Acceptance sampling plan for attributes

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A single-sample acceptance sampling plan is based on the parameters $(N,n,c)$; in the case of attributes, the lot is accepted if the number of defectives in a sample of size $n$, taken from a lot of size $N$, is no more than $c$. The operating characteristic curve (OC curve) of the single sample plan is the probability of lot acceptance, plotted as a function of the lot fraction defective. Points on the operating characteristic curve corresponding to the acceptable quality level (AQL) or lot tolerance percent defective (LTPD) are obviously of great interest. An important aspect is the behaviour of the operating characteristic curve as $n$ and $c$ vary. One important acceptance sampling procedure is the selection of a single sample plan with a specified operating characteristic curve; one requires that the latter pass through two designated points, usually the ones corresponding to the acceptable quality level and the LTPD. A nomograph (cf. also Nomography) or various statistical software packages may then be used to create the required plan. A single-sample plan assures high quality for both the consumer and the producer. The practice of rectifying inspection, i.e., 100% screening of rejected lots, with removal of bad items, leads to an overall increase in outgoing quality, to a level termed the average outgoing quality (AOQ).

Extensions of single-sample plans for attributes include double-, multiple-, and sequential sampling plans. A double-sample acceptance sampling plan accepts the lot if the number of defectives $d_1$ in a first sample is no more than $c_1$; it rejects the lot if $d_1>c_2$. If $c_1<d_1\leq c_2$, a further sample of size $n_2$ is taken ($d_2$ is the corresponding number of defectives) with the lot being accepted if $d_1+d_2\leq c_2$, and rejected if $d_1+d_2>c_2$. A double-sample plan usually (but not always) reduces the total amount of inspection. The operating characteristic curve is somewhat more complex, and the average sampling number curve (ASN curve) is of great interest to the quality engineer/statistician. The design of the double-sample plan with two specified points on the operating characteristic curve is detailed in [a2], as is the extension to multiple sample and sequential sampling plans.

Lot-sensitive compliance sampling plans (LTPD plans) are useful when one wishes to assure quality no worse than targeted, as may be the case in compliance testing or sampling for safety-related characteristics. The basic point is that the lot is rejected if any defectives are found in the sample ($c=0$). The procedure gives the proportion of the lot that must be sampled so that the fraction of defectives in the lot is less than $100(1-\alpha)$% with probability $1-\beta$. The above plan should clearly be used only in a near zero-defect environment.

The military standard 105D (MIL STD 105D) is the most commonly used acceptance sampling system for attributes. Civilian versions include ANSI/ASQC Z1.4 and ISO 2859. The background of MIL STD 105D can be found in [a1]. MIL STD 105D involves three types of sampling: single, double and multiple. Inspection in each case may be normal, tightened or reduced, and the focal point is the acceptable quality level, which makes the procedure appropriate when one wishes to maintain quality at a target. Details and tables may be found in [a2].

Other sampling plans include the Dodge sampling plan and the Romig sampling plan, and are based either on the lot-sensitive compliance sampling plans or average outgoing quality level criteria. See [a2] for more details.


[a1] G. Keefe, "Attribute sampling - MIL-STD-105" Industrial Quality Control (1963) pp. 7–12
[a2] D. Montgomery, "Introduction to statistical quality control" , Wiley (1991) (Edition: Second)
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Acceptance sampling plan for attributes. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.P. Godbole (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article