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# How to calculate Cp, Pp et Ppk

As we have just seen, Cp and Pp are calculated based on the same formula. What differentiates them is the time period for which variability is calculated.  There are several methods of calculating Cp and Pp.

### 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:

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

$$\sigma_{long \, term}=\sigma=1.3825$$
$$Pp=\frac{tolerance \, interval}{6\sigma_{long \, term}}=\frac{10}{6*1.3825}=1.45$$
$$Ppk=\frac{Min(\mu – Tol \, min, Tol \, max – \mu)}{3\sigma_{long \, term}}=\frac{9.9333-4}{3*1.3825}=1.43$$