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Non-Normal Effects on Cpk

Q: How will lack of normality affect my Cpk statistics for a process? Will the calculated statistics be higher or lower than they should be?

A: There is no one answer to this question. It depends on how different from normal the distribution is, whether the distribution is skewed to one side, how close to a specification the distribution is, and other considerations. Even moderate departures from normality that affect the tails of the process distribution may severely impact the validity of the process capability calculations.

The construction and interpretation of process capability statistics are based on the process being distributed as a normal distribution. For a normal distribution, approximately 99.73% of the observations should fall within 3 standard deviations (s) above the mean and 3 standard deviations below the mean. The Cp statistic is defined as:

The Cp statistic is designed to be equal to 1.0 when the process spread ( 3) is the same as the specification width. With a Cp equal to 1.0 for a normally distributed and centered process, we would expect about 0.27% of the output (2700 parts per million) to be beyond the specification limits.

The Cp statistic assumes that the process is centered, which may not be true. Therefore the Cpk statistic is typically reported and is calculated as:

The Cpk statistic should be equal to 1 when the 3 process spread coincides with one or both of the specification limits. With a Cpk equal to 1.0 for a normally distributed process, we would expect about 0.27% OR LESS of the output to be beyond the specification limits.

When the distribution differs significantly from a normal distribution, the calculated process capability indices will probably be incorrect. There may also be significant discrepancies between the predicted (theoretical) proportion and actual proportion of the process output that are beyond the specification limits. This is because the theoretical percentages are calculated based on the tail probabilities for a normal distribution.

Normal Distribution

First, examine the process capability chart for a normally distributed process. Note how the superimposed normal distribution line follows the histogram of observations fairly well. For this data and specification limits, the Cp is approximately 1 and the Cpk is a bit smaller due to the offset of the process from the center line. Also note that the values of the "theoretical" and "actual" proportions beyond the specifications match fairly well.

Uniform Distribution

Next, examine what happens when the process distribution is uniformly distributed instead of being normal. The superimposed normal curve no longer fits the data in the histogram well, especially the tails. Even though all of the process output is within the specification limits, the calculated Cpk only 0.578. In addition, the estimated proportion of product beyond the specifications is over 8%, although the actual proportion is zero.

The above analysis clearly illustrates a situation in which the Cpk does not truly capture what is happening in the process. Although the Cpk calculated appears "lower" than it "should be" in the above case, it is also possible to have scenarios where the calculated Cpk appears to be significantly better than the actual situation would warrant. Thus, it is important to be very careful that the normality condition is met or nearly met when doing Cpk calculations.

Statit provides nonparametric process capability indices by using the QC -> Variables Charts -> Process Capability Analysis. This analysis looks at the tail probabilities of the process itself rather than a theoretical normal distribution.

ANALYSES BASED ON EMPIRICAL DISTRIBUTION OF THE DATA:

If the extrapolations in both plots looks reasonable, then the estimated fraction of product outside of specification limits is
  Upper spec limit: 0.013042
  Lower spec limit: 0.007186
which equate to a total of 20227 parts per million out-of-spec.

Empirical versions of the Cpk or Ppk statistics are then
  Upper spec limit: CPU = 0.741656
  Lower spec limit: CPL = 0.815947