capability indices such as Cp, Cpk, Cpm, Pp, and Ppk were developed to provide a simple metric to assess process capability (the ability of a process to produce parts that meet specifications). In practice, they are often misused. Cp and Pp describe the potential of the process to meet specification if the process is centered. The index compares the tolerance to the range over which 99.73% of the individual values will fall (6σ for data following a normal distribution). Cp and Pp only differ in the method used to estimate the standard deviation.
Since Cp and Pp are not affected by the process being off center, the Cpk and Ppk were developed to produce indices which include a "penalty" for being off center. Again, Cpk and Ppk only differ in the method used to estimate the standard deviation.
In cases where the process target is not halfway between the upper and lower spec limits, the Cpm is a better alternative to the Cpk since it includes a "penalty" for being off target rather than off center.
Common mistakes when utilizing capability indices include:
Failure to Ensure Process Stability
Computing process capability for a process that is out-of-control (e.g. unstable) doesn’t provide any idea of what to expect in the future. Furthermore, process capability implies that a single process exists! Assessing process capability without evidence of stability (using an appropriate control chart) is irrational.
Failure to Appropriately Handle Non-Normal Data
Blindly utilizing the typical process capability formulas (for normally distributed data) can lead to incredibly misleading results. Since process capability assessment is performed on individual measurements, care must be taken to check whether an assumption of normality is reasonable. Much (if not most) industrial data does not follow a normal distribution. Several methods (such as data transformations or distribution fitting) for handling non-normal data are readily available in modern software packages.
Failure to Quantify Uncertainty in Estimated Capability Indices
Capability indices are computed from sample data which is subject to randomness. While confidence intervals are routinely utilized in many applications of statistical methods (such as reliability estimation), they are routinely ignored in process capability assessment. Simply comparing capability estimates to minimum acceptable levels of process capability (e.g. is Cpk > 1.33?) fails to account for the inherent uncertainty in our estimates. Utilizing appropriate sample sizes is a critical aspect of ensuring adequate precision in any estimates including capability indices.
Failure to Understand the Limitations of Capability Indices
When assessing process capability, both the process center (e.g mean or median) and the amount of variation (e.g. standard deviation) are important. Identifying a statistic such as Cp or Cpk that summarizes both is impossible. A process that produces only 50% (or 0%) in spec can have the same Cp as one that produces nearly 100% in spec since the Cp is not affected by the process location relative to spec limits. What about Cpk? Consider the two processes below:
Both of these processes have a Cpk = 1. The top process produces much more consistently. Furthermore, the bottom process will produce twice the proportion of parts that are out-of-spec than the top process will! It should be relatively easy to shift the top process to the right (if this is desired as it may be more cost effective to run near the lower spec limit).
While they are widely used to report process capability, the indices described here are often misapplied and misunderstood in practice. Estimating the actual data distribution and the proportion of parts exceeding each specification limit would serve us much better as it avoids the misleading nature of the process capability indices. Most importantly, we should focus on minimizing variation in key characteristics since it is critical to achieving high quality, reliability, and success of manufacturers. ME
This article was first published in the August 2011 edition of Manufacturing Engineering magazine. Click here for PDF.