Statline article from The Quality Magazine, 4(4), August 1995, 88.

"In control" doesn't mean "normally distributed"

Doug Shaw, John Field and Barbara Trudinger

Some training material on statistical process control (SPC) leaves the reader with the impression that statistical distributions of measurements must be made Normal (Gaussian or bell-shaped) for a process to be brought into control.

For a process to be in control, the statistical distribution of the measurements being used to monitor the process shoulod be the same whenever the process is observed - the measurement distribution should be 'repeatable and predictable'.  Provided the distribution is constant, it doesn't matter what form the distribution takes.

Some measurements are inherently non-normal in their distribution.  A good example is seen in measurements of times taken to complete services; the times cannot be less than zero, and can be very large (as we have probably all experienced!).  The distribution of such times is therefore skewed, and so disticntly non-normal.

As a further illustration, consider the distribution of times, in days, taken to repair breakdowns.  The histogram of 4688 of these times, recorded during August 1993, is shown as Figure 1.

Clearly the distribution is not normal.  We can use this data to set control limits for future observations on the process.  In this case, we took logarithms of the times.  The logarithms have a histogram which is very nearly normal in appearance.  Assuming that the logarithms are normally distributed, we can easily determine the value of log(time) which is three sample standard deviations above the mean, that is, we can determine the upper control limit.  That value is 1.97.  Note that we are only interested in an upper control limit in this case.

We can now say that 99.86% of repair times should, if the process remains in control, be lkess than exp(1.97) = 7.19.  This upper limit is shown in Figure 1.   Note that we have used the normal distribution of the logs of the repair times only to get an upper control limit for repair times.

Repair times were subsequently sampled on a weekly basis, and the histogram of each week's sample repair times are shown in Figure 2 for 10 successive weeks.  The dotted vertical line to the right of Figure 2 is the upper control limit (7.19) calculated from the historical data.

It is clear from Figure 2 that the weekly distributions are similar in shape to the historical distribution in Figure 1, and that in no week is there any repair time in excess of the upper control limit of 7.19.  We could be confident that, in terms of this measure, the repair process is in control.

The measurement we use to monitor a process doesn't have to have a normal distribution for the process to be in control.  Many processes will be monitored by measurements which cannot be normally distributed but, as with the repair process above, the process can be in control.