Process performance index: Difference between revisions

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Formally, if the upper and lower [[Specification (technical standard)|specifications]] of the process are USL and LSL, the estimated mean of the process is <MATH>\hat{\mu}</MATH>, and the estimated variability of the process (expressed as a [[standard deviation]]) is <MATH>\hat{\sigma}</MATH>, then the process performance index is defined as:

:<MATH>\hat{P}_{pk} = \min \Bigg[ {USL - \hat{\mu} \over 3 \times \hat{\sigma}}, { \hat{\mu} - LSL \over 3 \times \hat{\sigma}} \Bigg]</MATH>

<MATH>\hat{\sigma}</MATH> is estimated using the [[Unbiased estimation of standard deviation|sample standard deviation]]. P<SUB>pk</SUB> may be negative if the process mean falls outside the specification limits (because the process is producing a large proportion of defective output).

Some specifications may only be one sided (for example, strength). For specifications that only have a lower limit, <MATH>\hat{P}_{p,lower} = {\hat{\mu} - LSL \over 3 \times \hat{\sigma}}</MATH>; for those that only have an upper limit, <MATH>\hat{P}_{p,upper} = {USL - \hat{\mu} \over 3 \times \hat{\sigma}}</MATH>.

Practitioners may also encounter <MATH>\hat{P}_{p} = \frac{USL - LSL} {6 \times \hat{\sigma}}</MATH>, a metric that does not account for process performance not exactly centered between the specification limits, and therefore is interpreted as what the process would be capable of achieving if it could be centered and stabilized.

==Interpretation==