SIST ISO 28801:2014
Double sampling plans by attributes with minimal sample sizes, indexed by producer's risk quality (PRQ) and consumer's risk quality (CRQ)
Double sampling plans by attributes with minimal sample sizes, indexed by producer's risk quality (PRQ) and consumer's risk quality (CRQ)
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.
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)
Dvojni vzorčni načrti za opisne spremenljivke z minimalnimi velikostmi vzorcev, razvrščeni po stopnji kakovosti, ki ustreza določenemu tveganju proizvajalca in odjemalca
Ta mednarodni standard zagotavlja dvojne vzorčne načrte za opisne spremenljivke za pregled sprejemljivosti serij diskretnih elementov. Načrti so razvrščeni po stopnji kakovosti, ki ustreza določenemu tveganju proizvajalca in odjemalca, pri čemer tveganja nominalnega proizvajalca in odjemalca znašajo (5 %, 5 %), (5 %, 10 %) ali (10 %, 10 %). Načrti se zagotovijo za inšpekcijski pregled za popolno neskladnost in za inšpekcijski pregled za neskladnosti na 100 elementov. Serija je sprejeta, če ni neskladnih elementov (neskladnosti) v prvem naključnem vzorcu, in zavrnjena, če vsebuje dva ali več neskladnih elementov (neskladnosti). Če se v prvem vzorcu odkrije natanko en neskladen element, se izbere drug naključen vzorec; serija je sprejeta, če drugi vzorec ne vsebuje neskladnih elementov (neskladnosti), sicer pa je zavrnjena. Cilj tega mednarodnega standarda je določiti postopke, ki omogočajo hitro in ekonomično določitev odstranitve serije, če je kakovost zlasti dobra ali slaba. Za vmesno kakovost se izbere drug vzorec, da se omogoči zanesljivejše razlikovanje med sprejemljivimi in nesprejemljivimi serijami. Dve velikosti vzorcev se izbereta, da se zmanjša največja pričakovana velikost vzorca ob upoštevanju vhodne kakovosti, ki je predmet nominalnega tveganja, ki se ne preseže. Podobno se lahko načrti uporabijo za preskušanje hipoteze, da je stopnja kakovosti serije ali procesa enaka PRQ (tj. sprejemljiva), v primerjavi z alternativno hipotezo, da je stopnja kakovosti enaka CRQ (tj. nesprejemljiva). Načrti imajo prednost pred enojnimi vzorčnimi načrti, kadar je strošek inšpekcijskega pregleda visok, kadar sta zamuda in negotovost, ki ju povzroči morebitna zahteva po jemanju drugega vzorca, nepomembni, ter kadar je možno dopustiti razmeroma velik delež stopnje kakovosti, ki ustreza določenemu tveganju odjemalca, glede na stopnjo kakovosti, ki ustreza določenemu tveganju proizvajalca. Načrti so primerni za izolirane serije ali za kratke sklope serij, kadar vsota dveh velikosti vzorcev ni večja od približno 10 % velikosti serije. Načrti so primerni tudi za trajne sklope serij, kadar so serije, ki ne izpolnjujejo meril sprejemljivosti, 100-odstotno pregledane, vsi neskladni elementi pa so nadomeščeni s skladnimi elementi; vendar bi bilo treba pri trajnih sklopih serij razmisliti tudi o uporabi dvojnih vzorčnih načrtov iz standarda ISO 2859-1. Statistična teorija, na kateri temeljijo načrti, preglednice in vrednosti, je opisana v dodatku A.
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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 28801:2011(E)
©
ISO 2011
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ISO 28801:2011(E)
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ii © ISO 2011 – All rights reserved
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ISO 28801:2011(E)
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
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ISO 28801:2011(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 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.
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ISO 28801:2011(E)
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
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ISO 28801:2011(E)
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.
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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.
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ISO 28801:2011(E)
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]
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ISO 28801:2011(E)
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
otherwise
[ISO 3534-2:2006; 4.7.2]
2.1.9
nonconformity
non-fulfilment of a requirement
[ISO 9000:2005, 3.6.2; ISO 3534-2:2006; 3.1.11]
NOTE See notes to defect (2.1.11).
2.1.10
nonconforming item
item with one or more nonconformities (2.1.9)
[ISO 3534-2:2006; 1.2.12]
2.1.11
defect
non-fulfilment of a requirement related to an intended or specified use
[ISO 3534-2:2006; 3.1.12]
NOTE 1 The distinction between the concepts defect and nonconformity (2.1.9) is important as it has legal
connotations, particularly those associated with product liability issues. Consequently, the term “defect” should be used
with extreme caution.
NOTE 2 The intended use by the customer can be affected by the nature of the information, such as operating or
maintenance instructions, provided by the customer.
2.2 Symbols
α producer's risk
β consumer's risk
m size of the second sample in a double sampling plan
n size of the first sample in a double sampling plan
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ISO 28801:2011(E)
3 Choosing and operating a sampling plan
3.1 Choosing a plan
The appropriate table from among Tables 1 to 6 shall be selected in accordance with whether inspection is for
nonconforming items or nonconformities and the values of the nominal producer's and consumer's risks. The
selected table shall be entered with the PRQ and CRQ to obtain the sample sizes n and m of the double
sampling plan.
When the table indicates by asterisks that no double sampling plan with the prescribed acceptance and
rejection numbers is available with the required properties, consideration should be given to decreasing the
PRQ or increasing the CRQ, or both.
3.2 Operating a plan for nonconforming items
3.2.1 Acceptance sampling
A random sample of size n shall be drawn and inspected, where n is the first sample size given by the plan. If
no nonconforming items are found in this first sample, the lot shall be accepted. If the number of
nonconforming items found in this first sample is greater than or equal to two, the lot shall not be accepted.
If one nonconforming item is found in the first random sample, a second random sample of size m shall be
drawn and inspected, where m is the second sample size given by the plan. If no further nonconforming items
are found in this second sample, the lot shall be accepted. If one or more nonconforming items are found in
this second sample, the lot shall not be accepted.
3.2.2 Hypothesis testing
The double sampling plan may alternatively be used for hypothesis testing when it is required to test the null
hypothesis that the quality level is less than or equal to the PRQ against the alternative hypothesis that the
quality level is greater than or equal to the CRQ. A first random sample of size n shall be drawn and inspected,
where n is the first sample size given by the plan. If no nonconforming items are found in this first sample, the
null hypothesis shall be accepted. If the number of nonconforming items found in this first sample is greater
than or equal to two, the alternative hypothesis shall be accepted.
If one nonconforming item is found in the first random sample, a second random sample of size m shall be
drawn and inspected, where m is the second sample size given by the plan. If no further nonconforming items
are found in this second sample, the null hypothesis shall be accepted. If one or more nonconforming items
are found in this second sample, the alternative hypothesis shall be accepted.
3.3 Operating a plan for nonconformities
3.3.1 Acceptance sampling
In order to determine the acceptability of a lot in a nonconformities-per-hundred-items inspection, the
procedure specified in 3.2.1 for nonconforming inspection shall be used, except that the term “nonconforming
items” shall be replaced by “nonconformities”.
3.3.2 Hypothesis testing
In order to carry out a hypothesis test in a nonconformities-per-hundred-items inspection, the procedure
specified in 3.2.2 for nonconforming inspection shall be used, except that the term “nonconforming items” shall
be replaced by “nonconformities”.
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ISO 28801:2011(E)
3.4 Notation
A notation used to describe such plans is (n, 0, 2; m, 1, 2). This indicates that the acceptance and rejection
numbers for the first sample of size n are 0 and 2, and that the acceptance and rejection numbers for the first
and second samples combined are 1 and 2, where the second sample is of size m.
4 Operating characteristics
The operating characteristic curves, shown in Figures 1 to 6, indicate the probability that a lot will be accepted
under the various sampling plans for a range of quality levels.
The curves are based on the assumption that the sum of the two sample sizes is no more than about 10 % of
the lot size. If the sum of the two sample sizes exceeds 10 % of the lot size, then the probabilities of accepting
the lot are higher than those shown by the curves at all quality levels, so the producer's risk is reduced and
the consumer's risk is increased.
NOTE A formula for the operating characteristic curves is provided in A.1.3 for nonconforming items and in A.2.3 for
nonconformities.
5 Average sample sizes
5.1 Curtailed inspection
Under curtailed inspection, inspection is stopped as soon as the inspection results are sufficient to establish
whether or not the lot is acceptable or, in the case of hypothesis testing, as soon as it is clear which
hypothesis will be accepted. In the case of the double sampling plans in this International Standard, inspection
would cease as soon as two nonconforming items (nonconformities) were found in the first sample or as soon
as one nonconforming item (nonconformity) was found in the second sample. The operating characteristic
curves are unaffected by curtailment, but the average sample size is reduced. The reduction is small at good
quality levels but substantial at very poor quality levels. The disadvantage of curtailment is that it results in
less precise estimates of the lot or process quality level; this matters more in the case of a continuing series of
lots.
5.2 Average sample size (ASSI) for uncurtailed inspection
Average sample size curves for uncurtailed inspection for the double sampling plans of this International
Standard are shown in Figures 7 to 12. These curves show the average sample sizes that may be expected to
occur under the various sampling plans for a range of levels of process quality. For all the plans, the ASSI
⎯ begins at the value n for perfect quality (because only the whole of the first sample is inspected),
⎯ rises to a maximum where the process quality level p is equal to 1/n, i.e. 100/n % nonconforming or 100/n
nonconformities per 100 items,
⎯ falls more gradually to the value n as the process quality level p worsens beyond 1/n (again because
ultimately only the whole of the first sample is inspected).
NOTE A formula for the average sample size for uncurtailed inspection is provided in A.1.4.1 for nonconforming
items and in A.2.4.1 for nonconformities. Formulae for the corresponding maximum ASSI are given in A.1.5 and A.2.5.
The average sample sizes of the double sampling plans at the PRQ and CRQ for uncurtailed inspection are
given in Tables 7 to 12. Also given are the maximum average sample sizes.
EXAMPLE Suppose that it is required to carry out a test of whether the quality level in a large lot is no more than
0,25 % nonconforming. It has been agreed that the probability of acceptance should be at least 95 % if the lot has a
quality level of 0,25 % nonconforming, but no more than 5 % if the quality is as bad as 5 % nonconforming.
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ISO 28801:2011(E)
Thus the PRQ is 0,25 % and the CRQ is 5 %, with producer's and consumer's risks both equal to 5 %. Table 1 shows that
the appropriate plan has an initial sample of size n = 66 and a second sample size, if required, of m = 39. Table 7 shows
for this sampling plan that the ASSI is 71,5 at the PRQ and 70,6 at the CRQ, with a maximum of 80,5.
5.3 Average sample size (ASSI) for curtailed inspection
Average sample size curves for curtailed inspection for the double sampling plans of this International
Standard are shown in Figures 19 to 24. As in the case of uncurtailed inspection, for all the plans the ASSI
begins at the value n for perfect quality and rises to a maximum. For plans for nonconforming items the ASSI
then falls to the value 2 as quality worsens, because at least two nonconforming items are required for
non-acceptance; for plans for nonconformities the ASSI falls to the value 1 as quality worsens because, at a
quality level of an infinite number of nonconformities per 100 items, the first item will be certain to have more
than one nonconformity.
NOTE Formulae for the average sample sizes under curtailed inspection are provided in A.1.4.2 for nonconforming
items and in A.2.4.2 for nonconformities.
The average sample sizes of the double sampling plans at the PRQ and CRQ for curtailed inspection are
given in Tables 25 to 30. Also given are the maximum average sample sizes.
EXAMPLE Consider again the plan for the example in 5.2, but suppose that this time it is used with curtailment. The
relevant ASSIs are then given in Table 25. It can be seen that effect of curtailment is to reduce the ASSI at the PRQ from
71,5 to 69,1, at the CRQ from 70,6 to 38,2 and at the maximum from 80,5 to 73,7.
It is typical of curtailment that it leads to increasing reductions in the ASSI as the quality level worsens.
6 Actual producer's and consumer's risks
Because the sample sizes m and n are necessarily integers, the actual producer's and consumer's risks will in
general be less than their nominal values. These actual risks are presented in Tables 13 to 18. Note that,
whereas the consumer's risks are typically close to their nominal values, the producer's risks are in some
cases much smaller.
EXAMPLE For the data in the example of 5.2, Table 13 shows that the actual producer's risk is 2,510 % (i.e. about
half of its nominal value) and the actual consumer's risk is 4,978 % (i.e. just below its nominal value).
7 Average outgoing quality (AOQ)
For a continuing series of lots with rectification of all lots that fail to meet the acceptance criteria, it is of
interest to know the long-term AOQ at different levels of incoming quality. Tables 19 to 24 present the values
of the AOQ at the PRQ and at the CRQ, together with the maximum over all levels of incoming quality. This
maximum is called the average outgoing quality limit (AOQL). AOQ curves for the plans of this International
Standard are presented in Figures 13 to 18.
NOTE It is only by means of high (and generally uneconomical) average levels of inspection that good average
outgoing quality levels are achieved at poor incoming quality levels.
EXAMPLE For the data in the example of 5.2, Table 19 shows that the average outgoing quality is 0,244 % at the
PRQ and 0,249 % at the CRQ with a maximum of 0,869 %. The AOQ curve for this example is shown in Figure 13.
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ISO 28801:2011(E)
8 Examples
8.1 Example of sampling plan for nonconforming items
A retailer intends to purchase a single production lot of ten thousand low-energy light bulbs from an accredited
supplier. The supplier demonstrates by in-house records that only one bulb in 1 000 will fail on first use. The
retailer customarily audits such supplies by means of a product sampling plan agreed with the supplier on the
basis that
a) the cost of sampling is factored into the contract price, and
b) if the lot is not accepted, the supplier pays for the 100 % test and the associated cost of replacing the
nonconforming light bulbs.
The agreed sampling plan is a double sampling plan with nominal producer's and consumer's risks of 5 %, a
PRQ of 0,1 % and a CRQ of 2,5 %. Entering Table 1 with these parameters yields the sample sizes n = 133
and m = 80. A random sample of 133 light bulbs is drawn from the lot and tested. One light bulb in the sample
immediately fails, so a second random sample of 80 bulbs is drawn from the lot and tested. None of these
light bulbs fail, so the lot is accepted.
8.2 Example of sampling plan for nonconformities
2 000 metres of 17 cm wide weatherboarding is to be supplied to a builder for use in a prestigious barn
conversion. The owner of the barn has specifically requested that the weatherboarding be knot-free. From
past experience the builder decides that, on the basis of the lengths required, he can manage to comply with
this if the supplied weatherboarding contains knots at a rate of no more than 4 per 100 metres length on
average. The supplier claims that his premium weatherboarding is virtually knot-free and so he is prepared to
use a PRQ of 1 knot per 500 metres. Both supplier and builder decide to limit their risk to 5 %.
Table 4 is therefore used, and entered with a PRQ of 0,2 % and a CRQ of 4 %, yielding a double sampling
plan with first sample size n equal to 84 and second sample size m equal to 51. 84 one-metre lengths of
weatherboarding are selected at random from the consignment and checked for knots. Two are found, so the
consignment is rejected without a second sample being necessary, and the supplier agrees to carry out 100 %
inspection on-site, replacing any lengths of weatherboarding found to have knots.
9 Tables and figures
The numbering scheme and location of the tables and figures of this International Standard are given below:
Nonconforming items Nonconformities
Contents
αu 5 %, αu 5 %, αu 10 %, αu 5 %, αu 5 %, αu 10 %,
βu 5 % βu 10 % βu 10 % βu 5 % βu 10 % βu 10 %
Table 1 Table 2 Table 3 Table 4 Table 5 Table 6
Sample sizes of the plans
Page 8 Page 9 Page 10 Page 11 Page 12 Page 13
Average sample sizes at the
Table 7 Table 8 Table 9 Table 10 Table 11 Table 12
PRQ, maximum and CRQ,
Page 14 Page 15 Page 16 Page 17 Page 18 Page 19
uncurtailed inspection
Table 13 Table 14 Table 15 Table 16 Table 17 Table 18
Actual risks
Page 20 Page 21 Page 22 Page 23 Page 24 Page 25
Table 19 Table 20 Table 21 Table 22 Table 23 Table 24
Average outgoing qualities
Page 26 Page 27 Page 28 Page 29 Page 30 Page 31
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ISO 28801:2011(E)
Nonconforming items Nonconformities
Contents
αu 5 %, αu 5 %, αu 10 %, αu 5 %, αu 5 %, αu 10 %,
βu 5 % βu 10 % βu 10 % βu 5 % βu 10 % βu 10 %
Average sample sizes at the
Table 25 Table 26 Table 27 Table 28 Table 29 Table 30
PRQ, maximum and CRQ,
Page 32 Page 33 Page 34 Page 35 Page 36 Page 37
curtailed inspection
Figure 1 Figure 2
...
SLOVENSKI STANDARD
SIST ISO 28801:2014
01-januar-2014
'YRMQLY]RUþQLQDþUWL]DRSLVQHVSUHPHQOMLYNH]PLQLPDOQLPLYHOLNRVWPLY]RUFHY
UD]YUãþHQLSRVWRSQMLNDNRYRVWLNLXVWUH]DGRORþHQHPXWYHJDQMXSURL]YDMDOFDLQ
RGMHPDOFD
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)
Ta slovenski standard je istoveten z: ISO 28801:2011
ICS:
03.120.30 8SRUDEDVWDWLVWLþQLKPHWRG Application of statistical
methods
SIST ISO 28801:2014 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
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SIST ISO 28801:2014
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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 28801:2011(E)
©
ISO 2011
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SIST ISO 28801:2014
ISO 28801:2011(E)
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ii © ISO 2011 – All rights reserved
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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
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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 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.
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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
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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.
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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.
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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]
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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
otherwise
[ISO 3534-2:2006; 4.7.2]
2.1.9
nonconformity
non-fulfilment of a requirement
[ISO 9000:2005, 3.6.2; ISO 3534-2:2006; 3.1.11]
NOTE See notes to defect (2.1.11).
2.1.10
nonconforming item
item with one or more nonconformities (2.1.9)
[ISO 3534-2:2006; 1.2.12]
2.1.11
defect
non-fulfilment of a requirement related to an intended or specified use
[ISO 3534-2:2006; 3.1.12]
NOTE 1 The distinction between the concepts defect and nonconformity (2.1.9) is important as it has legal
connotations, particularly those associated with product liability issues. Consequently, the term “defect” should be used
with extreme caution.
NOTE 2 The intended use by the customer can be affected by the nature of the information, such as operating or
maintenance instructions, provided by the customer.
2.2 Symbols
α producer's risk
β consumer's risk
m size of the second sample in a double sampling plan
n size of the first sample in a double sampling plan
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3 Choosing and operating a sampling plan
3.1 Choosing a plan
The appropriate table from among Tables 1 to 6 shall be selected in accordance with whether inspection is for
nonconforming items or nonconformities and the values of the nominal producer's and consumer's risks. The
selected table shall be entered with the PRQ and CRQ to obtain the sample sizes n and m of the double
sampling plan.
When the table indicates by asterisks that no double sampling plan with the prescribed acceptance and
rejection numbers is available with the required properties, consideration should be given to decreasing the
PRQ or increasing the CRQ, or both.
3.2 Operating a plan for nonconforming items
3.2.1 Acceptance sampling
A random sample of size n shall be drawn and inspected, where n is the first sample size given by the plan. If
no nonconforming items are found in this first sample, the lot shall be accepted. If the number of
nonconforming items found in this first sample is greater than or equal to two, the lot shall not be accepted.
If one nonconforming item is found in the first random sample, a second random sample of size m shall be
drawn and inspected, where m is the second sample size given by the plan. If no further nonconforming items
are found in this second sample, the lot shall be accepted. If one or more nonconforming items are found in
this second sample, the lot shall not be accepted.
3.2.2 Hypothesis testing
The double sampling plan may alternatively be used for hypothesis testing when it is required to test the null
hypothesis that the quality level is less than or equal to the PRQ against the alternative hypothesis that the
quality level is greater than or equal to the CRQ. A first random sample of size n shall be drawn and inspected,
where n is the first sample size given by the plan. If no nonconforming items are found in this first sample, the
null hypothesis shall be accepted. If the number of nonconforming items found in this first sample is greater
than or equal to two, the alternative hypothesis shall be accepted.
If one nonconforming item is found in the first random sample, a second random sample of size m shall be
drawn and inspected, where m is the second sample size given by the plan. If no further nonconforming items
are found in this second sample, the null hypothesis shall be accepted. If one or more nonconforming items
are found in this second sample, the alternative hypothesis shall be accepted.
3.3 Operating a plan for nonconformities
3.3.1 Acceptance sampling
In order to determine the acceptability of a lot in a nonconformities-per-hundred-items inspection, the
procedure specified in 3.2.1 for nonconforming inspection shall be used, except that the term “nonconforming
items” shall be replaced by “nonconformities”.
3.3.2 Hypothesis testing
In order to carry out a hypothesis test in a nonconformities-per-hundred-items inspection, the procedure
specified in 3.2.2 for nonconforming inspection shall be used, except that the term “nonconforming items” shall
be replaced by “nonconformities”.
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3.4 Notation
A notation used to describe such plans is (n, 0, 2; m, 1, 2). This indicates that the acceptance and rejection
numbers for the first sample of size n are 0 and 2, and that the acceptance and rejection numbers for the first
and second samples combined are 1 and 2, where the second sample is of size m.
4 Operating characteristics
The operating characteristic curves, shown in Figures 1 to 6, indicate the probability that a lot will be accepted
under the various sampling plans for a range of quality levels.
The curves are based on the assumption that the sum of the two sample sizes is no more than about 10 % of
the lot size. If the sum of the two sample sizes exceeds 10 % of the lot size, then the probabilities of accepting
the lot are higher than those shown by the curves at all quality levels, so the producer's risk is reduced and
the consumer's risk is increased.
NOTE A formula for the operating characteristic curves is provided in A.1.3 for nonconforming items and in A.2.3 for
nonconformities.
5 Average sample sizes
5.1 Curtailed inspection
Under curtailed inspection, inspection is stopped as soon as the inspection results are sufficient to establish
whether or not the lot is acceptable or, in the case of hypothesis testing, as soon as it is clear which
hypothesis will be accepted. In the case of the double sampling plans in this International Standard, inspection
would cease as soon as two nonconforming items (nonconformities) were found in the first sample or as soon
as one nonconforming item (nonconformity) was found in the second sample. The operating characteristic
curves are unaffected by curtailment, but the average sample size is reduced. The reduction is small at good
quality levels but substantial at very poor quality levels. The disadvantage of curtailment is that it results in
less precise estimates of the lot or process quality level; this matters more in the case of a continuing series of
lots.
5.2 Average sample size (ASSI) for uncurtailed inspection
Average sample size curves for uncurtailed inspection for the double sampling plans of this International
Standard are shown in Figures 7 to 12. These curves show the average sample sizes that may be expected to
occur under the various sampling plans for a range of levels of process quality. For all the plans, the ASSI
⎯ begins at the value n for perfect quality (because only the whole of the first sample is inspected),
⎯ rises to a maximum where the process quality level p is equal to 1/n, i.e. 100/n % nonconforming or 100/n
nonconformities per 100 items,
⎯ falls more gradually to the value n as the process quality level p worsens beyond 1/n (again because
ultimately only the whole of the first sample is inspected).
NOTE A formula for the average sample size for uncurtailed inspection is provided in A.1.4.1 for nonconforming
items and in A.2.4.1 for nonconformities. Formulae for the corresponding maximum ASSI are given in A.1.5 and A.2.5.
The average sample sizes of the double sampling plans at the PRQ and CRQ for uncurtailed inspection are
given in Tables 7 to 12. Also given are the maximum average sample sizes.
EXAMPLE Suppose that it is required to carry out a test of whether the quality level in a large lot is no more than
0,25 % nonconforming. It has been agreed that the probability of acceptance should be at least 95 % if the lot has a
quality level of 0,25 % nonconforming, but no more than 5 % if the quality is as bad as 5 % nonconforming.
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Thus the PRQ is 0,25 % and the CRQ is 5 %, with producer's and consumer's risks both equal to 5 %. Table 1 shows that
the appropriate plan has an initial sample of size n = 66 and a second sample size, if required, of m = 39. Table 7 shows
for this sampling plan that the ASSI is 71,5 at the PRQ and 70,6 at the CRQ, with a maximum of 80,5.
5.3 Average sample size (ASSI) for curtailed inspection
Average sample size curves for curtailed inspection for the double sampling plans of this International
Standard are shown in Figures 19 to 24. As in the case of uncurtailed inspection, for all the plans the ASSI
begins at the value n for perfect quality and rises to a maximum. For plans for nonconforming items the ASSI
then falls to the value 2 as quality worsens, because at least two nonconforming items are required for
non-acceptance; for plans for nonconformities the ASSI falls to the value 1 as quality worsens because, at a
quality level of an infinite number of nonconformities per 100 items, the first item will be certain to have more
than one nonconformity.
NOTE Formulae for the average sample sizes under curtailed inspection are provided in A.1.4.2 for nonconforming
items and in A.2.4.2 for nonconformities.
The average sample sizes of the double sampling plans at the PRQ and CRQ for curtailed inspection are
given in Tables 25 to 30. Also given are the maximum average sample sizes.
EXAMPLE Consider again the plan for the example in 5.2, but suppose that this time it is used with curtailment. The
relevant ASSIs are then given in Table 25. It can be seen that effect of curtailment is to reduce the ASSI at the PRQ from
71,5 to 69,1, at the CRQ from 70,6 to 38,2 and at the maximum from 80,5 to 73,7.
It is typical of curtailment that it leads to increasing reductions in the ASSI as the quality level worsens.
6 Actual producer's and consumer's risks
Because the sample sizes m and n are necessarily integers, the actual producer's and consumer's risks will in
general be less than their nominal values. These actual risks are presented in Tables 13 to 18. Note that,
whereas the consumer's risks are typically close to their nominal values, the producer's risks are in some
cases much smaller.
EXAMPLE For the data in the example of 5.2, Table 13 shows that the actual producer's risk is 2,510 % (i.e. about
half of its nominal value) and the actual consumer's risk is 4,978 % (i.e. just below its nominal value).
7 Average outgoing quality (AOQ)
For a continuing series of lots with rectification of all lots that fail to meet the acceptance criteria, it is of
interest to know the long-term AOQ at different levels of incoming quality. Tables 19 to 24 present the values
of the AOQ at the PRQ and at the CRQ, together with the maximum over all levels of incoming quality. This
maximum is called the average outgoing quality limit (AOQL). AOQ curves for the plans of this International
Standard are presented in Figures 13 to 18.
NOTE It is only by means of high (and generally uneconomical) average levels of inspection that good average
outgoing quality levels are achieved at poor incoming quality levels.
EXAMPLE For the data in the example of 5.2, Table 19 shows that the average outgoing quality is 0,244 % at the
PRQ and 0,249 % at the CRQ with a maximum of 0,869 %. The AOQ curve for this example is shown in Figure 13.
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8 Examples
8.1 Example of sampling plan for nonconforming items
A retailer intends to purchase a single production lot of ten thousand low-energy light bulbs from an accredited
supplier. The supplier demonstrates by in-house records that only one bulb in 1 000 will fail on first use. The
retailer customarily audits such supplies by means of a product sampling plan agreed with the supplier on the
basis that
a) the cost of sampling is factored into the contract price, and
b) if the lot is not accepted, the supplier pays for the 100 % test and the associated cost of replacing the
nonconforming light bulbs.
The agreed sampling plan is a double sampling plan with nominal producer's and consumer's risks of 5 %, a
PRQ of 0,1 % and a CRQ of 2,5 %. Entering Table 1 with these parameters yields the sample sizes n = 133
and m = 80. A random sample of 133 light bulbs is drawn from the lot and tested. One light bulb in the sample
immediately fails, so a second random sample of 80 bulbs is drawn from the lot and tested. None of these
light bulbs fail, so the lot is accepted.
8.2 Example of sampling plan for nonconformities
2 000 metres of 17 cm wide weatherboarding is to be supplied to a builder for use in a prestigious barn
conversion. The owner of the barn has specifically requested that the weatherboarding be knot-free. From
past experience the builder decides that, on the basis of the lengths required, he can manage to comply with
this if the supplied weatherboarding contains knots at a rate of no more than 4 per 100 metres length on
average. The supplier claims that his premium weatherboarding is virtually knot-free and so he is prepared to
use a PRQ of 1 knot per 500 metres. Both supplier and builder decide to limit their risk to 5 %.
Table 4 is therefore used, and entered with a PRQ of 0,2 % and a CRQ of 4 %, yielding a double sampling
plan with first sample size n equal to 84 and second sample size m equal to 51. 84 one-metre lengths of
weatherboarding are selected at random from the consignment and checked for knots. Two are found, so the
consignment is rejected without a second sample being necessary, and the supplier agrees to carry out 100 %
inspection on-site, replacing any lengths of weatherboarding found to have knots.
9 Tables and figures
The numbering scheme and locat
...
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