# How to calculate Cp, Pp et Ppk

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

The first method for calculating short term and long term variability is to take several samples at regular intervals.

- Short-term variability: short term variability is calculate by using the intra-series standard deviation calculation for the entirety of the samplings:

\(\sigma_{short \, term} = \sigma_{intra \, series}\)

- Long term variability: we take a sample of 50 parts over a period characteristic of the process to take into account multiple variation sources of the process such as adjustments, tool replacement, material, etc…The long term variability is calculated by:

\(\sigma_{short \, term} = \sqrt{\sum{\frac{(x_i-\mu)^2}{n-1}}}\)

Example: We take the following samples

The tolerance interval is [1;10].

The intra-series standard deviation is calculated:

\(\sigma_{short \, term}=1.7321\)

The standard deviation for all the parts is calculated:

\(\sigma_{long \, term}=standard \, deviation(all \, parts)=2.6904\)

We therefore deduce:

\(Cp=\frac{tolerance \, interval}{6\sigma_{short \, term}}=\frac{9}{6*1.7321}=0.87\)

\(Pp=\frac{tolerance \, interval}{6\sigma_{long \, term}}=\frac{9}{6*1.7321}=0.87\)

\(Ppk=\frac{Min(\mu – Tol \, min, Tol \, max – \mu)}{3\sigma_{long \, term}}=\frac{4.333}{3*2.26904}=0.54\)

### Method 2 : By taking two distinct samplings

The second short term and long term variability calculation method is made by taking two distinct samplings.

- Short term variability: 50 consecutive parts are taken out without adjustment so that the process’ short term variability can be calculated. Short-term variability is calculated by:

\(\sigma_{long \, term}=standard \, deviation(all \, parts)\)

- Long term variability: 50 pieces are taken out distributed over a period characteristic of the process in order to take into account multiple process variation sources such as adjustments, tool replacements, material, etc…The long term variability is calculated by:
- \(\sigma_{long \, term}=standard \, deviation(all \, parts)\)

#### Example : the two following samplings are taken :

The tolerance interval is [1; 10]

### Short-term :

The standard deviation is therefore calculated as:

\(\sigma_{short \, term}=standard \, deviation(all \, parts \, of \, short \, term \, sample)\)

As Ellistat don’t know that this series was taken out over a short time period, we use:

\(\sigma_{short \, term}=\sigma=1.0426\)

Consequently:

\(Cp=\frac{tolerance \, interval}{6\sigma_{short \, term}}=\frac{10}{6*1.0426}=2.17\)

### Long-term :

The standard deviation of the entirety of the parts is therefore calculated:

\(\sigma_{long \, term}=standard \, deviation(all \, parts \, of \, long \, term \, sample)\)

We simply read

\(\sigma_{long \, term}=\sigma=1.3825\)

Consequently:

\(Pp=\frac{tolerance \, interval}{6\sigma_{long \, term}}=\frac{10}{6*1.3825}=1.45\)

\(Ppk=\frac{Min(\mu – Tol \, min, Tol \, max – \mu)}{3\sigma_{long \, term}}=\frac{9.9333-4}{3*1.3825}=1.43\)