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# What is the difference between Cpk and Ppk?

### Introduction

This is a question that comes up a lot. The people who ask typically want to hear something like, “Cpk is the potential process capability, the capability the process could have. Ppk is the actual capability of the process, what it is today.

This view, however, could mask a deeper difference: how these 2 indices are calculated. The IMPACT of which one is used - especially if employing as an acceptance criterion or for process improvement - can be profound.

• Cpk uses an estimate of the process (‘population’) standard deviation taken from the within-subgroup variation. It’s derived from the natural process variation over a short period of time. Cpk can be calculated using the sample standard deviation as well, but most statistical software use the within-subgroup variation by default.

• Ppk, on the other hand, uses the sample standard deviation from all observations as an estimate of the process standard deviation.

The difference between the two values for a given manufacturing process can be significant. To illustrate, we'll simulate a manufacturing process with the following parameters:

Normal Distribution

Process Std. Dev. = 1

t(min) = 0 Hrs.

t(max) = 24 Hrs.

Sampling frequency = every hour (including at tmin)

Subgroup size (n) = 5

Total number of subgroups = 25

Total number of samples = # subgroups x subgroup size = 100

### Drifting Process

We begin with a simulation of the process average increasing (drifting) from an initial value of 2.25 to a value of 6 at 24 Hrs. This could be the result of numerous factors, such as:

• Tool wear

• Environmental conditions

• Operator fatigue (manual operation)

• Etc.

The sample values, process average and calculated Cpk (using the average range) and Ppk are shown in the figure below.

### Improved Process

Next, we simulate an improvement in the process resulting from a smaller change in the average: from an initial value of 3.5 at t(min) to only 5 at 24 Hrs.

Perhaps we have:

• Installed better environmental controls

• Provided more frequent operator breaks

• Etc.

As we can see, the Cpk value stays about the same even for the improved process (1.34 in the drifting process vs. 1.38 in the improved process), while the Ppk value increases to reflect the improvement in the process (0.91 vs. 1.29).

### Non-Drifting Process

To complete our comparison, we also include the graph, Cpk, and Ppk values for the non-drifting process (fixed process average = target value of 4.25).

The Cpk and Ppk values are very close, as expected.

### Summary

The table below shows the calculated values of Cp, Cpk (both using the average range and the pooled standard deviation) and Pp, Ppk for the drifting, improved and non-drifting process.

Note: the calculation of Cp (avg. range) and Cpk (avg. range) in the table above includes the use of an unbiasing constant, while that of Cp (Spooled) and Cpk (Spooled) do not.

Using a single value (Cpk or Ppk) is not enough to fully summarize the performance of a complex manufacturing process. However, everything else being equal, I recommend using Ppk instead of Cpk.

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