Capability Study when the data is not normal but in control.

Semoi I suggest that you read the original article on Cpk - it describes how it’s not about statistical math or models. (Quality engineering methods rarely are - although classically trained statisticians cant see that - the appropriate quote is “if all you have is a hammer then everything looks like a nail”)

The original article is “Reducing Variability - a New Approach to Quality by L.P Sullivan. Quality Progress July 1984. You can request a copy from ASQ.
The critical quote from his paper is:
“In practice the absolute value of Cpk is not very important, as it is only relevant to the specification limits which in many cases are arbitrary (or not well engineered). The important thing is the change in Cpk values over time…”
This article is straight from the Japanese who pioneered the objective of continually reducing variation rather than settling for ‘in-spec’.

Unfortunately, I do not have access to Sullivans article. However, I infer that your critical quote emphasises the fact that the Cpk value is calculated from data and therefore "merely" is a point estimate. Thus, it is a random variable and is best described in combination with its uncertainty, Sd[Cpk]. If this is the key point of your critical quote, I "generally agree" with you -- see remark, why I do not perfectly agree. If this is not the key point, I really should get this article and learn what Sullivan says about the Cpk value

Remark: Commonly, we associate the standard deviation of a statistic to the confidence interval of that statistic. However, often the confidence interval does not provide the best answer to our question. Instead, in my experience often a more appropriate interval would be either the prediction interval or the tolerance interval.
This brings me back to my first post: Ensure that your calculation serve your purpose. Or, as you have stated many time in this forum: Think instead of blindly using mathematical formulas.

Warning: Long Reply

First I feel for the OP; they are obviously in a position where they have to check the box and are not in a position to challenge the insanity. Any advice to get them thru the exercise is welcome.

As I’ve continually stated those of us in a position to change things must keep fighting back against the insanity of ‘statistical alchemy’ and promote good quality improvement practices…and we can disagree about some of the specifics. This response is to the Cove audience at large and not just the OP or any individual responder…


In that vein, remember that my response was to Jim’s posting about Cpk from a US/EU perspective published by NIST that Cpk is based on the Normality of the data. My point is that the true originating intent of Cpk was never about predicting defect rates or the normal distribution, etc.

Yes, Semoi, you should get the article and read it several times. Also keep an open mind and proper mind set: it’s not pro-statistics or anti-statistics. It isn’t about statistics at all. It’s about continual improvement not prediction. Everyone should read it…

Then we need to remember that Cpk is a point estimate of both the average and the standard deviation. But like all statistical calculations and conclusions, the underlying assumptions (requirements) are more important than which formula you select. In this case the important thing is the sampling scheme: Are you using the within subgroup SD or the total SD, is your sample size sufficient and is it representative of the actual process variation? These inns do matter regardless of which purpose of Cpk you here to.

BUT of even more importance is that Cpk was NEVER intended by the originators to be a prediction of anything. It was only intended to be a rough quantification of the current state of a process’s variation. (Again we must remember Sullivan’s quote that the Cpk was not an absolute number but a relative one with respect to itself). The whole point of his article was the difference between the Japanese approach to quality (continual reduction of variation) vs the US and EU approach (in-spec is good enough). Once the Japanese use of Cpk became ‘known’, the US and EU stayed true to their approach, ignored the true intent and added the prediction of future defect rates as the critical aspect of Cpk thus perverting a somewhat useful index to an abomination of statistical cruelty and gamesmanship.

With the new statistical attempt to predict the future from past events came the never-ending addition of statistical manipulations based on theory rather than real life. Statisticians and those who played statistician at work kept adding things that would improve the precision of the mathematical calculation while reducing the robustness and focus of the underlying requirements. Customers don’t check the underlying assumptions/requirements, they just look at the number and compare it to their ‘spec limit’ of Cpk. Thus Cpk became nothing more than a mathematical exercise.

All of the statistical wringing of hands and clutching of pearls is meaningless. Statisticians can add all of the sophistication to the calculations they want but it is not helpful to quality improvement.

In other, cruder, words you can polish a turd as much as you want but it’s still a turd.