ISO 3951-1:2005

SLOVENSKI STANDARD
SIST ISO 3951-1:2006
01-april-2006
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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 la limite d'acceptation de qualité
(LAQ) pour le contrôle lot par lot pour une caractéristique de qualité unique et une LAQ
unique
Ta slovenski standard je istoveten z: ISO 3951-1:2005
ICS:
03.120.30 8SRUDEDVWDWLVWLþQLKPHWRG Application of statistical
methods
SIST ISO 3951-1:2006 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

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SIST ISO 3951-1:2006

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SIST ISO 3951-1:2006


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 3951-1:2005(E)
©
ISO 2005

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SIST ISO 3951-1:2006
ISO 3951-1:2005(E)
PDF disclaimer
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shall not be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In
downloading this file, parties accept therein the responsibility of not infringing Adobe's licensing policy. The ISO Central Secretariat
accepts no liability in this area.
Adobe is a trademark of Adobe Systems Incorporated.
Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-creation
parameters were optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In
the unlikely event that a problem relating to it is found, please inform the Central Secretariat at the address given below.


©  ISO 2005
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means,
electronic or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or
ISO's member body in the country of the requester.
ISO copyright office
Case postale 56 • CH-1211 Geneva 20
Tel. + 41 22 749 01 11
Fax + 41 22 749 09 47
E-mail copyright@iso.org
Web www.iso.org
Published in Switzerland

ii © ISO 2005 – All rights reserved

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SIST ISO 3951-1:2006
ISO 3951-1:2005(E)
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
© ISO 2005 – All rights reserved iii

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SIST ISO 3951-1:2006
ISO 3951-1:2005(E)
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

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SIST ISO 3951-1:2006
ISO 3951-1:2005(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 3951-1 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.
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SIST ISO 3951-1:2006
ISO 3951-1:2005(E)
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

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SIST ISO 3951-1:2006
ISO 3951-1:2005(E)
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.
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SIST ISO 3951-1:2006
ISO 3951-1:2005(E)
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.
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SIST ISO 3951-1:2006
ISO 3951-1:2005(E)
© ISO 2005 – All rights reserved ix
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.

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SIST ISO 3951-1:2006

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SIST ISO 3951-1:2006
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.
© ISO 2005 – All rights reserved 1

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SIST ISO 3951-1:2006
ISO 3951-1:2005(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 2859-1, 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
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SIST ISO 3951-1:2006
ISO 3951-1:2005(E)
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 lim
...

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 3951-1:2005(E)
©
ISO 2005

---------------------- Page: 1 ----------------------

ISO 3951-1:2005(E)
PDF disclaimer
This PDF file may contain embedded typefaces. In accordance with Adobe's licensing policy, this file may be printed or viewed but
shall not be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In
downloading this file, parties accept therein the responsibility of not infringing Adobe's licensing policy. The ISO Central Secretariat
accepts no liability in this area.
Adobe is a trademark of Adobe Systems Incorporated.
Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-creation
parameters were optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In
the unlikely event that a problem relating to it is found, please inform the Central Secretariat at the address given below.


©  ISO 2005
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means,
electronic or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or
ISO's member body in the country of the requester.
ISO copyright office
Case postale 56 • CH-1211 Geneva 20
Tel. + 41 22 749 01 11
Fax + 41 22 749 09 47
E-mail copyright@iso.org
Web www.iso.org
Published in Switzerland

ii © ISO 2005 – All rights reserved

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ISO 3951-1:2005(E)
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
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ISO 3951-1:2005(E)
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

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ISO 3951-1:2005(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 3951-1 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.
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ISO 3951-1:2005(E)
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
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ISO 3951-1:2005(E)
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.
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ISO 3951-1:2005(E)
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.
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ISO 3951-1:2005(E)
© ISO 2005 – All rights reserved ix
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.

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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.
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ISO 3951-1:2005(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 2859-1, 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
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ISO 3951-1:2005(E)
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]
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ISO 3951-1:2005(E)
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 t
...