# statistics simple random sampling

## Process Capability

Process capability can be defined as a measurable property of a process relative to its specification. It is expressed as a process capability index ${C_p}$. The process capability index is used to check the variability of the output generated by the process and to compare the variablity with the product tolerance. ${C_p}$ is governed by following formula:

## Formula

${ C_p = min[\frac{USL – \mu}{3 \times \sigma}, \frac{\mu – LSL}{3 \times \sigma}] }$

Where −

• ${USL}$ = Upper Specification Limit.
• ${LSL}$ = Lower Specification Limit.
• ${\mu}$ = estimated mean of the process.
• ${\sigma}$ = estimated variability of the process, standard deviation.

Higher the value of process capability index ${C_p}$, better is the process.

## Example

Consider the case of a car and its parking garage. garage size states the specification limits and car defines the process output. Here process capability will tell the relatonship between car size, garage size and how far from middle of the garage you can parked the car. If car size is litter smaller than garage size then you can easily fit your car into it. If car size is very small compared to garage size then it can fit from any distance from center. In term of process of control, such process with little variation, allows to park car easily in garage and meets the customer’s requirement. Let’s see the above stated example in terms of process capability index ${C_p}$.

• ${C_p = \frac{1}{2}}$ – garage size is smaller than car and can not accomodate your car.
• ${C_p = 1}$ – garage size is just sufficient for car and can accomodate your car only.
• ${C_p = 2}$ – garage size is two times than your car and can accomodate two cars at a time.
• ${C_p = 3}$ – garage size is three times than your car and can accomodate three cars at a time.

## Process Performance

Process performance works to check the conformance of the sample generated using the process. It is expressed as a process performance index ${P_p}$. It checks whether it is meeting customer requirement or not. It varies from Process Capability in the fact that Process Performance is applicable to a particular batch of material. Sampling method may need to be quite substancial to support of the variation in the batch. Process Performance is only to be used when a process control cannot be evaluated. ${P_p}$ is governed by following formula:

## Formula

${ P_p = \frac{USL – LSL}{6 \times \sigma} }$

Where −

• ${USL}$ = Upper Specification Limit.
• ${LSL}$ = Lower Specification Limit.
• ${\sigma}$ = estimated variability of the process, standard deviation.

Higher the value of process performance index ${P_p}$, better is the process.

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