by the way the Wikis are incorrect: a cursory look reveals the following errors:
- Xbar R and Xbar S charts are NOT dependent on an underlying Normal distribution. They ARE dependent on a homogeneous process stream but may have ANY shape.
- They are designed to 'catch' drifts of more than 1.5sigma
Interestingly, the article only made a connection between the
control chart constants and the normal distribution. In practice, they state:
"In the U.S., whether X is normally distributed or not, it is an acceptable practice to base the control limits upon a multiple of the standard deviation. Usually this multiple is 3 and thus the limits are called 3-sigma limits. This term is used whether the standard deviation is the universe or population parameter, or some estimate thereof, or simply a "standard value" for control chart purposes. It should be inferred from the context what standard deviation is involved. (Note that in the U.K., statisticians generally prefer to adhere to probability limits.)"
Nonetheless, Xbar R and Xbar S charts and the associated Western Electric rules are designed to detect variation from the mean. The control limits are a function of the mean, and the rules are concerned with activity about the mean. If that is appropriate action for the controlling the underlying distribution, then it is a great tool for any such processes. I would not say that is appropriate for
any distribution, but it is good for
many centralized distributions. There are the processes that the mean has no real value in understanding variation.
And that goes back to "Shewharts' book was entitled "the economic control of quality", and there is no specific tie from economic control and variation from the process mean. One tool he describes is appropriate for certain processes, but it is not the
only tool in the tool box.