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. In the table below, the upper and lower specification limits of the process are denoted as USL and LSL, the target process mean as T, the estimate of the mean of the process as <MATH>\hat{\mu}</MATH>, and the estimated standard deviation of the process as <MATH>\hat{\sigma}</MATH>.
| Index | Description |
|---|---|
| <MATH>\hat{C}_p = \frac{USL - LSL} {6 \times \hat{\sigma}}</MATH> | Estimates what the process would be capable of producing if the process could be centered. Assumes process output is approximately normally distributed. |
| <MATH>\hat{C}_{p,lower} = {\hat{\mu} - LSL \over 3 \times \hat{\sigma}}</MATH> | Estimates process capability for specifications that consist of a lower limit only (for example, strength). Assumes process output is approximately normally distributed. |
| <MATH>\hat{C}_{p,upper} = {USL - \hat{\mu} \over 3 \times \hat{\sigma}}</MATH> | Estimates process capability for specifications that consist of an upper limit only (for example, concentration). Assumes process output is approximately normally distributed. |
| <MATH>\hat{C}_{pk} = \min \Bigg[ {USL - \hat{\mu} \over 3 \times \hat{\sigma}}, { \hat{\mu} - LSL \over 3 \times \hat{\sigma}} \Bigg]</MATH> | 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, <MATH>\hat{C}_p</MATH> overestimates process capability. <MATH>\hat{C}_{pk} < 0</MATH> if the process mean falls outside of the specification limits. Assumes process output is approximately normally distributed. |
| <MATH>\hat{C}_{pm} = \frac{ \hat{C}_p } { \sqrt{ 1 + \left ( \frac{\hat{\mu} - T} {\hat{\sigma}} \right )^2 } }</MATH> | Estimates process capability around a target, T. <MATH>\hat{C}_{pm}</MATH> is always greater than zero. Assumes process output is approximately normally distributed. |
| <MATH>\hat{C}_{pkm} = \frac{ \hat{C}_{pk} } { \sqrt{ 1 + \left ( \frac{\hat{\mu} - T} {\hat{\sigma}} \right )^2 } }</MATH> | Estimates process capability around a target, T, and accounts for an off-center process mean. Assumes process output is approximately normally distributed. |
<MATH>\hat{\sigma}</MATH> is estimated using the sample standard deviation.
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 (<MATH>\hat{\mu}</MATH> 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 |
References
- ^ Montgomery, Douglas (2004). Introduction to Statistical Quality Control. New York: John Wiley & Sons, Inc., 776. ISBN 9780471656319.
