We've been showing the limitations of Cp, Cpk in our seminars for years using very similar graphics to what you have shown.  It's always eye opening for students to really understand what these indices fail to tell you.  Nice summary of the key differences and limitations.  I might also stress that far too often the indices are computed assuming the data is well described by a normal distribution and also assuming the process is stable.  Often these are not the case.  When normality is violated the shape of the data must be accounted for (various methods) and no capability assessment is meaningful for an unstable process.

It might help some folks to consider what "k" represents in Cpk. If we express Cpk as Cp times (1 - |k|) then I think about it a little differently compared to the typical expression minimum (Cpl, Cpu). It should be easy to calculate and display k (k for katayori, meaning deviation or offset). And yeah, I understand Cp needs two limits but the notion still holds.

What is the source and purpose of the fictitious distributions drawn over every graph?

Very good question!  Simply put; it makes people feel good about their false beliefs.  I have seen many data sets that satisfy the "goodness of fit" test for up to six different distributions!  For some reason, people desire the normal distribution.  Six Sigma practitioners are obsessed with the normal distribution.  At a Deming 4-day seminar in 1991, I heard Dr. Deming say, "Normal distribution?  I've never seen one."  At the time, I thought he was crazy!  Now I am older and enlightened (at least somewhat enlightened!) by Dr. Deming's teachings.

I appreciate all the astute comments here!

No doubt Cpm is another important stat to consider if you have a process target. The main point being, of course, that one should never get too fixated on any single stat but look at the data from as many angles as possible.

And yes, the normal distribution is generally overrated and people mistakenly think that if their data aren't normal, that it's "bad" or abnormal. The distribution curves for the illlustrations in this post were generated using a normal capability analysis, hence the bell curves. Certainly didn't mean to imply by using them that eveyrone should be using a normal distribution model to analyze their data!

It's true that some data sets can satsify goodness-of-fit tests for several distributions. It's important to remember, though, that those tests only detect statistically significant *departures* from the selected distribution. That is, the null hypothesis (the assumption) is that the data fit the distirbution, the alternative is that the data do not fit. So failure to reject the null only means that you do not have sufficient evidence to show that your data do NOT fit a given distribution. If the sample is small, that could simply due to a lack of sufficient power.  That's why it's also critical to apply process knowledge, subject matter expertise, historical studies, and so on, when selecting a distribution model for your data, in conjunction with the results from a goodnes-of-fit test.

Statistics is a powerful tool, but only when used synergistically with many other sources of one's brain capacity!

Hi Patrick,

Agree with the article. Just a small issue I have when this Capability issue is raised and used. Control Charts were invented by Walter Shewhart as we know and he took great pains to explain the +/- 3 standard deviations or Sigma's.

Your article says early on in the piece 'six sigma'. I think we all understand that but for Six Sigma (Trade and Service mark for the then 1987 Motorola Quality Program) fraternity they seem to forget it is derived from the above Shewart calculation. Secondly, from a Process in its raw form (not having to be 'normalised') and reflected in a the Control Chart which produces a Histogram. Something I find many have not made the connection. Weird. The AIAG SPC Manual provides such guidance.

I noticed you used process capability in its rightful place and did not go into the spurious Six Sigma's +/- 1.5 Std Deviation Process Shift Allowance for Six Sigma.

That was all, thanks again.

Michael 

Hi Michael,

Thanks for taking the time to point out these important distinctions. It's always a good to go back to the source and reinforce the concrete meaning of these fundamental concepts in SPC. They easily get lost in the fray. I appreciate your commenting.

Patrick