Process capability index

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In process improvement efforts, the process capability index or process capability ratio is a statistical measure of process capability: The ability of a process to produce output within specification limits. The concept of process capability only holds meaning for processes that are in a state of statistical control.

If the upper and lower specifications of the process are USL and LSL, the target process mean is T, the estimated mean of the process is $\hat{\mu}$ , and the estimated variability of the process (expressed as a standard deviation) is $\hat{\sigma}$ , then commonly-accepted process capability indices include:

Index	Description
$\hat{C}_p = \frac{USL - LSL} {6 \times \hat{\sigma}}$	Estimates what the process would be capable of producing if the process could be centered. Assumes process output is approximately normally distributed.
$\hat{C}_{p,lower} = {\hat{\mu} - LSL \over 3 \times \hat{\sigma}}$	Estimates process capability for specifications that consist of a lower limit only (for example, strength). Assumes process output is approximately normally distributed.
$\hat{C}_{p,upper} = {USL - \hat{\mu} \over 3 \times \hat{\sigma}}$	Estimates process capability for specifications that consist of an upper limit only (for example, concentration). Assumes process output is approximately normally distributed.
$\hat{C}_{pk} = \min \Bigg[ {USL - \hat{\mu} \over 3 \times \hat{\sigma}}, { \hat{\mu} - LSL \over 3 \times \hat{\sigma}} \Bigg]$	Estimates what the process is capable of producing if the process target is centered between the specification limits. If the process mean is not centered, $\hat{C}_p$ overestimates process capability. $\hat{C}_{pk} < 0$ if the process mean falls outside of the specification limits. Assumes process output is approximately normally distributed.
$\hat{C}_{pm} = \frac{ \hat{C}_p } { \sqrt{ 1 + \left ( \frac{\hat{\mu} - T} {\hat{\sigma}} \right )^2 } }$	Estimates process capability around a target, T. $\hat{C}_{pm}$ is always greater than zero. Assumes process output is approximately normally distributed.
$\hat{C}_{pkm} = \frac{ \hat{C}_{pk} } { \sqrt{ 1 + \left ( \frac{\hat{\mu} - T} {\hat{\sigma}} \right )^2 } }$	Estimates process capability around a target, T, and accounts for an off-center process mean. Assumes process output is approximately normally distributed.

$\hat{\sigma}$ is estimated using the sample standard deviation.

[edit] Recommended values

Process capability indices are constructed to express more desirable capability with increasingly higher values. Values near or below zero indicate processes operating off target ( $\hat{\mu}$ far from T) or with high variation.

Fixing values for minimum "acceptable" process capability targets is a matter of personal opinion, and what consensus exists varies by industry, facility, and the process under consideration. However, at least one academic expert recommends^[1] the following:

Situation	Recommended minimum process capability for two-sided specifications	Recommended minimum process capability for one-sided specification
Existing process	1.33	1.25
New process	1.50	1.45
Safety or critical parameter for existing process	1.50	1.45
Safety or critical parameter for new process	1.67	1.60
Six Sigma quality process	2.00	2.00

It should be noted though that where a process produces a characteristic with a capability index greater than 2.5, the unnecessary precision may be expensive^[2].

[edit] References

^ Montgomery, Douglas (2004). Introduction to Statistical Quality Control. New York: John Wiley & Sons, Inc., 776. ISBN 9780471656319.
^ Booker, J. M.; Raines, M.; Swift, K. G. (2001). Designing capable and reliable products. Oxford [England]: Butterworth Heinemann. ISBN 0-7506-5076-1.