The concept of capability refers to the width of the Gauss’ Bell that characterizes it. In a capacity study, the width of the normal distribution obtained (what we call the Process Voice) is compared with the limits of tolerances (the Voice of the Client).
Process capacity is traditionally defined as the distance of 3 times sigma from each side of the average. Therefore, it corresponds to a value equal to 6 times the standard deviation. In some cases, we want to cover more width of the bell so it is carried to include up to 6 times the distance sigma of each side (a total of 12 sigmas).
The formula of the capacity index Cp, as an indicator of quality, is the following:
(Upper tolerance limit – Lower tolerance limit) / 6 sigma
We want our process to be able to operate within the limits of specifications (customer requirements) so the value obtained with the formula must be large (at least greater than 1).
Of course, a capable process (Cp >> 1) could generate defects if it is off-centered. This is the reason why the Cpk index is associated with the Cp index.
The calculation formula of Cpk is as follows:
An indicator on the upper side is calculated: (upper limit – medium) / 3 sigma
A lower side indicator is calculated: (average – lower limit) / 3 sigma
You choose, as index Cpk, the minimum value of these two indicators calculated being this the worst case (the case in which the bell is closer to the limit with the risk of causing defects)
We understand by the formulas themselves that a perfectly centered process will have:
Cp = Cpk.
We must also bear in mind that all these formulas work in case of having some data that fit a normal distribution.