SIST ISO 28801:2014

INTERNATIONAL ISO
STANDARD 28801
First edition
2011-03-15
Double sampling plans by attributes with
minimal sample sizes, indexed by
producer's risk quality (PRQ) and
consumer's risk quality (CRQ)
Plans d'échantillonnage double par attributs, avec taille d'échantillon
minimale, indexés par la qualité du risque du fournisseur (QRF) et la
qualité du risque du client (QRC)

Reference number
©
ISO 2011
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ii © ISO 2011 – All rights reserved

Contents Page
Foreword .iv
Introduction.v
1 Scope.1
2 Terms, definitions, abbreviations and symbols.1
3 Choosing and operating a sampling plan.4
4 Operating characteristics .5
5 Average sample sizes .5
6 Actual producer's and consumer's risks.6
7 Average outgoing quality (AOQ).6
8 Examples.7
9 Tables and figures.7
Annex A (informative) Statistical theory underlying the plans, tables and figures .63
Bibliography.70

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
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International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of tehnical 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 28801 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods,
Subcommittee SC 5, Acceptance sampling.
iv © ISO 2011 – All rights reserved

Introduction
ISO 2859-1 provides double sampling plans for sampling by attributes. Those plans are indexed by
acceptance quality limit (AQL) and are thus designed for a continuing series of lots. For each preferred range
of lot sizes, i.e. for each sample size code letter, the first and second sample sizes of the double sampling
plans in ISO 2859-1 are constant and equal across AQLs, while the acceptance numbers increase with
the AQL.
As production processes and quality levels have improved during the latter half of the twentieth century, there
has been a shift of interest towards sampling plans with smaller acceptance and rejection numbers than most
of the plans in ISO 2859-1.
Moreover, in some industries, in an effort to focus on customers' more specific requirements, there has also
been a trend towards shorter production runs. Sometimes these production runs are too short for the
switching rules of AQL-indexed plans (such as those of ISO 2859-1) to operate effectively.
In order to address an evolving market need, this International Standard has been developed to provide
double sampling plans by attributes indexed by producer's risk quality (PRQ) and consumer's risk quality
(CRQ) and having the smallest possible acceptance and rejection numbers. No constraint has been placed on
the relative sizes of the first and second sample sizes; instead, the first and second sample sizes have been
derived to minimize the expected total amount of sampling subject to the nominal producer's risk, α, and
consumer's risk, β, not being exceeded. The combinations (α, β) of nominal risks provided in this International
Standard are (5 %, 5 %), (5 %, 10 %) and (10 %, 10 %).
Thus, the double sampling plans provided in this International Standard are of the following kind. In the case
of sampling for nonconforming items, a lot is acceptable if no nonconforming items are found in the first
random sample, and not acceptable if the sample contains two or more nonconforming items. If the first
random sample contains precisely one nonconforming item, a second, smaller random sample is taken; if no
nonconforming items are found in the second sample, then the lot is acceptable, otherwise it is not
acceptable. For each pair of nominal producer's and consumer's risk, up to 17 preferred nominal values of
CRQ and up to 17 preferred nominal values of PRQ are provided.
Similar plans are provided for nonconformities.
The double sampling sample sizes are minimal among sampling plans for acceptance inspection of isolated
lots or for short series of lots. However, because the ISO 28801 plans do not rely on the protection of
switching rules, the sample sizes are necessarily larger than those used for lot-by-lot inspection, such as
those of ISO 2859-1, for similar producer's and consumer's quality levels, where these exist. This is illustrated
by the following two examples, both for nonconforming items with nominal maximum producer's and
consumer's risks or 5 % and 10 % respectively.
EXAMPLE 1
Realized Realized Producer's Consumer's
Sample
Source producer's consumer's risk quality risk quality
sizes
risk risk (PRQ) (CRQ)
ISO 2859-1, code letter E, AQL = 1 % 5 % 10 % 0,394 % 20,6 % 8,8
ISO 28801, Tables 2 and 14 0,266 % 9,639 % 0,4 % 20 % 12,9

EXAMPLE 2
Realized Realized Producer's Consumer's
Sample
Source producer's consumer's risk quality risk quality
sizes
risk risk (PRQ) (CRQ)
ISO 2859-1, code letter F, AQL = 0,65 % 5 % 10 % 0,256 % 10,9 % 13,13
ISO 28801, Tables 2 and 14 0,435 % 9,920 % 0,25 % 10 % 26,16

A compensating feature of the ISO 28801 plans is that many of the realized producer's risks are much smaller
than their nominal values.
vi © ISO 2011 – All rights reserved

INTERNATIONAL STANDARD ISO 28801:2011(E)

Double sampling plans by attributes with minimal sample sizes,
indexed by producer's risk quality (PRQ) and consumer's risk
quality (CRQ)
1 Scope
This International Standard provides double sampling plans by attributes for the acceptance inspection of lots
of discrete items. The plans are indexed by the producer's risk quality (PRQ) and the consumer's risk quality
(CRQ) where the nominal producer's and consumer's risks are respectively either (5 %, 5 %), (5 %, 10 %) or
(10 %, 10 %). Plans are provided for inspection for percent nonconforming and for inspection for
nonconformities per 100 items. The lot is accepted if there are no nonconforming items (nonconformities) in
the first random sample, and rejected if it contains two or more nonconforming items (nonconformities). If
precisely one nonconforming item is found in the first sample, a second random sample is drawn; the lot is
then accepted if the second sample contains no nonconforming items (nonconformities) and rejected
otherwise.
The objective of this International Standard is to provide procedures that enable lot disposition to be
determined quickly and economically if quality is particularly good or bad. For intermediate quality, a second
sample is drawn in order to be able to discriminate more reliably between acceptable and unacceptable lots.
The two sample sizes are chosen to minimize the maximum expected sample size with respect to incoming
quality subject to the nominal risks not being exceeded.
Similarly, the plans may be used to test the hypothesis that a lot or process quality level is equal to the PRQ
(i.e. acceptable) against the alternative hypothesis that the quality level is equal to the CRQ (i.e.
unacceptable).
The plans are preferable to single sampling plans where the cost of inspection is high, where the delay and
uncertainty caused by the possible requirement for second samples is inconsequential and where a relatively
large ratio of the consumer's risk quality to the producer's risk quality can be tolerated.
The plans are suitable for isolated lots or for short series of lots, where the sum of the two sample sizes is no
larger than about 10 % of the size of the lot. The plans are also suitable for continuing series of lots when lots
that fail to satisfy the acceptance criteria are 100 % inspected and all nonconforming items replaced by
conforming items; however, for continuing series of lots, consideration should also be given to using double
sampling plans from ISO 2859-1.
The statistical theory underlying the plans, tables and figures is provided in Annex A.
2 Terms, definitions, abbreviations and symbols
2.1 Terms, definitions and abbreviations
The words “accept”, “accepted”, “acceptable”, etc., refer only to the use of the sampling plans contained in this
International Standard and do not imply an agreement to accept any product. Determination of acceptability by
the customer shall be as described in contractual documents.
For the purposes of this document, the following terms and definitions apply.
2.1.1
consumer's risk
CR
〈acceptance sampling〉 probability of acceptance when the quality level (2.1.5) of the process has a value
stated by the acceptance sampling plan as unsatisfactory
[ISO 3534-2:2006; 4.6.2]
2.1.2
consumer's risk quality
CRQ
〈acceptance sampling〉 quality level (2.1.5) of a lot or process which, in the acceptance sampling plan,
corresponds to a specified consumer's risk (2.1.1)
[ISO 3534-2:2006; 4.6.9]
2.1.3
producer's risk
PR
〈acceptance sampling〉 probability of non-acceptance when the quality level (2.1.5) of the process has a
value stated by the plan as acceptable
[ISO 3534-2:2006; 4.6.4]
2.1.4
producer's risk quality
PRQ
〈acceptance sampling〉 quality level (2.1.5) of a lot or process which, in the acceptance sampling plan,
corresponds to a specified producer's risk (2.1.3)
[ISO 3534-2:2006; 4.6.10]
2.1.5
quality level
〈acceptance sampling〉 quality expressed as a rate of nonconforming units or rate of number of
nonconformities (2.1.9)
[ISO 3534-2:2006; 4.6.16]
NOTE Rate (or fraction) can be expressed in the unit 1 or in percentage. Rate of nonconforming units can be read as
a numerical value or a percentage of nonconforming units. Rate of number of nonconformities can be read as a number of
nonconformities per unit or per 100 units.
2.1.6
average sample size
ASSI
〈acceptance sampling〉 average number of units in sample inspected per lot in reaching decisions to accept or
not to accept when using a given acceptance sampling plan
NOTE ASSI is dependent on the actual quality level (2.1.5) of the submitted lots.
[ISO 3534-2:2006; 4.7.3]
2 © ISO 2011 – All rights reserved

2.1.7
average outgoing quality
AOQ
〈acceptance sampling〉 expected average quality level (2.1.5) of outgoing product for a given value of
incoming product quality
[ISO 3534-2:2006; 4.7.1]
2.1.8
average outgoing quality limit
AOQL
〈acceptance sampling〉 maximum AOQ (2.1.7) over all possible values of incoming product quality level
(2.1.5) for a given acceptance sampling plan and rectification of all non-accepted lots unless specified
ot
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