A capability index question

The following question came to me from a student in my online Six Sigma Black Belt course:

QUESTION: I am calculating P_pk‘s for various processes and some values are, for example -1.2 and 1.2. Is it acceptable to take the absolute value for -1.2 and say that both process performances are equal even though the values are less than the desired value of 1.33? My second question is how to interpret various P_pk values of, say, 0.1 and 0.6? I would say the process is not capable of meeting the specification. Is it possible to differentiate in words small differences in P_pk values?

ANSWER

No, you can’t consider the absolute value to be equivalent. P_pk = -1.2 indicates that the process average is outside of one of the specification limits.P_pk = +1.2 indicates a process that is vastly better than P_pk = -1.2. Some texts assume that both (USL-xbar) and (xbar-LSL) are positive, but this would simply mean that you need not bother doing capability analysis if one of these is not positive because the process is so poor that capability analysis serves no purpose.

As for comparing capability indexes that are relatively close to one another in magnitude, you need to be able to calculate the confidence intervals of the capability indexes to make the comparison. This article describes the calculations. It gets a little deep and we don’t require that Black Belts learn this. It’s more in the domain of Master Black Belts, Quality Engineers, and Statisticians.

Rather than learning the advanced math for problems like this I like to suggest that Black Belts (and MBBs for that matter) use resampling. This involves setting up a spreadsheet with your raw data, calculating whatever statistic you’re interested in from the raw data, then randomly sampling the raw data with replacement and calculating the same statistics on the resampled data many times, say 100 or more. The range of the statistic in the resamples gives a valid confidence interval and it is more precise the more samples you use. To use this approach with your problem, create a spreadsheet with the raw data and calculate the difference in the two C_pks (0.6-0.1=0.5). Then resample and calculate the differences for each resample 100 or more times. The percent of times you get a difference greater than 0 is an estimate of the confidence you have that there is a real difference between the two processes.

Here’s a spreadsheet that uses resampling to estimate the confidence interval on C_pk. The pseudo-lower-confidence-limit on C_pk is in the C_pk Min cell, the upper is in the C_pk Max cell. I used a test data set with 50 values from a random normal universe (mean=100, sigma=10) and 100 resamples for these calculations. If you’re running excel 2003 you can repeatedly press F9 to see how these limits change with a new resampling set.