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Re: Process/Quality Conrtol & Acceptance Sampling for Small Uniqu...

From: Stan Hilliard shilliard@samplingplans.com
Date: 13 Aug 2001
Time: 16:07:57

Comments

Hi John,

Attribute acceptance sampling plans are designed to detect when a lot is unique with respect to quality. So the uniqueness of any lot is not a problem for the plan and does not require any special consideration.

On the other hand, small lots can present an issue, but the important thing is not the lot size but the ratio of the lot size to the sample size (N/n). The binomial distribution's assumption of "infinite population" becomes less accurate when N/n is less than 10.

In that case of N/n less than 10, the accurate method would be to use the hypergeometric distribution. To answer your question about references, a hypergeometric method is shown in "Acceptance Sampling in Quality Control", Edward G. Shilling, Page 119 to 121. The tables in that book only go up to N=11. For larger N, see "Tables of the Hypergeometric Probability Distribution (going up to N=100). It is published by Sanford University Press by G. J Lieberman and D. B. Owen.

Working with the hypergeometric is awkward. If N/n is less than 10 and you use the binomial anyway to develop an attribute sampling plan, the calculated sample size will be larger than required to achieve the sampling risks that you specify. (In most sampling applications, the sample is much smaller than one tenth of the lot so this is not usually a problem)

This is not as great a problem as it sounds because the error is on the side of "safety". Consider the most extreme case of violation of the N/n>10 rule. When N/n = 1, (100% inspection) the rule is violated but there is no sampling error at all.

When N/n is less than 10, you can either (1) Use the binomial distribution as an approximation (knowing that you are taking a larger sample than necessary to meet the risks that you specified) or

(2) Use the (accurate but awkward) hypergeometric distribution. (Most hand calculators will calculate the combinations necessary for the hypergeometric distribution.)

I would use the binomial distribution first to get a sample size and acceptance number. Software program TP105 will do this:

www.samplingplans.com/software_oc.htm

Then I would calculate N/n. If N/n is below 10, the consumer's risk will be smaller than calculations imply. That is safe -- not bad if the slight over-sizing of the sample does not increase costs too much.

You can double-check that your chosen plan is satisfactory by calculating its oc curve using the (exact) hypergeometric distribution.

Other considerations:

1) If the sample takes a high percentage of the lot -- consider 100% inspection.

2) If it is important to take as small a sample as possible, use a sequential sampling plan to reduce the average sample size. This could be a sequential plan for attributes. The advantages of the sequential would probably far exceed the slight loss from mildly violating the N/n>10 rule. Or,

3) If you are taking measurements, a variables plan could be much more efficient with respect to sample size -- either based on the lot fraction beyond specification limit, or based on the lot mean.

4) For SPC applications -- as opposed to acceptance sampling -- the goal is usually to analyze or adjust a process rather than to make a decision about a specific lot. For this purpose, procedures based on infinite population are always appropriate.

Stan Hilliard


Last changed: November 20, 2007