I am trying to understand the mathematical difference between Cpk and Ppk used in Statistical Process Control (SPC). I have gone through the web but all I found a confusing and more confusing theoretical implication of the two. Before I can try to induce their application, I will need to find out What mathematical difference are? And even that I am not able to find a good source. I have googled and wiki-ed.
My understanding is Ppk uses all data in calculating its standard deviation and Cpk uses all data but minus 1 in calculating its standard deviation. Is that it?
For example, if I have a set of data "5" , "4" , "5.5" , "4.5" , "3.5" what is their Cpk and Ppk?
Best Answer
The difference is in the way $\sigma$ is calculated. For an out of control process, AIAG (Automotive Industry Action Group) suggests using $\hat{\sigma}=S=\sqrt{\frac{1}{mn-1}\sum_{i=1}^{m}\sum_{j=1}^{n}(X_{ij}-\bar{\bar{X}})^2}$ as an estimator of the standard deviation (long-term variability).
where $\bar{\bar{X}}=\frac{1}{mn}\sum_{i=1}^{m}\sum_{j=1}^{n}X_{ij}$, for $m$ samples of size $n$.
In that case we are talking about process performance indexes ($P_p, P_{pk}$).
On the contrary, for an in control process, $\sigma$ should be estimated by $\frac{\bar S}{c_4}$ or $\frac{\bar{R}}{d_2}$ (short-term variability) and we are talking about process capability indices ($C_p, C_{pk}$).
Of course, for processes that are in control, process performance and process capability indexes will be almost identical. It should be noted that several prominent statisticians suggest not using any performance indicators when the process is out of control, because the process capability is meaningful only for processes that are in control.