## Tuesday, May 27, 2014

In a previous article, I showed how a control chart is a major tool in statistical process control. A control chart with the center line being the sample average to monitor the process mean, upper control limit and lower control limit, it's important to know the sample size that will give the size of the shift we wish to detect in the process. It's also very important to know the frequency in which we wish to sample.

Suppose we have a process where we are measuring the average ring diameter over many samples. Let's say the sample average which is the center line is 64 mm and the probability of detecting a shift from 64 mm to 64.02 mm increases as the sample size increases. If the process is large, we use smaller sample sizes than those if the shift of interest is relatively small. In general, larger samples will make it easier to detect small shifts in the process.For example, maybe a sample size of 15 will detect a shift of .01 thirty percent of the time. A sample size of 10 may only detect a shift of .01 twenty percent of the time.

As for the frequency of sampling, the most desirable situation is to take large sample sizes very frequently to best detect even the smallest shifts in the process. But that is usually not possible economically. What is general done is small sample sizes can be taken at short intervals, say sample of 10 every 15 minutes. Or take larger sample sizes at larger intervals, say sample of size 50 every hour.

Another way to handle sample size and frequency is through what is known as the "average run length" of a control chart, or ARL. It's calculation is simple, just 1 divided by p, where p is the probability that any point is out of control (outside of the upper control limit or lower control limit).

For example, if a control chart has limits of 3 standard deviations from the mean (center line), then ARL = 1/0.0027 = 370. Note that .0027 is found from a cumulative standard normal distribution chart. Many calculators will calculate this as well.

What does the ARL mean? It means that the average run length of any process in control is 370. In other words, even if the process is in control, an out of control signal will occur every 370 samples, on average.
There are other factors to be taken into account when trying to more accurately answer the questions of sampling frequency. Such factors include the cost of sampling, probabilities of various types of shifts in the process to occur, losses associated with an out of control process and more.