In this article we'll look at how to calculate the different capability indices, in particular Cp, Pp and Ppk.

To fully appreciate the capability of a process, the notion of time is particularly important, as there are two types of variability:

- Short-term variability: when two parts are produced consecutively, these two parts will not be completely equivalent due to intrinsic variations in the machine. This short-term variability depends mainly on the machine.
- Long-term variability: when the same process is produced over a longer period of time, the machine itself will go out of adjustment, and changes in series, changes in material batches, etc. will bring new sources of variation. This long-term variability depends on the machine, but also on numerous external sources of variability and the way the process is managed.

## Example

To illustrate this point, let's look at the following production diagram:

## Short-term Cp

Short-term CpShort-term capability: Short-term capability is used to characterise the ability of the process to produce good parts, taking into account only the intrinsic variability of the process (the variability between two consecutive parts). Short-term capability is denoted Cp and is calculated by :

Cp= \frac{\text{tolerance interval}}{6*\text{short term dispersion}}= \frac{\text{tolerance interval}}{6*\sigma_{\text{short term}}}

In general, we want :

## Long-term Pp

Long-term PpLong-term capability: Long-term capability is used to characterise the ability of the process to produce good parts over a long period of time, i.e. taking into account adjustments and process modifications that may occur. Long-term capability is denoted Pp and is calculated by :

Pp= \frac{\text{tolerance interval}}{6*\text{long term dispersion}}= \frac{\text{tolerance interval}}{6*\sigma_{\text{long term}}}

In general, we want :

Cp > 1.33

\text{long term dispersion} < \text{short term dispersion}

And so

Pp < Cp

## Methods for calculating Cp, Pp, Ppk

As we have just seen, Cp and Pp are calculated using the same formula. Cp and Pp are differentiated by the time periods over which the variability is calculated. There are several methods for calculating Cp and Pp.

### Method 1: By taking several samples at regular intervals

**Short-term variability :** Short-term variability is calculated using the intra-series standard deviation of all samples:

\sigma_{\text{short term}}=\sigma_{\text{intra sample}}

**Long-term variability :** 50 parts are sampled over a characteristic period of the process, to take into account multiple sources of process variation such as adjustments, tool changes, material changes, etc. Long-term variability is calculated by :

\sigma_{\text{long term}}=\sigma_{\text{all sample}}=\sqrt{\sum_{}^{}(\frac{^{(x_{i}-\mu)^{2}}}{n-1})}

The tolerance interval is [1 ;10].

The intra-series standard deviation is calculated :

\sigma_{\text{short term}}=\sigma_{\text{intra sample}}=1.7321

\sigma_{\text{long term}}=\sigma_{\text{all sample}}=\sqrt{\sum_{}^{}(\frac{^{(x_{i}-\mu)^{2}}}{n-1})}=2.6904

Cp= \frac{\text{tolerance interval}}{6*\text{short term dispersion}}= \frac{\text{tolerance interval}}{6*\sigma_{\text{short term}}= \frac{9}{6*1.7321}=0.87

Pp= \frac{\text{tolerance interval}}{6*\text{long term dispersion}}= \frac{\text{tolerance interval}}{6*\sigma_{\text{long term}}= \frac{9}{6*2.6904}=0.56

### Method 2: By taking two separate samples

- Short-term variability: 50 consecutive parts are taken without adjustment to calculate the short-term variability of the process. Short-term variability is calculated by :

\sigma_{\text{short term}}=\sigma_{\text{all sample}}=\sqrt{\sum_{}^{}(\frac{^{(x_{i}-\mu)^{2}}}{n-1})}

- Long-term variability: 50 parts are sampled over a characteristic period of the process, to take into account multiple sources of process variation such as adjustments, tool changes, material changes, etc. Long-term variability is calculated by :

\sigma_{\text{long term}}=\sigma_{\text{all sample}}=\sqrt{\sum_{}^{}(\frac{^{(x_{i}-\mu)^{2}}}{n-1})}