How do we determine process capability if the process isn't normal?
Cp, Cpk, Pp, and Ppk do not necessarily rely on the normal distribution. If any of these indices increases, you know that the process capability has improved. What you do not know is how that improvement translates into good product. This requires knowledge of the distribution of the individual units produced by the process. The Central Limit Theorem refers to averages and this works for the control chart, but it doesn't work for the histogram. Therefore, we generally make the assumption of the normal distribution in order to estimate the percent out of specification (above, below, and total).
For non-normal distributions, we first estimate some parameters using the data. We then use these parameters and follow a Pearson curve fitting procedure to select an appropriate distribution. Since the relationship between the standard deviation and the percent within CAN vary differently from the normal distribution for distributions that are not normal, (e.g., plus and minus one sigma may not equal 68.26%, plus and minus 2 sigma may not equal 95.44%, etc.), we try to transform the capability indices into something comparable. With this distribution equation, we integrate in from the tails to the upper and lower specifications respectively. Once the percent out-of-spec above and below the respective spec limits are estimated, the z values (for the normal with the same mean and standard deviation) associated with those same percents are determined. Then, Cpk and Ppk are calculated using their respective estimates for the standard deviation. This makes these values more comparable to those that people are used to seeing.