lnternational Sundard
2859 12
TNTERNAl|ONAL ORGANTZATTON FOR STANOAROIZATTON.MEXAyHAPOAXAB OPfAH113ALlt48 nO CfAHAAPTU3AUt4T4.ORGANISATION INTERNATIONALE DE NORMALISATION
Sampling procedu Part 2: Sampling isolated lot ins Rdgles d'Cchantillonnage pour les indexds d'aprCs la qualitd limib (AL)
First edition
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Website: www,tcvninf o'org'vn
ffi1[ti
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fi'fi i i o ffi 4tu6'6t'ffi memr
contrilles de lots isol6s,
genter tor Smn0ar0s,llletrolosy 0nil llualittt
1985-0!r-01
UDC 311.213.2 : 60.113.4
lity (LOl for
:
658.562.012.7
Ref. No. ISO 2859/2-1985 (E)
Descriptore : statistical analysis, quality control, inspection by attributes, sampling, sampling tables. acceptability.
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Price based on 2'l pages
:
Gontents Page
0 lntroduction,...,. 1
1
Scope and field of application . . . .
.
1
2 DeJinitions
2
3
Choice of sampling plan .
4
Bulesforacceptanceand non-acceptance
.
.
2
,....
3
5 Doubleandmultiplesamplingplans .......
4
6
Examples illustrating how to use this part of ISO
4
7
Compatibility with ISO 2859
Tables
.
/1
...
.
2859
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p pling ppro Dv a ttributes by ures or lnsoecuon Samplinq Part 2: Sampling plans indexed by limiting quality (LOl for isolated lot inspection 0
Consequently, in c€rtain of the above cases consumer protection may need to be attained or measured by other methods" This part of ISO 2859 uses the limiting quality to measure consumer protection, lt should be remembered, however. that prior information on the supplier's quality assurance system and its effectiveness may play a major part in deciding whether or not to accept a single lot.
lntroduction
0.1
General
ISO 2859 comprises four parts: Part
0: General introduction.
Part 1: Sampling plans indexed by acceptable quality level (AOL) for lot-by-lot inspection" Part 2: Sampling plans indexed by limiting quality (LO) for isolated lot inspection. Part
3: Skip lot sampling plan.
ISO 2859/1 sampling plans, indexed in terms of AOL, which is defined as a process average, were primarilv designed for the assessment of a continuing series of lots. This enables switchinq rules to be employed which not only give protection to the consumer (by the switch to tightened inspection and discontinuation where necessary) but also provide an incentive to the producer (by the switch to reduced inspection) with a reduction to test and inspection costs (when consistently good quality is achieved). However, there is little doubt that in many industrial situations today the switching rules are not applied for a variety of reasons or excuses, ilbt all of which may be valid:
a)
individual ISO 2859/1 plans are used alone but "AOL" protection is still claimed or AOL re-defined, to suit "socalled unique products";
b)
"our industry... product is soecial so ISO 2859/l standard plans need not apply to us";
c) d)
production is intermittent (not continuousl;
e)
purchases are from stock-holders
production is from several different sources in varying quantities, i.e. "job lots";
available;
f)
-
no source data
lots are "small" (use of hypergeometric distribution re-
quired);
g)
lots are "isolated";
h)
lots are re-submitted after initial rejection.
0.2
Objectives
ln an attempt to reconcile the somewhat diverse requests for assistance made over the past few years by cornmittees of international standardizing bodies representing various product sectors, this part of ISO 2859 was drawn up in accordance with the following principles:
a)
the new LQ plans can be easily integrated with the existing AOL plans in ISO 285911;
b)
the LO indexing uses a preterred series of values that cannot be confused with the preferred series of AOL values;
c)
the five basic numbers associated with a single sarnpling plan, i.e. lot size, sample size, acceptance number, AOL (or quality accepted with probability 0,951 and LO, appear in the same table, whenever possible"
0.3
Summary
The problems associated with acceptance sampling inspection involve defining unambiguously the criteria used to judge discrete individual items supplied in quantity, the quality level expected from the manufacturing process, the discrimination offered by the plans and the procedure to be followed when a lot is not accepted, Above afl, however, it is necessary to design the sampling scheme so that it may easily be invoked in a purchasing contract. The plans in this part of ISO 2859 make maximum use of the established plans given in ISO 2859/1, so that sub-clause 12.6 of ISO 2859/1 {see 1.1) can be made directly operational, by providing a rationalized series of plans indexed in terms of limiting quality (LO)'
rso 2859/2-198s (El
Scope and field of application
1
3.2
Scope
1.1
This part of ISO 2859 establishes LQ sampling plans and procedures for inspection by attributes compatible with ISO 285g/l
that can be used when the switching rules given in
8N/lare
ISO not applied. for example, when lots are of an isolated
nature.
The plans in this part of ISO 2859 are indexed by a preferred series of limiting qualities (LO with a consumer's risk usually below 10 %, but always below 13 %. This method of indexing perrnits the "ad-hoc" procedurel) mentioned in sub-clause 12.6 of ISO 2859/1 to be implemented as a standard procedure.
-
NOTE The plans in ISO 2859/1 are indexed by a preferred series of AOL values and inspection levels. During the inspection of a continuing series of lots the application of switching rules helps to ensure that the process average in that series of lots is kept below the specified AOL. l-imiting quality does not have the same direct relationship with the process average (see 3.5.11.
1.2
Field of application
Alternative procedures are provided in this part of ISO 2859 to cater for two situations often met in practice:
a!
Procedure A, to be used when the supplier and consumer both wish to regard the lot in isolation. The tables are based on random sampling from finite lots for both consumer and producer risks. This orocedure shall be used unless there is a soecific instruction to use orocedure B,
bl
Procedure B, to be used when the supplier regards the lot as one of continuing series but the consumer considers the lot received in isolation. The tables are based on random sampling from a finite lot for consumer's risk at the limiting quality, but random sampling from a process for the pro-
ducer's risk and the tabulated values
maintain consistent procedures for customers whether or not they receive individual lots or a continuing series of lots. This procedure is suitable for inclusion in product standards
or
specifications
Although procedure A is based on the hypergeometric distribution for sampling results, this distribution is well approximated by the binomial distribution for plans with non-zero acceptance numbers in procedure A. Hence the OC curves for these plans are well approximated by the tabulated performance of the same plans in procedure B. However, procedure A uses plans with acceptance number zero and with sample sizes based on the hypergeometric distribution of sampling results while procedure B excludes accept zero plans, replabing them with 100
% inspection.
The choice between the procedures is dominated by the attitude towards accept zero plans. Procedure A uses accept zero plans linking the sample size and the lot size by the hypergeometric distribution until the proposed plan corresponds to the accept zero plan in ISO 2859/1 corresponding to that limiting quality. Thereafter the progression of sampling size and acceptance number with lot size is based on the inspection levels available in ISO 2859/1. The overall effect of procedure A resembles inspection level ll for limiting quality less than 8 % and level I for limiting quality greater than 8 7o with 8 % being intermediate between these levels. Procedure B provides greater flexibility in the choice of inspec-
tion levels. The tabulated OC curves are based on the probability of producing a nonconforming unit so thgy are approximately.correct for small sampling fractions but as the sampling
fraction increases the curves (and tables) underestimate the probability of acceptance for good quality and underestimate the probability of rejection for poor quality. For sufficiently small lots. procedure B requires 100 % inspection.
of the operating
characteristic (OCl curves. Tie plans used are a selection of the plans available in ISO 2859/1 so thai a producer can
that contain sampling clauses"
The
manufacturer will be concerned with all of his production, but the individual consumer onlv with the particular lot received.
2
Choice of sampling proceduro
Both procedures treat the limiting quality (LOl as the actual percent nonconforming units in the submitted lots and the probability of acceptance at this limiting quality can be found in table Dl for procedure A and tables 81 to B10for procedure B. Procedure B is indicated as the appropriate procedure if the manufacturer is concerned with a continuing series of lots; whether or not these go to the same consumer. Procbdure A is indicated as the appropriate procedure when the lot is a single lot and shall be used if accept zero plans are required to be part of the sampling scheme.
Definitions
The definitions given in ISO 2859/1 shall apply.
3
Choice of sampling plan
3.1
Speclfying a sampling ptan
When reference is made to this part of ISO 2959 in a contract or specification, it is necessary to specify the procedure to be used (i.e" either A or B). ln the absence of any reference to either procedure, then procedure A shall be used.
3.3
Procedure A
(use table A)
A plan is identified by the lot size and the limiting quality (LO).
with the specified lot
size and limiting quality as indexing values, the sampling size (nl and acceptance number (Ac) are given in table A.
1l ISO 2859/ 1 sub-clause 12.6.2. states : "lf the lot or batch is of an isolated nature. it may be desirable to limit the selection of sampling plans to those, associated with a designatod AOL value, that provide not less than a specified limiting quality protectioh. Sampling plans for this puipose can be selocted by choosing a limiting quality (LQl and a consume/s risk to be associated *ith it. Tables 6 and 7 give piocess levels for which the probabifities of lot acceptance under various sampling plans are 10 lo and 5 %, respectively." This part of ISO 2859 esiablishes preferred values for the LQ and allows the consumer,s risk to vary as indicated above.
2
tso
Although the primary index to these tables is the limiting quality (LO), the producer/supplier needs guidance on the quality level required if lots are to have a high probability of acceptance. lnformation on the producer's risk point is contained in table Dl. lnformation on the probability of acceptance ol relatively good lots by accept zero plans is given in table D2.
3.4
Procedure B
A plan is identified by the lot size, limiting quality (LO) and the inspection level (unless otherwise specified, level ll shall be used).
The specified limiting quality is used to select the appropriate table from tables Bl to 810. Within each table the specified lot size and inspection level indicate the appropriate sample size (nl and acceptance number (Ac). Although the primary index to the table is the limiting quality (LO), the producer/supplier needs guidance on the quality level required if lots are to have a high probability of acceptance. Each table gives information on
the equivalent AOL and details of the OC curves. The OC curves are indexed by the sample size code letter used in ISO 2859/1 and the acceptance number. The OC curves given under tables 81
to 810 are based on the Poisson/binomial distribution; the actual operating characteristic will be more discriminating than these curves by accepting with greater probability when the indicated probability is > 0,90 and with a smaller probability when the indicated probability is < 0,10.
The limiting quality has previously been set at 3,5 %. This is not a preferred value and the tables shall be entered by 3,15 06, since using the nominal limiting quality LO 3,5 % lies in the range 2,5 % < LO <4,0 %.
=
Choice of parameters for the sampling procedures
Unlike the AOL, which prpvides a guide for the producer on the level or quality he needs to produce so that hB can satisfy the
acceptance criteria (sampling clauselrmost of the time, the limiting quality does not provide a reliable guide for the consumer as to the true quality of the accepted lots. For this reason the limiting quality should be chosen realistically at a minimum of three times the desired quality.
provide lots
4
Rules for acceptance and non-acceptance Sampling
Once the sampling plan has been determined as in clause 3, the specified sample units shall now be drawn at random from the lot, and all units in the sample inspected.
(LQ)
to
ln the ISO 2859/1 procedures, increased sample size corresponds to a greater protection for the consumer, ln this part of ISO 2859, consumer protection is held approximately constant and the effect of increasing sample size is to permit the supplier greater latitude in the permitted proceSs averages, lf the consumer is satisfied by the protection provided against an occasional poor lot by the nominal limiting quality, then the inspection level is primarily of interest to the supplier, especially if the costs of the sampling inspection are borne by him. A process average well below the limiting quality (better quality) would allow the use of smaller sample sizes. Conversely, if the consumer is concerned with actual rather than limiting quality, or if he pays for the sampling costs, then greater inspection levels are not necessarily advantageous. For this reason, inspection level lll provided by ISO 2859/1 has not boon incorporated in this part of ISO 2859 for limiting qualities greater than 5 %. When sample sizes have to be kept srnall and other considerations are secondary, specifying level S-2 has the effect of providing a fixed sample size for all lot sizes, the sample size depending only on the limiting quality.
4.1
3.5
This enables the producer/supplier
Example:
3.5,2 lnspectlon level
(use tables B)
3.5.1 Limiting quality
2859/2-1985 (El
of
the
desired quality and still have a reasonable probability of acceptance for the submitted lots, at least for acceptance numbers 3. 5, 10 and 18. For acceptance number 1 plans the lots have to have a quality better than LOlg and for acceptance number zero plans the lots have to be perfect or nearly perf€ct before the probability of acceptance rises to 0,95 or greater. The limiting values are restricted to a set of preferred values and the plans are intended to be used with these preferred values, lf these plans are introduced where a non-standard limiting quality is already specified, the plans should be entered with the prefer-
red value {LO} corresponding to the interval containing the non-standard value (Ll listed in column 4 of table C. Columns 2 and 3 of table C contain information on the consumer's risk qualities (CRO) found in tables Bl to 810 for consumer's risks of 10 % and 5 0,6.
4.2
Acceptance
lf the number of nonconforming units. or the total number of nonconformities, found in the sample is equal to or less than the acceptance number (Ac) specified in the plan. the lot shall be accepted.
4.3 Nonconformingunits Notwithstanding that the lot is accepted, any nonconforming units found during inspection, whether forming part of the sample or not, shall be rejected.
4.4 Non-acceptanceand
resubmission
lf the number of nonconforming units found in the sample is greater than the acceptance number (Ac), the lot is unacceptable. An unacceptable lot may not be resubmitted for inspection unless
a) b)
the Responsible Authority agrees. and
all units in the lot have been re-examined or retested and all nonconforming units have been removed or replaced by go6d units or nonconformities have been corrected.
tso 285!l/2-1S
(E)
lf the Responsible Authoritv agrees to the resubmission of
an
unacceptable lot, he shall determine the method of acceptance inspection to be applied (i.e. LO and/or inspection levell and whether re-inspection shall include all types or classes of nonconformities or only those that caused the initial rejection'
5
Double and multiple sampling plans
Tables D3 and D4 provide the sample sizes and acceptance patterns for the double and multiple sampling plans in ISO 2859/1 equivalent to the single sampling plans given in tables 81 to 810. The sample size code agrees with that in ISO 2859/1 and the acceptance pattern code is the acceptance number in the corresponding single size plans. Since procedure A and procedure B plans have similar OC curves for non-zero acceptance numbers, these double and multiple plans can also be used in procedure A to replace the corresponding single sampling plan. fhe user is referred to 11 .1 .2 and 11 .1.3 in ISO 2859/ 1 for the operation of these double and multiple sampling plans.
6
Examples illustrating how
of lso
to use this part
2859
6.1 A consumer wishes to purchase prepacked sets of 10 screws to include in the self-assembly bookcase kits he plans to sell" While he prefers each set to contain exactly 10 screws, he can tolerate I % of packs with fewer screws but he does not want to risk accepting a much higher percentage of deficient packs. He plans to produce 5 000 kits in lots of I 250.
The supplier agrees to use procedure A with nominal limiting quality 3,15 Yo. For lots of size 1 250, the selected plan has
n=
125,
Ac
=
N,
The single lot requires proportionately fewer tested items and yet the sampling plan still provides a high probability of rejection for quality as poor as 3.15 %, while increasing the probability of acceptance for a lot of quality 1 % from 0,04 to 0,86.
The same consumer wishes
to
purchase the wooden
components of his self-assembly bookcase kit as standard-size, plastic-faced chipboard panels. The supplier produces these panels as part of his regular production and regards the 7 500 panels needed for each lot of 1 250 kits as single lots in the general stream of supply to DIY shops. Scars to the plastic facing occur with probability 0,025 according to quality control checks. The consumer can tolerate some scarred panels since these can be detected and set aside during the processing to
4
processing.
The consumer and supplier agree that procedure B is appropriate and select nominal limiting quality 5,0 % with inspection level S-4. The parimeters of the sampling plan for a lot size of 7 500 is a = 80, Ac = 1. \Mth this plan the probabiliry of acceptance with the current process averags is less than 0,5' A lot which is unacceptable has to be 100 % inspected before use and this high probability of non-acceptance implies inspection costs larger than desirable' lnspection level lll would have provided a sampling plan with average would produce lots with a probability of acceptance great€r than 0,80. A supplier with a better process average, say 1 %, would obtain a similar probability of acceptance with inspection level S-4, This illustrates the ability of better suppliers to work with smaller sample sizes whilst still meeting the same limiting qual-
a = 315 and Ac = 10. The current process
ity criterion,
7
CompatibilitY wath ISO ?f6lgll
7.1
General
Within the probability constraints inherent in attribute sampling. the LO indexed plans standardized in table A and
tables B provide a rationalized selection from the existing AOL indexed plans of ISO 2859/1' Similar rules for acceptance and non-acceptance and the ISO 2859/1 lot size categories have also been retained for compatibility. Significant deviations are given in 7.2and7.3.
1.
The supplier offers to provide the packs needcd for all 5 000 kits Ac = 3. as a single lot. The new sampling plan has n =
6"2
produce the bookcase kit, but he decides that if 5 % of the panels were scarred this would present problems during the
7.2
Procedure A
(see table A)
For unique lots with a relatively high sample size/lot size ratio. becomes necessary to use the hypergeometric distribution'
it
Consequently, there are an additional 39 (Ac = 0l plans in table A for which operating characteristic (OC) curve data is given in table D2. The remaining 80 plans of table A'have been taken from ISO 2859/1.
7,3
Procedure B
(see tables Bl
All the plans in tables B have been selected from ISO 2859/1 by using a sliding scale for the consumer's risk (usually helow 10 %) at the specified limiting quality (LO). ISO 2859/1 inspection levels are also included (see 3.5.2) but Ac = 0 plans have
not been included in tables B, as table A can be used, if Ac = 0 plans are considered essenthl.
tso
Table A
-
2859/2-1985 (E)
Single sampling plans indexed by limiting quality (LOl (Procedure Al Limlting quality ln percent (LOl
Lot size 16
26
51
to
90
150
280
500
to
1 201
to
3 201
to to
1 200
3 200
10 000
35 000
150 000
00t to 5m 000
>
l) i
50
to
501
150
to
to
281
35 001
25
to
151
10 0O1
to
to
91
0,5
When
500 (n0
r
n
Ac
0.8
1,25
2,0
3,15
5,0
8,0
12,5
20
32
0
9 0
6 0
25l|
17 1t
13
0
o
n
501r
50ll
28ll
u.
l5
10
6
Ac
0
0
0
0
0
0
0
n
g0 1l
50
M
u
24
16
l0
I
Ac
0
0
0
0
0
0
0
0
80 0
55
38 0
26
18
13
13
0
0
0
1
65 0
42
28 0
20
20
13
0
1
n
150il
Ac
0
90 0 r30
0
n
200il
170 l)
Ac
0
0
0
95 0
280
20
155
105
20
20
0
0
0
32 0
g2
0
80 0
50
0
1
1
3
380
255
170
125
125
80
50
32
9
32
0
0
0
0
1
1
1
1
430
280
M
2W
125
125
80
50
50
50
0
0
0
1
1
3
3
3
5
10
450 0
315
315
200
125 5
80
1
200 5
80
1
200 3
80
0
5
10
18
500
315
315 3
315
315
2W
125
125
80
Ac
s00 0
5
10
10
10
18
18
n
800
500
500
500
500
315
2N
125
80
1
1
500 3
5
10
18
18
18
18
t8
n
800
800
800
800
800
500
315
2@
125
80
Ac
I
3
5
10
18
18
18
18
18
18
1m
1 250
l2w
125p
800
500
315
2@
125
80
3
5
10
18
18
18
18
18
18
18
n
Ac n
Ac n
Ac n
Ac n
Ac
n
Ac
exceeds the lot size, use 100
1
I
0
% inspection with zero acceptance number.
Limiting guality implies less than one nonconforming item in the lot. Use first available plan for higher LO.
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tso 2859/2-1985
(E)
Table C Preferred value of llmlting quality lLOl %
-
Relation between limitlng quality lLOl and'consumer's risk quality (CRO) Correspondlng range of CRO in tables 81 to 810 lat 10 % consumcr's rlskl
Correspondlng range of CRO in tables Bl to 810 (at 5 96 consumers rlskl
0,54
1,25
0,46
1,34
0,52
2"4
l,€Ei
2,12
2,10
3,15
2,94
3,34
3,34
3,88
5,35
8,16
8,4't
9,39
0,5 0,8
5,0 8,0 12,5 20,0 32,0
0,84
11,3
6,20
12,7
Correspondlng lnterval 1l for non-standard value of limlting quality (Ll
6,5
15,0
1) These plans are intended to be used with rhe preferred series of limiting qualities. lf thege plans are introduced where a non-standard limiting quality is already spacified, the plans should be entered with the preferred value (LO) corresponding to the interval containing the non-standard value (L).
16
tso 2859/2-1985
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rso 2859/2-1985 (Et
Table D2
Selected hypergeometric data for OC curves for zoro acceptance plans
-
r is drawn from a lot containing R nonconforming items. The lot is accepted if there are nonconforming items in the sample (Ac = 0), the probability of acceptance for the lot {P.} is shown below for the minimum and maximum lot sizes in the lot size range and the various sampling plans ,,n/0', (printed in boldtype). A sample of size
Llmitlng quallty (LOl 0,5
0,8
1,25
2,0
380/0
ffito
170to
lEt0
Lot 3lze
Lot size
1200
Lot size 501 r 200
PPP _" RP,P, 'a 0 1,00 l,m 0 1,o 1,@ x 0,24 0,68 | 0,rt9 0,79 2 0,06 0,47 2 0,24 0d2. 3 0,01 0,32 3 0.12 0,49 4 0,@ 0.2. 4 0,06 0,38 5 0,00 0,15 5 o,cB 0,c) I 0,00 0,10 10 0.m 0,09 I 0,00 0,05 13 0,@ 0,(x
RP"Pr 0 1,00 l,m 'l 0,66 0,s 2 0,44 0,74 3 0,29 0,63 6 0,m 0,40 7 0,05 0,31 ,5 0,m 0,10 ,9 0.q) 0,05
RP"P, 0 1,00 t,@ 1 0,75 0,$ 2 0,66 0,&) 3 0,42 0,72 I 0.10 0,4t 't0 0,(E 0,3}
Lot slze 501 1 2m
mt
{30/0 Lot sizo
auo
1201 32m
1
Lot gize 20t
3 200
RFrP. RPrP" 0 r,@ t,m 0 t,@ 1,00 1 0,64 0,r, 1 0,n 0,91 2 0,41 0,75 2 0,50 0,&| 3 0,26 0,66 3 0,,15 0,76 5 0,11 0,49 I 0,(E 0,44 7 0,04 0,36 11 0,05 0,38 16 0,00 0,10 25 0.00 0,10 21 0,m 0,05 3 0.00 0,06 4F0to
315tO
Lot slze
Lot size
3201 1000(
RP"Pa 0 1,00 1,00 I 0,86 0,96 2 0,74 0,91 3 0,63 0,S/ 15 0,10 0,50 20 0,6 0,4 50 0,m 0,10 €6 0,m 0,05 500/0
Lot size t0ml 35m
RP8P" 0 1,@ t,m r 0,95 0.99 2 0,90 0,97 3 0,86 0,96 46 0,10 58 0,06
lm 0,00 26 0.@
18
0,52 0,4it
0,r0 0.06
3
501
20 0,m 27 0,00
3,15
I
5,0
8,0
17t0 Lot siz8
't6 25 RP.P! 0 l,m 1 0,32 2 0,@ 3 0,@ 4 0,01 5 0,m 6 0,m 7 0,m
M
0,1t 0,05
2frto
a/o
!-ot slze
Lot slze
1
RPsP' 0 l,m l,m 1 0,&l 0.94 2 0.61 0,88 3 0,58 0,82 13 0.@ 0,€ t6 0,05 0,36 0,'t0 0,0s
50/0 Lot size
201 t0 0m
5'l
RP.P, 0 1,00 t,@ 1 0,90 0,g, 2 0,81 0.94 3 0,73 0,9t 2. 0,10 0.4
9o
RP"P" 0 t,m t,o 1 0,a2 0,44 2 0,m 0,1s 3 0,00 0,(B 4 0,m 0,04 5 0,m 0,01 6 0.00 z t,m o.o
29 0,05 0,39 7? 0,00 0,10 93 0.@ 0.6
0_01
90/0 Lot size 91
150
20,0
9t0
Gto
Lot slze
Lot slze
Lot 3lze
80/0 Lot slzc 91
150
1,,t0 Lot slze
22to Lot size
5t
90
RP"P8 0 t,@ t,m 1 0,r4 0,51 2 0.@. 0,23 3 0,00 0,'t3 4 0,m 0,06 5 0,m o,Gt 6 0,m 0,02 7 0.m 0.0t 55/0
Lot size 91
150
uto
21tO
Lot aiz6
Lot size
5t
90
51
10t0 Lot size
15/0
Lot size
16tO
Lot slze
g)
5r
9l
t50
ftto
18/0
Lot slze
Lot riza
9l
9l
r50
RPsP" RP!P! RPaP. RP"Pa RP.P. 0 l,m 1,00 0 t,m t,0 0 1,@ 1,m 0 l,m l,m 0 ,,@ 1,00 1 0,01 0,40 1 0,12 0,47 1 0,40 0,60 1 0,58 0.75 t 0,7t 0,&l 2 0,00 0,16 2 0,01 0,2 2 0,t5 0,4{, 2 0,34 0,56 2 0,5r 0,68 3 0,@ 0,06 3 0,@ 0,10 3 0,06 0,25 3 0,19 0,41 3 0,$ 0,56 4 0,@ 0,o2 4 0,m 0,6 il 0,(2 0,16 4 0,11 0,31 7 0.@ 0,26 5 0,m 0,01 5 0,00 0,(2 5 0,01 5 0,(E 0,23 I 0,6 0,21 6 0,@ 0,00 6 0,00 0,0.1 I 0,m 0,10 I 0,01 0,c, 12 0,01 0,(B 7 0.m 0.m 7 0,m 0.m 7 0,o 0,06 0,04 l0 0.00 0.05 15 0,m 0,05
26
50
P,
0,51
0,r0 0,04
0,fi1
22 0,m
0,68 0,27 0,18 0,10 0,05
0,76
0,38 0,S' 0,21 0,a 0.12 0.31 0,06 0,2 0,01 0,10
0,m
0,05
6to Lot size 26
50
Lot slzo
O,A 0,{0
0,2.
0,(B 0,@
0,24
0.m 0.05 19 0,m 0.6 't0/0 . 8/0 90
18 0,01 0,€ 22 0,m 0,05 13t0 Lot slzo 9t
t50
3 0,d} 0,76 14 0,10 0,26 17 0,05 0,20 23 0,C2 0,10 s
2 3 4 5 7 9
Lot sizo
0,9
0,12
0,62 0,&
RP"P, RP"P. 0 t,@ l,m 0 1,00 t,m r 0,0 0,8E r 0,86 0,91 2 0,u qn 2 0,73 0,&i
3. 10 13 17
%
0,r1
t,oo
0,t7
RP"P. 0 t,m .|,00 'I 0,c) 0,89 2 0,64 0,79 3 0,51 0,70 10 0,€ 0,29 't2 q6 0,2.
150
16
RP"Pa 0 t,m t,m r 0,63
RPrPs 0 l,m 1.@ 1 0,, 0,88 2 0,8 0,n 3 0,i4 0,68 7 0,12 0.3 9 0,05 0,28 t5 0,@ 0,10
P" l,m
5t
90
RPaPs RPrP! RP"P" 0 1,@ 1.@ 0 I,m l,m 0 't.@ t,m 't 0.*, 0.62 1 0,53 0,73 1 0,6S 0,82 2 0,1t 0,38 2 0,8 0,4 2 0,17 0,O 3 0,(B 0,24 3 0.r4 0.39 3 0,31 .o,s 0,55 4 0,01 0,t4 4 0,@ 0,n 6 009 5 0,00 0,@ 5 o,ql 0,20 I 0,04 0,19 6 0,m 0,05 7 0,0t 0,10 11 0,01 0,10 7 0,m o,qt 9 0,m 0,05 14 0,00 0,06 38/0 Lot size
32,0
't6 t6 6 6 RPaP" RP"P" 0 l,m l,m 0 t,@ 1,@ 'I 0,19 0,48 I 0,44 0,64 2 0,8 0,2. 2 0.r8 0,lo 3 0,@ 0,10 3 0,06 0.2i1 4 0,m 0,04 4 0,02 0,14 5 0,m 0,01 5 0,m 0.6 6 0,m 0,0r 6 0,@ 0,05 7 0,m 0.00 7 0.@ o.@.
26 50 26 60 8 50 RPrPo RP!P. RP"P8 R 0 l,m 0 1,00 ,,m 0 l,m l,m 0 I 0,44 1 0,15 0.56 I 0,42 0,70 r 2 0,'t9 2 0,@ 0,31 2 0,17 0,€ 2 3 0,G 3 0,m 0,t7 3 0,00 0,3l 3 4 0,00 4 0,@ 0,G 4 0,@. 0,8 4 5 0,01 5 0,00 0,06 5 0,0t 0,r5 6 I 0,m 6 0,00 0,m 6 0,00 0,10 I 7 0,m 7 0.00 0,01 8 0,o 0,04 12
20r 32m
35 0,m 46 0,@
12,5 1310
0.00 0.6
51
s
RP!Pd 0 1,00 t,fi 1 0,84 0.91 2 0,71 0,8 3 0,5S 0,75 12 0,r0 0,r 15 0,6 0,22 22 0,0r 0,10 ? 0,m 0,G
rso 2859/2-1985 {E}
Table D2
-
Selected hypergeometric data for OC curyes for zero acceptance plans lconcludedl Limitins quality lLOl
0,5
0,8
1,25
170t0
130/0
Lot slzo t51 2i)
RP"P" 0 1 2 3 4 s I 7
1.00 0,39 0,15 0,06 0,02 0,01
0,@ 0,00
2,0
3,15
5,0
65/0
Lot size
95/0 Lot size
42t0 Lot slza
t6r
t51
151
280
280
mt0
156/0
r05/0
Lot size
Lot sizo
Lot sizs
Lot siz6
81
2et
81
sul
RP.P. 0 1,00 r,o0 1 0,m 0,,r4 2 0,@ 0,r9 3 0,00 0.08 4 0,m 0.04 5 0,00 0,02 6 0,@ 0,01 7 0,00 0,00
5m
RP.P" 0 1.00 1,@ r 0.22 0,56 2 0,05 0,3r 3 0,0t 0,17 4 0,00 0,t0 5 0,@ 0,06 6 0.@ 0,03 7 0,m 0,02
2&
151
m
RP8P" RPrPa RP"Pa RP"Pa 0 r,00 l,@ 0 1,00 1,00 0 r,@ t,m 0 1,00 t,00 r 0,14 0.54 I 0,37 0,66 1 0,67 0,n 'I 0,72 0,85 2 0,02 0,29 2 0,'t4 0,4 2 0,32 0,5S 2 0,52 0,72 3 0,00 0,r5 3 0.05 0,29 3 0,18 0,45 3 0,37 0,6t 4 0,00 0,08 4 0,02 0,r9 4 0,10 0,36 7 0,10 0.32 5 0,00 0,04 s 0,01 0.r2 5 0.06 0,26 I 0,06 0,23 6 0,00 0,02 6 0.m 0,08 I 0,0t 0.0{, 14 0,01 0,10 7 0,00 0,01 7 0,@ 0,06 ll 0,00 0,05 18 0,@ 0,05
2&/0 281
Lot size
RP.Pe 0 t,m | 0,45 2 0.20 3 0,09 4 0,04 5 0,02 6 0,01 8 0.m
500
r,00 0,69
0,{8 0,33 0,23
0,!6
0,rt
0.05
5@
80/0 Lot size
8t RP"P" 0 t,m I 0,72 2 0,51 3 0,36 7 0.0!, 9 0,05 t3 0,0'r
500
RPePr 0 t,m 1,00 1.@ 'I 0,63 0,79 0,U 2 0,3S 0,62 0.7t 3 0,24 0,49 0,59 5 0,G 0,3r 0,29 6 0,06 0,24 0,2r r0 0.0r 0,09 0,r0 13 0.m 0.04 17 0.@ 0,06
12,5
?ft0
w0
Lot size 151 8t
Lot size t51
20,0
32,0
m
RPaPa RP.Pe 0 l,m 1,00 0 1.00 t,@ t 0,81 0,90 I 0"87 0,93 2 0,86 0,81 2 0,75 0,8 3 0,54 0,73 3 0,85 0,&) ll 0.r0 0,31 15 0,t1 0,32 14 0,O5 0,U N 0,6 0,2. 21 0,0r 0,10 29 0,01 0,r0 ? 0,00 0,05 38 0,m 0,05
ut0
50/0
Lot gize 81
8.0
500
RP.P. 0 1,00 t,m 1 0,82 0,90 2 0,68 0.8r 3 0,55 0,73 12 0,09 0,28 't5 0,05 0,20 2t 0.01 0,10 28 0,m 0,05
Lot 8ize
n1
5m
RPrPa 0 1,00 r,00 1 0,& 0,94 2 0,78 0,U 3 0,69 0,82
18 0,t1 24 0.05 3t 0,0r
rlil 0,00
0,30 0,20
0,r0 0,06
19
rso 2859/2-1985 {El
Table D3
-
Equivalent sample sizos for single, double and multiple sampllng plans Sample sizc code lettor and cumulativo eample slzesll ln accordance with ISO 2859/1
Type of
sampling plan
Double
'tzfi
630
1(m
1 600
2 500
M
n n
50
80
125
200
315
500
32
50
80
125
2W
315
64
1m
160
2n
400
8
13
20
16
26
40
39 52
60 80
32 64 96 128
Q
6s
100
160
48
78
120
192
56
91
144
224
;::
13
16
26
lst
3
5
2nd
6
10 15
24
12 18
20 25 30
21
35
15
2 000
800
L
20
I
1 250
500
K
I
4th 5th 6th 7th
800
J
13
3rd
Multipla
R
H
F
Slngle
o
G
E
q
9
N
P
50
80
't%
2W
315
500
100
160
250 375 500 625
400
dt0
1 000
600 800 1 000
945
1 500
1 260
2 000
1 575
?fi
12@
I
875
1 400
2 500 3 000 3 500
150
24
2N
320
300 350
400 480 560
2il
890
2tu'
)
I For double and multiple sampling plans, the tabulated entries are the cumulative sample sizes. ln each case the successive stages of sampling take a fresh sample equal in size to that taken in the first stage. This sample is aggregated with the samples from previous stages and the combined sample
is then tested by the criterion in table D4.
Table D4 Type of sampling plan
-
Equivalent acceptance numberc for single, double and multiple sampling plans
Approximata relativo
Slngle Double
Multiple
5
10
182t
Re
Ac
Re
Ac
Re
Ac
Re
Ac
Re
1
1
2
3
4
5
6
10
11
18
19
0,63
0
4
2
9
I
14
4
5
6
5 7
5
1
2 2
1
0,trt
12
13
23
24
0,25
3l
3l
4
3t
0
3 3
3t
0,25 0,25 0,25
1
5
0 0
2
1
4
2
6
3
2
5
3
7
13
0,25
I
3 4
6 6
5
1
3 3
2
3
6
7
18 6 .12 11 17 16 22 2. 25 2729 32 33
0,25
!}
0,25
Discrimination ratio (Procedure Bl PlolPss Probability of acceptance at AOL (Procedure Bl
20
3
1
Ac
n
1) 2) 3)
Acceptance number codes
sample sizes
at each stagel)
2 2
I I
7
I
10
05 38
610 .8 11 14 18
15 17 19
10,9
4,89
3,55
2,50
2l
0,91
0.96
0,98
0,98
2t
These relative samples are approximate only. The exact values are given in table D3 for each sample size bode letter. This acceptance number is only used in procedure A. Acceptance of the lot is not permitted on the samples examined.
rso 2859/2-1985 (E)
Table D2
-
Selected hypergeometric data for OC curves for zero acceptance plana lconcludedl Limiting quality {LOl
0,5
0,8
1,fi
2,0
110t0
1ilt/0
95/0
Lot slza
Lot size
rst N RP"P" 0 1,00 1 0,39 2 0,15 3 0,06 4 0,02 5 0,01 6 0,@ 7 0,m ?gJto Lot size 281
5m
t6r
Lot size
m
t5l
m
3,15
5,0
8,0
12,5
65/0 Lot size 15t N
42t0
8to
mt0
RP8P8 RPaPa RPeP. r,00 t,@ 0 1,00 1,@ 0 l,m 0 '| 0,14 0.5.1 1 0,37 0,66 1 0,67 2 0,02 0,29 2 0,14 0,4 2 0,32 3 0,00 0,t6 3 0,05 0,29 3 0,18 4 0,@ 0,08 4 0,02 0,19 4 0,10 5 0,00 0,04 5 0,0r 0,t2 5 0,06 6 0,@ 0,02 6 0,m 0,08 I 0,01 7 0,00 0,01 7 0,00 0,05 rr 0.@
l,m 0,n 0,5S
0,45 0,35
0,m 0,09 0,05
Lot slze 151
RP!Pa 0 l,m 1 0,72 2 0,62 3 0,37 7 0.10 I 0.05 't4 0,0t r8 0,@
Lot size
M t,00
0,6
0,72
0,6r 0,32
0,a 0,10 0,05
t5r
frn
156/0
10Et0
fit0
50t0
Lot size
Lot size
Lot size
Lot size
Lot size
Lot gize
281
281
D1
BI
5@
81
s00
241
0,72
0,U
0,82 0,68 0,56
0,90
5@
500
500
32.0
Lot size t51
RPrP. RPrPs 0 1,00 1,@ 0 t,00 1 0:81 0.90 1 0,87 2 0.68 0,81 2 0,75 3 0,54 0,73 3 0,65 11 0,r0 0,3r t5 o,il 14 0,06 0,u m 0,05 21 0,01 0,10 29 0,0t 27 0,@ 0,05 38 0,m
mt0
n,0
m 1,@ 0,93 0,86 0,80 0,32
0,2. 0,10 0,05
p.to 500
RP.P. RP.Pa RP.Pa RP"P. RPrPa RP.P. RP.Pa 0 1,00 r,@ 0 t,@ 1,00 0'I r,@ t,00 0 r,@ l,m 0 1,00 r,00 0 t,@ l,@ 0 1,@ 1,@ 1 2 3 4 5 6 7
0.00 0,00
0.@
0,m
0,00
0,@ 0,00
0,it4 0.19 0,08 0,04 0,02 0,01
0,00
1 2 3 4 5 6 7
0,22 0,05
0.0r 0,00 0,00 0,00
0.@
0,56
0,3t 0,17
0,!0 0,05 0,03 0.02
2 3 4 5 6 8
| 2 3 5 6 0,0r 0,ll r0
0,45 0,20
0,@ 0,48
0,@ o,sl 0,04 0,23 0,@ 0.16 0,00
0,05
0.63 0,31 0,24
0,@ 0,06 0,0't
13 0.m
0,79 0,62 0,/tS
0.3t 0,24
0,@ 0.04
1 2 3 7 9 t3
0,5r 0,38 0,09 0,05
0,0't 17 0,@
0,71
0,5s 0,29 0,21 0,10 0.05
I 2 3 t2
0,@ 15 0,05
21 0.01 28 0,m
0,8t 0,73 0,28 0,20 0,10 0,05
I 2 3 I8 24 3
0,89 0,94 0,78 0,8 0,69 0,82
0,Il 0,6
0,0r
43 0,@
0.30
0,m 0,r0 0,05
19
rso 2859/2-198s
(E)
Table D3
-
Equivalent sample sizes for single. double and multiple sampling plans Sample slze code lotter and cumulative sample aizeal) in accordance with ISO 2859/1
Type of
sampling plan
Multiple
J
F
G
13
n
a
;::
I
13
20
16
26
lm
lst
3
5
Znd 3rd 4th 5th 6th 7th
6 9
10 15
60
52
18
20 25 30
24 32 40
39
12
I
21
35
56
Single
Double
H
E
15
K
t
M
N
P
o
R
1m
2 000 2 500
50
80
125
200
315
500
800
32 64
50
&)
125
3't5
500
100
160
250
200 400
trn
1(m
800 1 600
I
13
20
315
500
26
40
&i0
1 000
945
80
32 64 96 128
200
16
1 500 2 000
65 78
100 120
91
140
1 250
50
80
125
100
160
2m
tm
150
2q 320
160
zn 2fi
1 260 1 575
300
400 480
600 800 000
192
375 s00 625 750
120o
't &10
3 000
224
350
s60
875
14m
2205
3 500
I
2 500
1)
For double and multiple sampling plans, the tabutated entries are the cumulative sample sizes. ln each case the successive stages of sampling take a fresh sample equal in size to that taken in th6 first stage. This sample is aggregated with the samples from previous stages and the combined sample is then tested by the criterion in table D4.
Table D4 Type of sampling plan
-
Equivalent acceptance numbers for single, double and multipte sampling plans
Approximate relative
Acceptance number codee
sample sizes
at each stagel)
Slngle Double
Multlple
3
1
Re
Ac
Re
Ac
Re
Ac
Re
1
I
2
3
4
5
6
10
11
18
19
0,63
0
4
14
5
59 12
9
4
2 6
5
1
2 2
1
0,63
13
23
24
3 3 4 5 6 6 7
3'
4
1
5
2
6
3
7
13
18 6 11 16 2. 27n 32
0,25 0,25
3)
2
3t
3t
0
0,25 0,25
0 0
2 2 3
2
0,25
1 1
3 3
4
2
3
6
0,25
(Procedure B) PlolPss
1
3
7
I I
5
7
I
10
05 38
610 ,8 11 't4 18
15 17 19
10,9
4,8['
3,ss
2,W
2t
0,91
0,96
0,98
0.98
2l
Probability of acceptance at AOL {Procedure Bl
20
182t
Ac
Discrimination ratio
tr
10
Re
0,25 t
) 2) 3)
5
Ac
n
These relative samples are approximate only. The exact values are given in table D3 for each sample size bode letter. This acceptance number is only used in procedure A. Acceptance of the lot is not permitted on the sampl€s examined.
.12 1?
22 25
fi!
tso
Table D5
-
Correspondonce between lot sizes, AOL values in ISO (lnspection levels I and ll, procedure Bl
Use letterll
Lot size
Limitins quality lLQl
size
2to90
and LO values
AOL valuos in ISO 2859/1 for limitins qualitios (LOl2r
Sample
code
&9/1
28s9/2,1985 (El
0,5 0,8 1,25 2,0 3,15 5,0 8,0 12,5 20,0
t3
I I
9'l to 151
150
to
1 201
I
I
I
2,5
6,5
1,5
4,0
6,5
1,0
2,5
4.0
10,0
0,65
1,5
2,5
6,5
1, 0
1,5
4,0
2,5
t
32
1 200
to 3
50
200
80
0,()
3 201 to 10 000
125
10 (X)1
to 35 000
200
0.25 0.65
1, 0
to
315
0.15 0,40 0,65
1, 5
35 001 150 001
>
i 4,0
n
281 to 500 501
I
I
t3
to 2&
150 Ofi)
to 500 000
,10 0,25 0,()
0,
1 500
500 000
A,O
0.06s
800
r,0
,15 0,25 0,65 t
0,
I
illl
12W
0,10 0,,15 0,40
2 000
0,10 +
)
I
The correspondence between lot size and sample size is determined by the inspection level. The sample size code letter to use is indicated by the head of the arrow corresponding to the given inspection level, usually up one row for level I and down one row for level ll. I
2)
When no AOL value appears in the table for a selected code letter and LO value, an arrow will indicate that the sample size has to be increased if it points downwards and decreased if it points upwards. The sample size and AOL value to be used correspond to the entry at the head of the arrow" when this sample size is greater than the lot size all items in the lot have to be inspected.
Table D6
Lot sizo
2to90 91 to 1g) 151 to 280 281 to 500 501 to t 200 1 201 to 3 200 3 201 to 10 000 10 (X)1 to 35 000 35 001 to 150 000 150 001 to 500 000 > 500 000
between lot sizes and samplo size code lettors levels S-1 to S-4 and I to l!!, procedure Bl
Ccfrespondence - (lnspection
General lnspection levela il
Special inspection levels S-1 and
S-2 I
S-3
I
54
ilt
E
E
E
E
E
F
E
E
E
E
F
G
E
E
E
E
E
E
F
G H
H
E E
E
F
J
K
E
E
K
L
E
F
G G H
G H
J
L M
tvl
E
F
E
b
E E
G
J J
H
K
K
J
NJ
M
N P
o
N
o
R
L
P
NOTE - The sample size code letter indicates the appropriate row in table D5; it does not always indicate the sampl&'size to be used directly since this can depend on the limiting quality. Levels S-1 and S-2, for example, give sample sizes independent of lot size but dependent on limiting quality, ranging from l3 to &)0.
21
