# What is acceptance-rejection sampling? How does it work?

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What is acceptance-rejection sampling? How does it work?

In: Mathematics

The basic idea is that if your manufacturing line is operating at a fairly constant quality level but with some variation, you can identify batches made when things are too out of control by inspecting/measuring a random sample of units out of each batch. Ideally this is cheaper than inspecting every single unit that closely, but still provides reasonable output quality.

You have to first decide on a couple of things, namely what are the odds that you want for finding a defective batch, and what risk you want to assume for the possibility of rejecting a batch that isn’t all *that* defective (consumers’ risk and producer’s risk, respectively).

Once you’ve decided on those, you can use some statistics (or look it up in tables) to determine what sample size you should measure out of every batch, and how many defective units out of that sample will cause you to reject the entire batch. What you do with the batch generally depends on the nature of the product….some things can be reworked, some things simply have to be scrapped. Of course, if your sample doesn’t exceed the reject limits, then the entire batch moves on, excepting any possible rejects found during sampling.

Doing such sampling is just one part of a quality control system, and is generally a fairly crude system if not combined with other techniques. But it can provide a fair method to ensure that the producer doesn’t ship too much non-conforming product.

There are some things that can go wrong with such a system. In particular, you need to truly do random sampling or you risk not achieving the goals you decided upon. And if you are striving for very high quality levels then the required sample sizes can become very large; in some cases, so large that you aren’t really “sampling” any more, you’re measuring every unit.