CONTROL CHARTS AND HYPOTHESIS TESTING

It is enlightening to think of control charts in the context of hypothesis testing. We let the null and alternative hypotheses correspond to in-control and out-of-control conditions, respectively. As we monitor the process with control charts, there are two types of errors that we can make. The first, a Type I error, occurs when we react to an out-of-control indication when, in fact, the process is still in control. We call this a false alarm. For example, there is some chance that a process that is operating in control will produce a point beyond the control limits. In this case, we might begin a search for assignable causes when there are none. We might also make an unnecessary adjustment to the process to bring it back into control (unnecessary because the process is still in control).

We want to make the probability of a Type I error fairly small. If it is too large, we react to too many false alarms, and in Deming's terminology, we tamper with the process. Not only could this be costly, but it could actually cause an increase in the variability of the process. Therefore, we set the control limits fairly far apart ”typically three standard deviations from the centerline ”so that the chance of observing a point beyond them is very small.

To pursue this a bit further, assume that the 's are normally distributed. (Because each Xbar is an average of several observations, the central limit theorem suggests that this normality assumption is reasonable.) Then we know that the probability of any Xbar's being more than three standard deviations from the mean is 0.0027. From this, we can calculate the mean number of subsamples, called the average run length, or ARL, until an in-control process produces a point beyond the control limits. It is simply

ARL = 1/0.0027 ‰… 370.

In other words, false alarms will be few and far between if the process remains in control. Of course, the flip side is a Type II error. This means that the process has gone out of control, but the control charts do not indicate it. As usual, it is difficult to calculate the probability of a Type II error, because there are many types of out-of-control conditions that could occur. However, let's concentrate on one possible type of out-of-control condition, in which the process variation remains constant but the mean shifts from ¼ to ¼ + k ƒ where k is some fixed constant. For example, if k = 1, then the process mean has shifted upward by one standard deviation. We would like to spot this shift immediately, but we will not spot it until an Xbar falls above the upper control limit. How long, on average, will this take?

Assuming that the Xbar chart has centerline ¼ , the upper control limit is ¼ + 3 ƒ / , and the mean of the process has shifted up to ¼ + ƒ , we first calculate the probability that an is above the upper control limit. Because Xbar now has mean ¼ + ƒ and standard deviation ƒ / , the calculation is a typical normal probability calculation, where we subtract the mean and then divide by the standard deviation:

Here, Z is normal with mean 0 and standard deviation 1.

We would like to keep both Type I and Type II errors to a minimum. That is, we would like to minimize the number of false alarms, but at the same time we would like to spot out-of-control conditions quickly. One strategy is to sample more frequently. Instead of sampling every half hour , we could sample every 15 minutes. Another strategy is to increase the subsample size n from, say, 5 to 10. Both of these strategies are intended to decrease the ARL when the process goes out of control.