Studying the Capability of Capability Studies - Part 2

In the previous post I listed three items to watch out for when evaluating capability study results.

• Cp versus Pp
• The distribution of the data
• Sample size

In part one, I discussed Cp versus Pp and hopefully helped cast some light on the differences between these two measures of capability. 

In this blog post I will deep dive deep into the distribution of data. As a reference you may want to review my earlier blog post Normal Data is Nearly Never Normal.

The thing to remember is that the standard capability study assumes your data is normally distributed. This assumption of normality, while not so critical in other statistical tools, is very important in capability studies. 

Cp and Pp are giving us predictions based on a sample of how our population will behave out in the far tails of the normal curve. These measures are using mean and standard deviation to create a normal distribution, and from this predict how many of our parts, over the entire population of parts, will fall outside our tolerance limits. 

If we assume our process is normally distributed, and measure Pp = 2.0 (6 sigma process), this means we are predicting extremely low ppm levels of defects, less than 0.002 ppm (3.4 ppm with the irritating  1.5 shift). 

What this means (regardless of your position on the 1.5 shift) is that if the data is not actually normal we could either over estimate or under estimate our risk. 

Below is an example of this. Using Minitab, I generated 1000 rows of random data using a Weibull distribution (Shape = 1, Scale = 100, Threshold = 10). I somewhat arbitrarily chose 0 and 400 and spec limits. I performed a Capability study using three variations.  (1) assuming the data was normal (2) assuming the data was Weibull, and (3) assuming the data was normal but setting the zero lower limit as a lower boundary. 

 

Weibull data calculated as Normal. Ppk =  0.38, Expected ppm outside limits = 127,006

Weibull data calculated as Weibull. Ppk =  0.69, Expected ppm outside limits =102,069

Weibull data calculated as Normal. 0 = LB. Ppk =  1.02, Expected ppm outside limits =1132

This example, while maybe not the best, does illustrate several things.  Firstly, that one can get many different calculations for Ppk and 'expected ppm defective' depending on your assumptions. In this example, if the engineer did not check the distribution, but did realize that the lower spec limit was a lower boundary, he or she would assume the process had a Ppk of about 1.02. But if he or she checked the distribution and ran the study using a Weibull distribution, the Ppk would be worse -- closer to 0.69.

Bottom line is that one must be practical. I often attempt to find a distribution or transformation that fits and if so, I use that to calculate Pp, etc.

But if I cannot find a good fit (for instance data that is heavily left leaning toward zero) we may find that the fit to normal is horrible, but the fit to other distributions is only marginally better. 

I am not sure if a Doctor of Statistics would approve, but in these cases I do two things. One, I pick the distribution or transformation with the lowest Anderson DaArling value and run the Capability study using that. I then create a Time Series plot (if the data is in time order) or a Box Plot and add the spec limits to that. 

This gives me a "practical" view of the range of the data in relation to the specification limits and I can gage whether my estimate of Ppk makes sense.

Have fun. Use the comments to add, comment, correct, spindle or mutilate.

Next time. "Sample Size and Pp"