Question 20.2: CHECKING THE QUALITY OF GASKETS MADE ON DIFFERENT MACHINES I......

The observations that comprise a particular subsample are labeled (in the file) M1Obs1, M1Obs2, M2Obs1, and M2Obs2. The first two are from machine 1; the last two are from machine 2. The \bar{X} and R charts for these data, where the centerline and control limits are based on all 50 subsamples, appear in Figures 20.11 and 20.12. The R chart looks perfectly well within control, and the \bar{X} chart looks even better. In fact, it looks suspiciously too good, with almost no points outside the one standard deviation band, let alone the three standard deviation band. The process appears to be in control, but is the small amount of variation in the \bar{X} chart (relative to the control limits) telling us something?

A simple look at the data shows that the observations from machine 1 are consistently below those from machine 2. The variability in the data from each machine is roughly the same, but they are varying around different means. Think of what this does to the control charts. First, each R is probably a large value from machine 2 minus a small value from machine 1. So the R’s are fairly large. This causes the control limits on the \bar{X} chart to be fairly far apart. However, each \bar{X} is an average of two typical machine 1 observations and two typical machine 2 observations. Such averages are not only fairly stable through time, but the highs tend to cancel the lows. The result is the unusually low variability we see in Figure 20.11.

For the sake of illustration, we assume four observations were taken from each machine each half hour. (These are labeled M1Obs1–M1Obs4 and M2Obs1–M2Obs4 in the file.) Only the first two observations from each machine were used in the above control charts. A rational subsample philosophy would suggest separate control charts for each machine. It turns out (you can check this) that the control charts for machine 1, based on the subsamples of size 4, indicate perfect in-control behavior. The charts for machine 2, again based on subsamples of size 4, appear in Figures 20.13 and 20.14. As you can see from the R chart, the variability in machine 2 suddenly increased shortly after subsample 25. This causes one out-of-control point in the \bar{X} chart and nearly another. Machine 2 should be checked for assignable causes.

The problem here is that when the observations from the two machines are combined into subsamples, the out-of-control behavior is masked by the mixing of highs and lows. We are unable to learn about each machine separately. Therefore, the lesson from this example is that observations within any particular subsample should come from a single process, not the mixture of two or more processes.