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Once a process is put into a state of statistical control, process capability is described using [[Process capability index|process capability indices]], which are formulaically identical to P<SUB>pk</SUB> (and P<SUB>p</SUB>). The indices are named differently to call attention to whether the process under study is believed to be in control or not.
==Example==
Consider a quality characteristic with target of 100.00 [[Micrometre|μm]] and upper and lower specification limits of 106.00 μm and 94.00 μm respectively. If, after carefully monitoring the process for a while, it appears that the process is out of control and producing output unpredictably (as depicted in the [[run chart]] below), we can't meaningfully estimate its mean and standard deviation. In the example below, the process mean appears to drift upward, settle for a while, and then drift downward.
[[File:ProcessPerformanceExample.svg]]
If <MATH>\hat{\mu}</MATH> and <MATH>\hat{\sigma}</MATH> are estimated to be 99.61 μm and 1.84 μm, respectively, then
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! Index
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| <MATH>\hat{P}_p = \frac{USL - LSL} {6 \hat{\sigma}} = \frac{106.00 - 94.00} {6 \times 1.84} = 1.09</MATH>
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| <MATH>\hat{P}_{pk} = \min \Bigg[ {USL - \hat{\mu} \over 3 \hat{\sigma}}, { \hat{\mu} - LSL \over 3 \hat{\sigma}} \Bigg] = \min \Bigg[ {106.00 - 99.61 \over 3 \times 1.84}, { 99.61 - 94 \over 3 \times 1.84} \Bigg] = 1.02</MATH>
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The fact that the process mean appears to be unstable is reflected in the relatively low values for P<SUB>p</SUB> and P<SUB>pk</SUB>. The process is producing a significant number of defectives, and, until the [[Common-cause and special-cause|cause]] of the unstable process mean is identified and eliminated, we really can't meaningfully quantify how this process will perform.
==See also==
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