Process Capability for Non-Normal Data Cp, Cpk

08/17/2007:

How is process capability (Cp, Cpk) estimated for non-normal data?
Andy

First, we should discuss some general requirements for Process Capability Indices (Cp, Cpk)
1. You need to know the underlying shape of the process distribution to calculate a meaningful Process Capability index. The standard calculations apply only to a process whose observations are normally distributed. To properly calculate a capability index for non-normal data, you either need to transform the data to normal, or use special case calculations for non-normal processes, such as found in more advanced SPC software.
2. You should never do a transformation, or calculate Process Capability, until you have determined the process is in the state of statistical process control. If the process is not in control, then it is not stable, and cannot be predicted using capability indices. Likewise, an out of control situation is evidence that multiple distributions are in place, so a single transformation for all the process data would be meaningless.

So how do you handle this data?
1. First investigate the process stability using a control chart. We could use an X-Bar chart with a subgroup size of 5. Why five? The Central Limit Theorem tells us the average of five observations from even pretty non-normal processes will tend to be normally distributed. You can do a Normality test on these averages to verify. You might also go with a subgroup size 3 if that works, which it often does. A better approach is to use a Moving Average chart (cell width of 3 or 5, for same reasons as above) or an EWMA chart with your original subgroup size of one (a lambda of 0.4 works well). This chart should handle even non-normal data well.
2. If the process is out of control, stop there and improve the process. Do not bother with a capability analysis or with transformation, as they will be meaningless.
3. If the process is in control, then you can estimate capability. You could either transform the data to Normal and use the standard calculations for capability applied to the normalized data, or fit a distribution to the data and calculate the capability using the percentiles of the distribution. The Johnson technique applies this latter approach.