I thought I understood Cp and Cpk, then I see Pp and Ppk and it looks the same to me. Help me understand this Cp, Cpk, Pp, Ppk stuff. And then there is long-term and short-term capability. Can you help me sort this out?
I feel your pain and confusion regarding these Capability Indices – we are not alone. Not all literature is in agreement and to further complicate matters, not all software treats them the same. You need to know the differences whenever you use these indices.
Let me say it again, I have not found a definitive source that all can agree is the “right” way to calculate any of the indices.
A little history about capability indices: prior to 25 years ago, Cr and Cp were the two predominant indices used. Cr (Capability Ratio) is 1/Cp – they were convenient ways to describe the natural variation of the process compared to the tolerance. Their origins go back much farther and were associated with control charts. Since control charts used Rbar/d2 to estimate standard deviation, the calculation of Cp used Rbar/d2.
The original definition and interpretation of Capability was from Statistical Quality Control Handbook Capability by the Western Electric Company (1956). In the book, capability is defined as “the natural or undisturbed performance after extraneous influences are eliminated. This is determined by plotting data on a control chart.” This means that engineers used process capability studies to eliminate all special causes from a process before using that process on the “shop floor” and proved it using Rbar/d2 and control charts. The index Cpk is a more recent index that also accounts for the centering of the process, and is predominately utilized today.
If statistical accuracy is required, it is best to use Cpk. If repeatability or trend improvement is the primary concern then Ppk is adequate enough. In other words, if you prefer stable results where long-term trends are of more interest, then Ppk is preferable. If you prefer quick notification of variation, then Cpk may be preferable.
The majority of literature and software (but not all of them) say to use Rbar/d2 when calculating Cp and Cpk, and use the sample standard deviation for Pp and Ppk. Among the literature that promote this are: AIAG; and “The Certified Six Sigma Green Belt Handbook” published by ASQ.
Here is how I think about it:
Compute both estimates of standard deviation. If they are close to the same, it doesn’t matter which one you use (there is probably more error from the sample size than the computation method). If they are different, then your sample probably includes some special cause variation (shift, drift, or outlier) – Ppk is probably the best choice.
When I am making the capability calculations:
If the data comes from a sequence (i.e. they are consecutive samples from the process – or you have a strong believe that there are no special causes acting on the sequence of samples drawn), use Rbar/d2 to calculate Cp and Cpk. Otherwise use the sample standard deviation reported as Pp and Ppk.
When I am reading a report of a capability study:
I should not accept the index value unless I know both how the data was collected and the method used to estimate the standard deviation.
So, the bottom line is – you need to know how the standard deviation was calculated regardless of whether it is called Cp, Cpk, Pp or Ppk.
Remember why we perform capability studies (Cpk or Ppk): to compare tolerance to processes and determine how many good and bad parts the process will produce. Under "ideal" conditions the result of Cpk and Ppk should be the same. Ideal conditions being: Normal distribution, sufficient data set size, no autocorrelated data, and data in statistical control.
I am going to leave the discussion about long-term and short-term and how it may relate to a one-and-a-half sigma shift for another day. Happy calculating!
p.s. A thanks to Bob Einwalter, my editor on this reply.