# Acceptance Sampling Attributes

**Acceptance sampling attributes**

In acceptance sampling by attributes each item tested is classified as conforming or non-conforming. (Items used to be classified as defective or non-defective but these days no self respecting manufacturing firm will admit to making defective items.)

A sample is taken and if it contains too many non-conforming items the batch is rejected, otherwise it is accepted.

For this method to be effective, batches containing some nonconforming items must be acceptable. If the only acceptable percentage of non-conforming items is zero this can only be achieved by examing every item and removing any which are nonconforming. This is known as 100% inspection and is not acceptance sampling. However the definition of non-conforming may be chosen as required. For example, if the contents of jars of jam are required to be between 453 g and 461 g, it would be possible to define a jar with contents outside the range 455 g and 459 g as non-conforming. Batches containing up to, say 5% nonconforming items, could then be accepted in the knowledge that, unless there was something very unusual about the distribution, this would ensure that virtually all jars in the batch contained between 453 g and 461 g.

**Operating characteristics**

For any particular plan the operating characteristic is a graph of the probability of accepting a batch against the proportion nonconforming in the batch. Provided the sample is small compared to the size of the batch and the sampling is random, the probability of each member of the sample being non-conforming may be taken to be constant. In this case the number of non-conforming items in a batch will follow a binomial distribution.

One possible acceptance sampling plan is to take a sample of size 50 and to reject the batch if 3 or more non-conforming items are found. If two or less non-conforming items are found the batch will be accepted. This plan is often denoted by n = 50, r = 3. For a batch containing a given proportion of non-conforming items the probability of the sample containing two or less nonconforming items may be read directly from tables of the binomial distribution ( or may be calculated). For example, if the batch contained 4% non-conforming items, the probability of any particular item in the sample being classified non-conforming is 0.04 and the probability of the batch containing two or less nonconforming items and therefore being accepted is 0.6767. The table below shows the probability of acceptance for a range of other cases.

Operating characteristics for n = 50,r = 3

Ideally, if up to 4% non-conforming is accceptable, the probability of accepting a batch containing less than 4% nonconforming should be one and the probability of accepting a batch containing more than 4% non-conforming should be zero. If this were the case, the shape of the operating characteristic would be as shown opposite. The larger the sample size the steeper the graph. That is, the larger the sample size, the better the plan discriminates between good batches (i.e. batches with a small proportion of non-conforming items) and bad batches (i.e. batches with a large proportion of nonconforming items). Note that, provided the batch is large enough for the binomial distribution to give a good approximation to the probabilities, it is the number of items inspected which determines how good the sampling plan is. The proportion of the batch inspected is not important. Provided the sampling is random it will be better to test say 100 items from a batch of 5000 than to test 10 items from a batch of 500.