Reddit reviews SPC Methods for Quality Improvement

Here's a link to some course notes on the subject (from a former professor of mine). The main textbook reference is Montgomery, 6th ed., which could be found through Google. A more in depth textbook is Quesenberry.

If you go to page 228 of Montgomery, it notes the "theoretical" basis for control charts. I'm using them a lot in my current project (which may turn into a thesis for my PhD). I put theoretical in quotation marks because I'm also a little bit uneasy about them. For the intended purpose (e.g. a collection of statistics from identical processes with), they are exactly what they should be. Using them in any other context is not statistically sound, but seems to be done fairly often.

Basically, you have a collection of means that are assumed to be from the same process. Under the null hypothesis (Normal with mean mu_xbar and constant standard deviation sigma), approximately 95% of these means should be within the control limits (mu_xbar + 3*sigma_xbar).

Unfortunately, this amounts to a large amount of hypothesis tests, and multiple comparison/researcher degrees of freedom become a large issue (especially since the control limits are meant to be re-evaluated every time a point falls outside the control limits - i.e. is rejected by a hypothesis test).

They are, however, very practical. They provide a nice little heuristic approach to find something like outliers or points that should be investigated further. They can also be easily used and understood by researchers. IN a lot of situations they lack statistical rigor, but they work as practical methods.