Role of Sigma in Cpk and Ppk calculations

Cpk and Ppk are calculated using the same formula.
So what’s the difference between the two?
The difference between Cpk and Ppk is the way sigma is estimated.

For Cpk, sigma is estimated using the formula…
Rbar/d2 for Xbar-R chart where Rbar is the average Range and d2 is a statistical constant
Sbar/c4 for Xbar-S chart where Sbar is the average standard deviation and c4 is a statistical constant

For Ppk, sigma is estimated using the formula



Where
X – Individual reading

- Process average

n – Sample size

So why use two separate indices?
Cpk and Ppk are designed for use under specific conditions.

Sigma used in the Cpk calculation explains inherent variation in the process which is caused by common causes. The control chart for such a process would be statistically stable.

On the other hand, sigma used in Ppk calculation explains the total variation caused by common as well as special causes. Control charts for this process would have points out of control i.e. the process is unstable.
New process design or pre production (pilot) runs for new products would typically have special and common causes present. Ppk is used for determining capability of such processes.