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The Formulas of Ppk and Cpk in VDA 4.1


Rob Nix

Cpk and Ppk are virtually the same thing, but for different purposes, there is no target to make one equal the other. Regarding the long term vs. short term issue, that has created a lot of debate (do a seearch on this site), but I stand by my previous post.
I was wondering when this would come up. :)

This has always caused me confusion as well - which one is long term? which one is short term? And then, the calculations themselves......

And the problem is, depending on which Google result you click first, you get a different description.

First - the k's. In both, Cpk and Ppk are like Cp and Pp EXCEPT they take centering into effect. What I mean is, in both indices, if the process is centered, then Cpk = Cp and Ppk = Pp. So the maximum they can be is the overall statistic. To me that's clear. the sub_ks are centering indices, the parents are variance.

Now, let's talk Cp and Pp then. We know they measure "fit". They are the same index, but one is called "short term" and the other "long term."

Both are expressed as (total tolerance) / (6 * sigma) so what's the difference between the two? Well, the difference is in the sigma.

When conducting the study, one takes subgroups. 3, 5, 6 doesn't matter. You are taking subgroups of size n and you do this N times. So your total number of parts you measure is n*N. Some people want 30. Some 100.

Let's look at the sigma.

In the Pp version, sigma is what Excel would give you if you asked for STDEV of all the measurements. In other words, it is the sigma for the n*N parts, regardless of subgroup. The whole sample.

In the Cp version, you are calculating the sigma for each subgroup, then averaging all these up (basically).

So here's the thought experiment..... Let's say take a subgroup of 3 and do this twice. My first subgroup have very little variation and I am near my lower limit. Then, I wait a while. Weeks. My second subgroup ALSO has little variation, just like the first, BUT ... my process has drifted due to tool wear and I am near the upper limit.

In other words 3 close together near the low limit, then 3 close together near the upper limit.

If I calculate Pp, my sigma will be big, because my "spread" will encompass most of my tolerance. And, since it is the denominator, my Pp will be low.

If I calculate Cp, my sigma will not be so big. Because I'm going to calculate it on subgroup 1 (very small), then separately calculate it on subgroup 2 (also very small) and average them. Resulting in a small variation. And, my Cp will be high.

Think about it - Cp is excluding subgroup to subgroup variation. It is only looking at within subgroup variation, which is then averages across all the subgroups. This is a SHORT TERM look at things. How noisy is it in and around a few parts, basically excluding drift (a long term effect).

Then Pp is LONG term. It maps out more of the variation in a long term effect.

Because of this TYPICALLY Cp > Pp and Cpk > Ppk. This is not ALWAYS the case, but normally it is.

Leastwise, that's what makes sense to me from looking at how we arrive at the numbers, regardless of what the internet tells me.

Bev D

Heretical Statistician
Staff member
Super Moderator
This was my response on the Quality Forum Online forum almost a year ago..
On Cpk, Ppk and long & short term capability. A mathematical conundrum.

Besides the fact that Cpk and Ppk are worse than useless and an abomination on the face of the earth (IMHO), the original definitions (and the ones most reputable statistical software use) are this:

Cpk = the ratio of the process spread to the tolerance using the within subgroup standard deviation. This assumes a homogenous process where the between subgroup variation is nothing more than sampling error. It is also referred to as the 'capability' index or short term capability because there is no subgroup to subgroup variation in the index.

Ppk = the ratio of the process spread to the tolerance using the total standard deviation; within subgroup and between subgroup. This is essential when we have a non-homogenous process where the between subgroup variation is more than sampling error. It is also referred to as the 'performance' index or long term capability because there is subgroup to subgroup variation.

IF the process is homogenous then Ppk and Cpk will yield very similar numbers...

Non-homogenous processes can be stable and predictable and capable. The appropriate control chart would require a rational subgroup schema different than the traditionally taught subgrouping based on sequential parts. In these cases, yes Ppk can be predictive if the other assumptions are met. (random representative sampling, Normal distribution, yada yada.)

There is a corruption of these original definitions perpetuated by some automotive and aerospace companies where the Ppk formula is used for short term studies in development phases where 30-60 sequential parts are made under the same conditions (same equipment, material, operators, etc.). (This is a valid method for short term capability)

Adding to the confusion, some companies will specify the calculation of "Cpk" but provide the formula for Ppk. This goes back to the original index which used the total standard deviation and was called Cpk. There was no short term capability at that time. (see “Reducing Variability: A New Approach to Quality”, L. P. Sullivan, Quality Progress, July 1984 and “Letters” Quality Progress, April, 1985).

The idea of short term capability seems to have ‘matured’ with the Six Sigma explosion. And most likely stems from, the idea that if you could get rid of all assignable causes (between subgroup variation in SPC) you would be left with only common cause variation and that was the ‘inherent’ capability of the process. The process could then only be improved through designed experimentation and fundamentally changing the process. However, the terms assignable and common cause are both operational definitions and not laws of physics. The difference between within subgroup and between subgroup variation is related to the non-homogeneity of the process. The so called ’short term’ capability has relatively little value in understanding or improving process performance.

So the ‘names’ Cpk and Ppk do not correlate to the phrases “short term” and “long term”. Nor do the formulas, as the Ppk formula using the total standard deviation is good for both short term and long term studies, depending on the study design.
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