Capability Indices Can Be Deceiving

While they are widely used to report process capability, process capability indices such as Cp, Cpk, Pp, and Ppk are almost always misleading.  The intent of process capability assessment is to quantify the ability of a process to produce acceptable parts - parts that meet specification.  However, indices such as Cp and Cpk do not adequately describe processes and their ability to meet specifications.  As a result, the common practice of simply asking suppliers to show evidence of acceptable values (e.g. Cp >1.67, Cpk >1.33) is not enough.

Consider the two normally distributed processes on the left:

The only difference between them is that the lower process is shifted to the right. Each of these processes has a Cp=1. In fact, even if the lower process were shifted so far to the right that it was representing parts that were almost 100% out of spec, the Cp would not change. Cp only tells us the ratio of the tolerance to the range that our process uses most (99.73%) of the time.

What about Cpk? Consider the two processes below:

Both of these processes have a Cpk=1. If you were able to procure parts from either the top supplier or the bottom supplier, which one would you choose? Should you be indifferent? The top process produces good parts much more consistently. Furthermore, the bottom process will produce twice the proportion of parts that are out-of-spec than the top process! And it should be relatively easy to shift the top process to the right (if this is necessary).

When assessing process capability, both the process center (median) and the amount of variation (standard deviation) are important. Trying to compute a statistic (such as Cpk) that summarizes both effectively is impossible.

Other issues contribute to the ineffectiveness of capability indices as a measure of process capability. For one thing, analysts often compute these indices without first establishing process stability. Computing process capability for a process that is out-of-control (unstable) doesn't provide any idea of what to expect in the future. Furthermore, process capability implies that a single process exists! Asking a supplier for a process capability number without evidence of stability (e.g. a control chart) is asking for trouble!

Clearly, process capability estimates of an unstable process are arbitrary, and depend on where the process happens to be running when capability is measured!

Also, many analysts computing process capability assume that the data follow a normal distribution. This is frequently not the case, and the assumption of normality often grossly exaggerates the reported capability numbers. Contributing to this behavior is the fact that the widely published formulas for Cp and Cpk show 6s and 3s (where s is the standard deviation) in the denominators respectively. This is only true when the individual values measured happen to follow a bell-shaped curve.

Simple and superior methods for assessing capability are available. For example, estimating the actual data distribution and the proportion of parts exceeding specification limits would serve us much better. Widely available statistical software handles this task with ease. Reporting process capability in terms of proportion out-of-spec is a direct measure of process capability, and it avoids the misleading nature of the process capability indices. More importantly, we should focus on the estimated standard deviation. It is the best indicator of quality, because reducing variation is critical to the success of manufacturers.

About the Authors:

Allise and Steven Wachs are President and Principal Statistician, respectively, of Integral Concepts, Inc. Integral Concepts is Pinnacle Enterprise Group's partner in the development and delivery of statistical methods training and consulting services. Integral Concepts is the premier provider of optimal solutions to R&D, product design, and manufacturing issues. Using mathematics, statistics, engineering, and scientific methods, Integral Concepts delivers significant improvements in quality, reliability, customer satisfaction, profitability, and "concept to customer."