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# Cpk Formulas

Process capability index i.e. Cpk is an important statistical tool. It is used to measure the ability of a process to produce desired output within the customer’s specification limits. In other words, it measures the producer’s capability to give a product within the tolerance range. Also, it is used to estimate how close we are to a given target as well as about the consistency around the average performance. We will learn the basic concept of cpk as well as Cpk formulas with examples. Let us learn it!

## Cpk Formulas

### Concept of Cpk Formulas

Cp and Cpk both are used for Process Capability measurement. Generally, we may use this when a process is under statistical control. This generally happens with a mature process that will be around for a while. Process capability will use the process of sigma value determined from either the Moving Range, Range or Sigma control charts.

Therefore, Process capability index i.e. cpk is the measurement of process capability. It shows how closely a process is able to produce the required output in comparison to its overall specifications. It also decides about the consistency around the average performance.

Cpk gives us the best-case scenario for the existing process. It may also estimate the future process performance, assuming performance is consistent over time. In Six Sigma we have to describe the quality of processes in terms of sigma. This is because it gives us an easy way to talk about how capable different processes are with a common mathematical model.

1. Cpk is a short term process index that describes the potential capability of a process assuming it was analyzed and stays in the control.
2. It’s an option along with z-score in statistics.
3. We may use time-series and SPC charts to determine process control. If the process is out of control, then assessing the current process is unlikely to reflect the long term performance.
4. The Cp is the best a process can perform if that process is centered on its midpoint.
5. The addition of “k” in Cpk quantifies the amount of which a distribution is centered. A perfectly cantered process where the mean is the same as the midpoint will have a “k” value of 0.
6. The minimum value of “k” is 0.0 and the maximum is 1.0. A proper centered process will have Cp = Cpk.
7. An estimate for Cpk = Cp(1-k). Since the max value for k is 1.0, so the value for Cpk will always be less or equal to Cp.
8. Input is required from the customer regarding the lower specification limit (LSL) and the upper specification limit (USL).

### The Formula for Cpk

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

Where,

 Cpk Process capability index mean Mean value USL Upper specification limit, LSL Lower specification limit. $$\sigma$$ Standard deviation

## Solved Examples for Cpk Formulas

Q.1: Food served at a restaurant should be between 38^{\circ}C and 49^{\circ}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^{\circ}C and the mean value for these temperatures is 40. Find out the process capability index of the process?

Solution:

USL (Upper Specification Limit) =$$49^{\circ}C$$

LSL (Lower Specification Limit) =$$39^{\circ}C$$

Standard Deviation =$$2^{\circ}C$$

Mean = 40

Cpk formula is:

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

$$=min \left (\frac{49-40}{3 \times 2},\frac{40 – 39}{3\times 2} \right) \\$$

$$= min (\frac {9}{6} , \frac {1}{6})\\$$

$$= \frac {1}{6} \\$$

= 0.166

Therefore, the process capability index i.e. Cpk is 0.166.

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