 # Cpk Formula

Process capability index (cpk) is the measure of process capability. It shows how closely a process is able to produce the output to its overall specifications. It decides how consistent we are to our average performance.

A person may perform with minimum variation, but he can be away from his target towards one of the specification limits that indicates the Cpk will be lower, but Cp will be high.

$\large Cpk=min \left (\frac{USL-mean}{3\sigma},\frac{mean-LSL}{3\sigma} \right)$

Where,
$$\begin{array}{l}\sigma\end{array}$$
is a standard deviation,
USL is the upper specification limit,
LSL is the lower specification limit.

### Solved Examples

Question 1: Food served at a restaurant should be between 39°C and 49°C when it is delivered to the customer. The process used to keep the food at the correct temperature has a process standard deviation of 2°C and the mean value for these temperatures is 40. What is the process capability index of the process?
Solution:

USL (Upper Specification Limit) =49°C
LSL (Lower Specification Limit) =39°C
Standard Deviation =2°C
Mean = 40

Cpk is given by,

$$\begin{array}{l}Cpk=min \left (\frac{USL-mean}{3\sigma},\frac{mean-LSL}{3\sigma} \right)\end{array}$$
Now, seperate the formula into two parts, and find the solution:
Solution of part 1: (USL – Mean)/ 3σ
Substitute the values:
= (49-40)/3 ×2
= 9/6
Solution of part 1= 1.5
Solution of part 2: (Mean – LSL)/ 3σ
= (40-39)/3 ×2
= 1/6
Solution of part 2= 0.166
Now, substitute the solutions in the formula, we have:
Cpk = min (part 1, part 2)
Cpk = min (1.5, 0.166)
Since the mininum value is 0.166,
The process capability index, Cpk is 0.166.