# Six Sigma Acceptance Sampling

## What is acceptance Sampling

Acceptance sampling is a method in which we use statistical sampling to determine whether to accept or reject an outcome.

## Operating Characteristic Curve or The OC Curve

The OC curve quantifies the α and β risks of an attribute sampling plan. Below is an ideal OC curve (the bold line) for a situation in which we might want to accept all lots that are, say, ≤ 1% defective and reject all lots that are > 1% defective:

With this ideal (no risks) curve, all batches with ≤ 1% defective incoming quality level would have a probability of acceptance (Pa) of 1.0. And, all lots with > 1% defective would have a Pa of 0. The Pa is the probability that the sampling plan will accept the lot. It is the long-run % of submitted lots that would be accepted when many lots of a stated quality level are submitted for inspection. It is the probability of accepting lots from a steady stream of product having a fraction defective P.

### Typical OC Curve

Since there will always be some risks, a more typical looking OC curve looks more like the one listed in the next page. It is based on the Poisson distribution* (with the defective rate < 10% and n is relatively large compared to N).

## Acceptance Quality Level (AQL), Rejectable Quality Level (RQL) and Lot Tolerance Percent Defective (LTPD)

### AQL – Acceptance Quality Level

The AQL (Acceptance Quality Level), the maximum % defective that can be considered satisfactory as a process average for sampling inspection, here is 1%. Its corresponding Pa is about 89%. It should normally be at least that high.

### RQL – Rejectable Quality Level

The RQL (Rejectable Quality Level) is the % defective, here at 5%, that is associated with the established β risk (which is usually standardized at 10%). It is also known as the Lot Tolerance Percent Defective (LTPD).

### LTPD – Lot Tolerance Percent Defective

The LTPD of a sampling plan is a level of quality routinely rejected by the sampling plan. It is generally defined as that level of quality (percent defective, defects per hundred units, etc.) which the sampling plan will accept 10% of the time.

* The hyper geometric and binomial distance are also used. The alpha risk is the probability of rejecting relatively good lots (at AQL). The beta risk is the probability of accepting relatively bad lots (at LTPD/RQL). It is the probability of accepting product of some stated undesirable quality; it is the value of Pa at that stated quality level. The OC curves are a means of quantifying alpha and beta risks for a given attribute sampling plan. The Pa value obtained assumes that the distribution of defectives among a lot is random – either the underlying process is in control, or the product was well mixed before being divided into lots. The samples must be selected randomly from the entire lot. The alpha risk is 1 − Pa. The shape of the OC curves is affected by the sample size (n) and accept number (c) parameters. Increasing both the accept number and sample size will bring the curve closer to the ideal shape, with better discrimination.

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