reserved

Contents Page
Foreword. v
Introduction . vii
1 Scope. 1
2 Normative references . 2
3 Terms and definitions. 2
4 Symbols . 5
5 Acceptance quality limit (AQL). 7
5.1 Principle . 7
5.2 Use. 7
5.3 Specifying AQLs . 7
5.4 Preferred AQLs. 7
5.5 Caution. 7
5.6 Limitation . 7
6 Switching rules for normal, tightened and reduced inspection. 7
7 Relation to ISO 2859-1 . 8
7.1 Similarities . 8
7.2 Differences. 8
8 Limiting quality protection. 9
8.1 Use of individual plans. 9
8.2 Consumer's risk quality tables . 9
8.3 Producer's risk tables. 9
8.4 Operating characteristic curves . 9
9 Planning . 10
10 Choice between variables and attributes .10
11 Choice between the “s” and “σσσσ” methods . 11
12 Choice of inspection level and AQL. 11
13 Choice of sampling plan. 11
13.1 Standard plans . 11
13.2 Special plans . 12
14 Preliminary operations . 12
15 Standard procedure for the “s” method. 12
15.1 Obtaining a plan, sampling and preliminary calculations . 12
15.2 Acceptability criteria for single specification limits. 13
15.3 Graphical method for a single specification limit. 14
15.4 Acceptability criterion for combined control of double specification limits . 15
16 Standard procedure for the “σσσσ” method. 21
16.1 Obtaining a plan, sampling and preliminary calculations . 21
16.2 Acceptability criteria for a single specification limit. 21
16.3 Acceptability criterion for combined control of double specification limits . 22
17 Procedure during continuing inspection.23
18 Normality and outliers . 24
18.1 Normality. 24
18.2 Outliers. 24
19 Records .24
19.1 Control charts.24
19.2 Lots that are not accepted.24
20 Operation of switching rules.24
21 Discontinuation and resumption of inspection.25
22 Switching between the “s” and “σσ” methods.25
σσ
22.1 Estimating the process standard deviation.25
22.2 State of statistical control .26
22.3 Switching from the “s” method to the “σσσσ” method.26
22.4 Switching from the “σσσσ” method to the “s” method.26
23 Chart A — Sample-size code letters of standard single sampling plans for specified
quality levels .26
24 Charts B to R (Figures 5 to 19) — Operating characteristic curves and tabulated values for
sample-size code letter B to R: “s” method .28
25 Charts s-D to s-R (Figures 20 to 32) — Acceptance curves for combined control of double
specification limits: “s” method.56
Annex A (normative) Tables for determining the appropriate sample size.69
Annex B (normative) Form k single sampling plans for the “s” method .71
Annex C (normative) Form k single sampling plans for the “σ” method .75
Annex D (normative) Values of f for maximum sample standard deviation (MSSD) .79
s
Annex E (normative) Values of f for maximum process standard deviation (MPSD) .83
σ
Annex F (normative) Estimating the process fraction nonconforming for sample size 3: “s”
method.84
Annex G (normative) Type p* single sampling plans.87
Annex H (normative) Values of c for upper control limit on the sample standard deviation .88
u
Annex I (normative) Supplementary acceptability constants for qualifying towards reduced
inspection.89
Annex J (normative) Procedures for obtaining s and σ .90
Annex K (informative) Consumer's risk qualities.92
Annex L (informative) Producer's risks .96
Annex M (informative) Operating characteristics for the “σ” method.100
Annex N (informative) Estimating the process fraction nonconforming for sample sizes 3
and 4 — “s” method.101
Bibliography.103

iv © ISO 2005 – All rights reserved

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 3951-1 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods,
Subcommittee SC 5, Acceptance sampling.
This first edition of ISO 3951-1 cancels and replaces ISO 3951:1989, of which it constitutes a technical
revision. The most significant differences between ISO 3951-1:2003 and ISO 3951:1989 are as follows.
 The acronym AQL now stands for Acceptance Quality Limit rather than Acceptable Quality Level, in order
to reflect more accurately its function.
 The coverage of this part of ISO 3951 is constrained to a single, normally distributed variable with a single
class of nonconformity. This part of ISO 3951 includes the case of combined control of double
specification limits, but procedures for separate or complex control of double specification limits are
deferred to ISO 3951-2. More general procedures that can be used for multiple characteristics and/or
multiple AQLs are also given in ISO 3951-2.
 The plans have been modified so that their operating characteristic curves more closely match those of
the plans in ISO 2859-1. The sample sizes for both the “s” method and the “σ ” method are constant along
rows of the master tables.
 All acceptability constants (see Annexes B, C, G and I) have been revised and tabulated to three decimal
places for an extended range of AQLs corresponding to ISO 2859-1:1999.
 All tabulated values of operating characteristics have been recalculated and related directly to reduced
inspection as well as to normal and tightened inspection.
 The annex containing the general statistical theory has been removed. It is planned ultimately to
reintroduce this within a guidance document to sampling procedures for inspection by variables.
 Tables that are required for implementing the procedures have been relocated into annexes.
 The annex dealing with the “R” method has been eliminated, now that the availability of calculators with a
standard deviation function key is so widespread. Data for acceptance sampling by variables is often
substantially more expensive to acquire than data for sampling by attributes, and the “s” method makes
more efficient use of this data.
ISO 3951 currently 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
The following parts are under preparation:
 Part 3: Double sampling schemes indexed by acceptance quality limit (AQL) for lot-by-lot inspection
 Part 5: Sequential sampling plans indexed by acceptance quality limit (AQL) for inspection by variables
vi © ISO 2005 – All rights reserved

Introduction
This part of ISO 3951 specifies an acceptance sampling system of single sampling plans for inspection by
variables. It is indexed in terms of the Acceptance Quality Limit (AQL), and is designed for users who have
simple requirements. A more comprehensive and technical treatment is given in ISO 3951-2. This part of
ISO 3951 is complementary to ISO 2859-1.
The objectives of the methods laid down in this part of ISO 3951 are to ensure that lots of an acceptable
quality have a high probability of acceptance, and that the probability of not accepting inferior lots is as high as
practicable. This is achieved by means of the switching rules, which provide:
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) if consistently good quality is achieved.
In this part of ISO 3951, the acceptability of a lot is implicitly determined from an estimate of the percentage of
nonconforming items in the process, based on a random sample of items from the lot.
This part of ISO 3951 is intended for application to a continuing series of lots of discrete products all supplied
by one producer using one production process. If there are different producers or production processes, this
part of ISO 3951 is applied to each one separately.
This part of ISO 3951 is intended for application to a single quality characteristic that is measurable on a
continuous scale. For two or more such quality characteristics, see ISO 3951-2.
It is assumed in this part of ISO 3951 that measurement error is negligible. For information on allowing for
measurement error, see Reference [17] in the Bibliography.
For double specification limits, this part of ISO 3951 treats combined control. For other types of control, see
ISO 3951-2.
Inspection by variables for percent nonconforming items, as described in the present document, includes
several possible modes, the combination of which leads to a presentation that may appear quite complex to
the user:
 unknown standard deviation, or originally unknown then estimated with fair precision, or known since the
start of inspection;
 a single specification limit, or combined control of double specification limits;
 normal inspection, tightened inspection or reduced inspection.
Fourteen annexes are provided. Annexes A to I provide the tables needed to support the procedures. Annex J
indicates how the sample standard deviation, “s”, and the presumed known value of the process standard
deviation, “σ”, should be determined. Annex K provides the statistical theory underlying the calculation of the
consumer's risks, together with tables showing these risks for normal, tightened and reduced inspection as
well as for the “s” and “σ” methods. Annex L provides similar information for the producer's risks. Annex M
gives the general formula for the operating characteristic of the “σ” method. Annex N provides the statistical
theory underlying the estimation of the process fraction nonconforming under the “s” method for sample
sizes 3 and 4, which for technical reasons are treated differently from the other sample sizes in this part of
ISO 3951.
Table 1 is intended to facilitate the use of this part of ISO 3951 by directing users to the paragraphs and tables
concerning any situation with which they may be confronted. Table 1 only deals with Clauses 15, 16, 20, 21
and 22; in every case, it is necessary to have first read the other clauses.
viii © ISO 2005 – All rights reserved

Table 1 — Summary table
Inspection type Single specification limit Double specification limits with combined control
“s” method “σ” method “s” method “σ ” method
Clauses or Tables Charts Clauses or Tables Charts Clauses or Tables Charts Clauses or Tables Charts
subclauses subclauses subclauses subclauses
Normal inspection 15.1, 15.2, A.1, A.2, B to R 16.1, 16.2 A.1, A.2, B to R * 15.1, 15.4 A.1, A.2, s-D to s- 16.1, 16.3 A.1,A.2, B to R *
15.3 and 20.1 B.1, and 20.1 C.1, and 20.1 D.1, R, B and 20.1 C.1, E,
B to R B to R * F (for n = 3), to R * B to R *
G (for n = 3
or 4), B to R*
Switching between 20.2, 20.3 B.1, B.2 B to R 20.2, 20.3 C.1, C.2 B to R * 20.2, 20.3 D.1, D.2 s-D to s- 20.2, 20.3 C.1, B to R *
normal and tightened R, B C.2, E
inspection to R *
Switching between 20.4, 20.5 B.1, B.3 B to R 20.4, 20.5 C.1, C.3 B to R * 20.4, 20.5 D.1, D.3 s-D to s- 20.4, 20.5 E B to R *
normal and reduced R, B
inspection to R *
Switching between 21 B.2 B to R 21 C.2 B to R * 21 D.2 s-D to s- 21 E B to R *
tightened and dis- R, B
continued inspection to R *
Switching between 22 Annex K 22 Annex K 22 Annex K 22 Annex K
the “s” and “σ”
methods
* But see 8.4.
INTERNATIONAL STANDARD ISO 3951-1:2005(E)

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
CAUTION —The procedures in this part of ISO 3951 are not suitable for application to lots that have
been screened previously for nonconforming items.
1 Scope
This part of ISO 3951 specifies an acceptance sampling system of single sampling plans for inspection by
variables, in which the acceptability of a lot is implicitly determined from an estimate of the percentage of
nonconforming items in the process, based on a random sample of items from the lot.
This part of ISO 3951 is primarily designed for application under the following conditions:
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 only a single quality characteristic x of these products is taken into consideration, which must be
measurable on a continuous scale;
c) where the measurement error is negligible, i.e. with a standard deviation no more than 10 % of the
process standard deviation;
d) where production is stable (under statistical control) and the quality characteristic x is distributed
according to a normal distribution or a close approximation to the normal distribution;
e) where a contract or standard defines an upper specification limit U, a lower specification limit L, or both;
an item is qualified as conforming if and only if its measured quality characteristic x satisfies the
appropriate one of the following inequalities:
1) x W L (i.e. the lower specification limit is not violated);
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).
Inequalities 1) and 2) are called cases with a single specification limit, and 3) a case with double specification
limits.
If double specification limits apply, it is assumed in this part of ISO 3951 that conformance to both
specification limits is equally important to the integrity of the product; in such cases it is appropriate to apply a
single AQL to the combined percentage of product outside the two specification limits. This is referred to as
combined control.
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 2859-1, Sampling procedures for inspection by attributes — Part 1: Sampling schemes indexed by
acceptance quality limit (AQL) for lot-by-lot inspection
ISO 3534-1, Statistics — Vocabulary and symbols — Part 1: Probability and statistical terms
ISO 3534-2, Statistics — Vocabulary and symbols — Part 2: Applied statistics
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 2859-1, ISO 3534-1, and
ISO 3534-2 apply.
3.1
inspection by variables
inspection by measuring the magnitude of a characteristic of an item
[ISO 3534-2]
3.2
sampling inspection
inspection of selected items in the group under consideration
[ISO 3534-2]
3.3
acceptance sampling inspection
acceptance sampling
sampling inspection (3.2) to determine whether or not to accept a lot or other amount of product, material or
service
[ISO 3534-2]
3.4
acceptance sampling inspection by variables
acceptance sampling inspection (3.3) in which the acceptability of the process is determined statistically
from measurements on a specified quality characteristic of each item in a sample from a lot
3.5
process fraction nonconforming
rate at which nonconforming items are generated by a process, expressed as a proportion
3.6
acceptance quality limit (AQL)
worst tolerable process fraction nonconforming (3.5) when a continuing series of lots is submitted for
acceptance sampling (3.3)
NOTE See Clause 5.
3.7
quality level
quality expressed as a rate of occurrence of nonconforming units
2 © ISO 2005 – All rights reserved

3.8
limiting quality
LQ
quality level (3.7), when a lot is considered in isolation, which, for the purposes of acceptance sampling
inspection (3.3), is limited to a low probability of acceptance
[ISO 3534-2]
NOTE 1 In this part of ISO 3951, the probability of acceptance is limited to 10 %.
NOTE 2 See 13.1.
3.9
nonconformity
nonfulfilment of a requirement
[ISO 9000]
3.10
nonconforming unit
unit with one or more nonconformities
[ISO 3534-2]
3.11
“s” method acceptance sampling plan
acceptance sampling (3.3) plan by variables using the sample standard deviation
[ISO 3534-2]
NOTE See Clause 15.
3.12
“σ” method acceptance sampling plan
acceptance sampling (3.3) plan by variables using the presumed value of the process standard deviation
[ISO 3534-2]
NOTE See Clause 16.
3.13
specification limit
conformance boundary specified for a characteristic
[ISO 3534-2]
3.14
lower specification limit
L
specification limit [3.13] that defines the lower conformance boundary
[ISO 3534-2]
3.15
upper specification limit
U
specification limit (3.13) that defines the upper conformance boundary
[ISO 3534-2]
3.16
combined control
requirement when both upper and lower limits are specified for the quality characteristic and an AQL (3.6) is
given that applies to the combined percent nonconforming beyond the two limits
NOTE 1 See 5.3.
NOTE 2 The use of combined control implies that nonconformity beyond either specification limit (3.13) is believed to
be of equal, or at least roughly equal, importance to the lack of integrity of the product.
3.17
acceptability constant
k
constant depending on the specified value of the acceptance quality limit (3.6) and the sample size, used in
the criteria for accepting the lot in an acceptance sampling (3.3) plan by variables
[ISO 3534-2]
NOTE See 15.2 and 16.2.
3.18
quality statistic
Q
function of the specification limit (3.13), the sample mean, and the sample or process standard deviation,
used in assessing the acceptability of a lot
[ISO 3534-2]
NOTE 1 For the case of a single specification limit (3.13), the lot may be sentenced on the result of comparing Q with
the acceptability constant (3.17) k.
NOTE 2 See 15.2 and 16.2.
3.19
lower quality statistic
Q
L
function of the lower specification limit (3.14), the sample mean, and the sample or process standard
deviation
NOTE 1 For a single lower specification limit (3.14), the lot is sentenced on the result of comparing Q with the
L
acceptability constant (3.17) k.
[ISO 3534-2]
NOTE 2 See Clause 4, 15.2 and 16.2.
3.20
upper quality statistic
Q
U
function of the upper specification limit (3.15), the sample mean, and the sample or process standard
deviation
NOTE 1 For a single upper specification limit (3.15) the lot is sentenced on the result of comparing Q with the
U
acceptability constant (3.17) k.
[ISO 3534-2]
NOTE 2 See Clause 4, 15.2 and 16.2.
4 © ISO 2005 – All rights reserved

3.21
maximum sample standard deviation
MSSD
s
max
largest sample standard deviation for a given sample-size code letter and acceptance quality limit (3.6) for
which it is possible to satisfy the acceptance criterion for the combined control of double specification limits
(3.13) when the process variability is unknown
NOTE See 15.4.
3.22
maximum process standard deviation
MPSD
σ
max
largest process standard deviation for a given sample-size code letter and acceptance quality limit (3.6) for
which it is possible to satisfy the acceptance criterion for the combined control of double specification limits
(3.13) under tightened inspection when the process variability is known
NOTE See 16.3.
3.23
switching rule
instruction within an acceptance sampling (3.3) scheme for changing from one acceptance sampling (3.3)
plan to another of greater or lesser severity based on demonstrated quality history
[ISO 3534-2]
NOTE 1 See Clause 20.
NOTE 2 Normal, tightened or reduced inspection, or discontinuation of inspection, are examples of “greater or lesser
severity”.
3.24
measurement
Set of operations to determine the value of some quantity
[ISO 3534-2]
4 Symbols
c factor for determining the upper control limit for the sample standard deviation (see Annex H)
U
f factor that relates the maximum sample standard deviation to the difference between U and L
s
(see Annex D)
f factor that relates the maximum process standard deviation under tightened inspection to the difference
σ
between U and L (see Annex E)
k acceptability constant for the “s” method or Annex C for the “σ” method (see Annex B)
L lower specification limit (as a suffix to a variable, denotes its value at L)
µ process mean
N lot size (number of items in a lot)
n sample size (number of items in a sample)
pˆ estimate of the process fraction nonconforming
ˆ
p estimate of the process fraction nonconforming below the lower specification limit
L
pˆ estimate of the process fraction nonconforming above the upper specification limit
U
p* maximum acceptable value for the estimate of the process fraction nonconforming
P probability of acceptance
a
Q quality statistic
Q lower quality statistic
L
NOTE 1 Q is defined as ()xL− /s when the process standard deviation is unknown, and as ()xL− /σ
L
when it is presumed to be known.
Q upper quality statistic
U
NOTE 2 Q is defined as ()Ux−/s when the process standard deviation is unknown, and as ()Ux−/σ
U
when it is presumed to be known.
s sample standard deviation of the measured values of the quality characteristic (also an estimate of the
standard deviation of the process), i.e.
n
()xx−
∑ j
j
s =
n −1
(See also Annex J.)
s maximum sample standard deviation (MSSD)
max
σ standard deviation of a process that is under statistical control
NOTE 3 σ the square of the process standard deviation, is known as the process variance.
σ maximum process standard deviation (MPSD)
max
U upper specification limit (as a suffix to a variable, denotes its value at U)
x measured value of the quality characteristic for the jth item of the sample
j
x arithmetic mean of the measured values of the quality characteristic in the sample, i.e.
n
x
∑ j
j=1
x =
n
6 © ISO 2005 – All rights reserved

5 Acceptance quality limit (AQL)
5.1 Principle
The AQL is the quality level that is the worst tolerable process fraction nonconforming when a continuing
series of lots is submitted for acceptance sampling. Although individual lots with quality as bad as the
acceptance quality limit may be accepted with fairly high probability, the designation of an acceptance quality
limit does not suggest that this is a desirable quality level. The sampling schemes found in this part of
ISO 3951, with their rules for switching and for discontinuation of sampling inspection, are designed to
encourage suppliers to keep the process fractions nonconforming consistently better than the respective
AQLs. Otherwise, there is a high risk that the inspection severity will be switched to tightened inspection,
under which the criteria for lot acceptance become more demanding. Once on tightened inspection, unless
action is taken to improve the process, it is very likely that the rule requiring discontinuation of sampling
inspection will be invoked pending such improvement.
5.2 Use
The AQL, together with the sample-size code letter, is used to index the sampling plans in this part of
ISO 3951.
5.3 Specifying AQLs
The AQL to be used will be designated in the product specification, contract or by the responsible authority.
Where both upper and lower specification limits are given, this part of ISO 3951 addresses only the case of an
overall AQL applying to the combined percent nonconforming beyond the two limits; this is known as
“combined control”. (See ISO 3951-2 for “separate” and “complex” control of double specification limits.)
5.4 Preferred AQLs
The sixteen AQLs given in this part of ISO 3951, ranging in value from 0,01 % to 10 % nonconforming, are
described as preferred AQLs. If, for any product or service, an AQL is designated other than a preferred AQL,
then this part of ISO 3951 is not applicable (see 13.2).
5.5 Caution
From the above definition of the AQL, it follows that the desired protection can only be assured when a
continuing series of lots is provided for inspection.
5.6 Limitation
The designation of an AQL shall not imply that the supplier has the right to supply knowingly any
nonconforming product.
6 Switching rules for normal, tightened and reduced inspection
Switching rules discourage the producer from operating at a quality level that is worse than the AQL. This part
of ISO 3951 specifies a switch to tightened inspection when inspection results indicate that the AQL is being
exceeded. It further specifies a discontinuation of sampling inspection altogether if tightened inspection fails to
stimulate the producer into rapidly improving his production process.
Tightened inspection and the discontinuation rule are integral, and therefore obligatory, procedures of this part
of ISO 3951 if the protection implied by the AQL is to be maintained.
This part of ISO 3951 also provides the possibility of switching to reduced inspection when inspection results
indicate that the quality level is stable and reliable at a level better than the AQL. This practice is, however,
optional (at the discretion of the responsible authority).
If there is sufficient evidence from the control charts (see 19.1) that the variability is in statistical control,
consideration should be given to switching to the “σ” method. If this appears advantageous, the consistent
value of s (the sample standard deviation) shall be taken as σ (see Clause 22).
When it has been necessary to discontinue acceptance sampling inspection, inspection under this part of
ISO 3951 shall not be resumed until action has been taken by the producer to improve the quality of the
submitted product.
Details of the operation of the switching rules are given in Clauses 20, 21 and 22.
7 Relation to ISO 2859-1
7.1 Similarities
a) This part of ISO 3951 is complementary to ISO 2859-1; 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 part of ISO 3951 are
identical with those given for percent nonconforming in ISO 2859-1 (i.e. from 0,01 % to 10 %).
c) In both International Standards, lot size and inspection level (inspection level II in default of other
instructions) determine a sample-size code letter. Then general tables give the sample size to be taken
and the acceptability criterion, indexed by the sample-size code letter and the AQL. Separate tables are
given for the “s” and “σ” methods, and for normal, tightened and reduced inspection.
d) The switching rules are essentially equivalent.
7.2 Differences
a) Determination of acceptability: Acceptability for an ISO 2859-1 attributes sampling plan for percent
nonconforming is determined by the number of nonconforming items found in the sample. Acceptability
for a plan for inspection by variables is based on the distance of the estimated value of the process mean
from the specification limit(s) in terms of the estimated or presumed value of the process standard
deviation. In this part of ISO 3951, two methods are considered: the “s” method for use when the process
standard deviation σ is unknown, and the “σ” method for use when σ is presumed to be known. In the
case of a single specification limit, the acceptability may be calculated from a formula (see 15.2 and
16.2), but for the “s” method it is also easily established by a graphical method (see 15.3). In the case of
combined control of double specification limits under the “s” method, this part of ISO 3951 provides only
for a graphical method of determining acceptability (see 15.4); for combined control of double
specification limits under the “σ” method, a numerical method is given.
b) Normality: In ISO 2859 there is no requirement relating to the distribution of the characteristics. However,
in this part of ISO 3951 it is necessary for the efficient operation of the plans that the measurements be
distributed according to a normal distribution or a close approximation to a normal distribution.
c) Operating characteristic curves (OC curves): The OC curves of the variables plans in this part of
ISO 3951 are not identical to those of the corresponding attributes plans in ISO 2859-1. The curves have
been matched as closely as possible subject to a number of pragmatic constraints, such as keeping the
sample size the same for a given code letter, and inspection method regardless of the AQL.
d) Producer's risk: For process quality precisely at the AQL, the producer's risk that a lot will not be
accepted tends to decrease with one-step increases in sample size coupled with one-step decreases in
AQL, i.e. down diagonals of the master tables running from top right to bottom left. The progressions of
probabilities are similar, but not identical, to those in ISO 2859-1. (The producer's risks of the plans are
given in Annex L.)
8 © ISO 2005 – All rights reserved

e) Sample sizes: The variables sample sizes corresponding to given code letters are usually smaller than
the attributes sample sizes for the same letters. This is particularly true for the “σ ” method.
(See Table A.2.)
f) Double sampling plans: Double sampling plans are presented separately, in ISO 3951-3.
g) Multiple sampling plans: No multiple sampling plans are given in this part of ISO 3951.
h) Average Outgoing Quality Limit (AOQL): The AOQL concept applies when 100 % inspection and
rectification is feasible for non-accepted lots. It follows that the AOQL concept cannot be applied under
destructive or expensive testing. As variables plans are generally used under these circumstances, no
tables of AOQL have been included in this part of ISO 3951.
8 Limiting quality protection
8.1 Use of individual plans
This part of ISO 3951 is intended to be used as a system employing tightened, normal and reduced
inspections on a continuing series of lots to provide consumer protection, while assuring the producer that
acceptance will be very likely to occur if quality is better than the AQL.
Some users may select specific individual plans from this part of ISO 3951 and use them without the switching
rules. For example, a purchaser may be using the plans for verification purposes only. This is not the intended
application of the system given in this part of ISO 3951 and its use in this way should not be referred to as
“inspection in compliance with ISO 3951-1”. When used in such a way, this part of ISO 3951 simply
represents a collection of individual plans indexed by the AQL. The operating characteristic curves and other
measures of a plan so chosen shall be assessed individually from the tables provided.
8.2 Consumer's risk quality tables
If the series of lots is not long enough to allow the switching rules to be applied, it may be desirable to limit the
selection of sampling plans to those, associated with a designated AQL value, that give a consumer's risk
quality not worse than the specified limiting quality protection. Sampling plans for this purpose can be selected
by choosing a consumer's risk quality (CRQ) and a consumer's risk to be associated with it. Annex K gives
values of consumer's risk quality for the “s” method and “σ” methods corresponding to a consumer's risk of
10 %.
However, application of this part of ISO 3951 to isolated lots is deprecated, as the theory of sampling by
variables applies to a process. For isolated or short series of lots, it is appropriate and more efficient to use
plans for sampling by attributes, such as from ISO 2859-2. (See Reference [5].)
8.3 Producer's risk tables
Annex L gives the probability of non-acceptance under the “s” and “σ” methods for lots produced when the
process fraction nonconforming equals the AQL. This probability is called the producer's risk.
8.4 Operating characteristic curves
The tables for consumer's risk quality and producer's risk provide information about only two points on the
operating characteristic curves. The degree of consumer protection provided by an individual sampling plan at
any process quality may, however, be judged from its operating characteristic (OC) curve. OC curves for the
normal inspection “s” method sampling plans of this part of ISO 3951 are given in Charts B to R, which should
be consulted when choosing a sampling plan. Also given are tables of process qualities at nine standard
probabilities of acceptance for all of the “s” method sampling plans in this part of ISO 3951.
These OC curves and tables apply to a single specification limit under the “s” method. Most of them also
provid
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