SC QUALITY GUILDVariables sampling plan: the k-method
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Enter the sample size and acceptability constant from your plan (for example ANSI/ASQ Z1.9), the specification limits and the measurements. Get the quality index, the accept or reject decision, the Form 2 estimate of percent nonconforming, and the OC curve of the plan.
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Q-BotClick for an explanation of this tool.For educational use. This page is a study aid. The method behind it has far more depth and detail than one page and one tool can cover, and what you see here is not everything the method offers or everything this site offers. For full understanding and application, use the governing standards and texts, and seek further education, training or qualified advice. Results depend on the data you enter, so check them before relying on them.
Variables or attributes?
An attributes plan (such as ANSI/ASQ Z1.4) counts the defectives in the sample. A variables plan measures each unit and uses the sample mean and standard deviation. Because a measurement carries more information than a pass/fail call, a variables plan gives the same protection with a much smaller sample. The price is an assumption: the characteristic must be normally distributed, and each plan covers one characteristic. A lot can be rejected even though every unit in the sample is within specification, because the plan judges how much of the lot is likely beyond the limit.
Form 1: the k-method
With the variability unknown (the usual case, the standard deviation method), compute the sample mean x̄ and standard deviation s, then the quality index for each limit:
- upper limit: QU = (U − x̄) / s
- lower limit: QL = (x̄ − L) / s
Accept the lot if Q is at least the acceptability constant k from the plan. Q is the number of standard deviations between the mean and the limit, so the rule says the mean must sit at least k standard deviations inside. With the variability known from a long, stable record, use σ in place of s; the plans for known variability need fewer units.
Form 2: the M-method
Form 2 turns Q into an estimate of the percent of the lot beyond the limit, p̂, and accepts if it is no more than the maximum allowable percent M. For an unknown σ the estimate is the minimum-variance unbiased one, read from the beta distribution with parameters (n − 2)/2; for a known σ, it is the normal tail area beyond Q√(n/(n − 1)). For a single limit, Forms 1 and 2 of the same plan always agree. For two specification limits, Z1.9 uses Form 2: estimate p̂U and p̂L and accept if their sum is no more than M.
Where n, k and M come from
The standard’s tables give a sample size code letter from the lot size and inspection level, then n, k and M for the AQL, with normal, tightened and reduced inspection and switching rules between them. This page does not reproduce those tables; read the values from your copy of the standard or from the plan agreed with your supplier or customer.
What the plan protects against
The OC curve shows the chance of accepting a lot against the percent of it beyond the limit. The point with a 95% chance of acceptance is near the AQL, the quality the producer can expect to pass; the 10% point is the quality the consumer can expect to be rejected (the LTPD, or limiting quality). The curve is computed from the noncentral t distribution for an unknown σ and from the normal distribution for a known σ.
On the exam
The CQE Body of Knowledge asks you to identify, interpret and apply ANSI/ASQ Z1.4 and Z1.9 for attributes and variables sampling (IV.C.2), with OC curves, AQL and LTPD (IV.C.1). The CQPA covers variables sampling and the OC curve (III.C.1, III.C.2), and the CMDA asks when to use a variables plan instead of an attributes plan (V.C.2). Expect to compute QU or QL and compare it with k, and to explain why a variables plan needs a smaller sample.