# Cpk, Cp, Cr

Potential capability (Cp). This is the simplest and most straightforward indicator of process capability. It is defined as the ratio of the specification range to the process range; using ± 3 sigma limits we can express this index as:

Cp = (USL-LSL)/(6*Sigma)

Put into words, this ratio expresses the proportion of the range of the normal curve that falls within the engineering specification limits (provided that the mean is on target, that is, that the process is centered).

Non-centering correction (k). We can correct Cp for the effects of non-centering. Specifically, we can compute:

k=abs(D - Mean)/(1/2*(USL - LSL))

Where D = (USL+LSL)/2. This correction factor expresses the non-centering (target specification minus mean) relative to the specification range.

Demonstrated excellence (Cpk). Finally, we can adjust Cp for the effect of non-centering by computing:

Cpk = (1-k)*Cp

If the process is perfectly centered, then k is equal to zero, and Cpk is equal to Cp. However, as the process drifts from the target specification, k increases and Cpk becomes smaller than Cp.

Capability ratio (Cr). This index is equivalent to Cp; specifically, it is computed as 1/Cp (the inverse of Cp).

Estimate of sigma. When the data set consists of multiple samples, such as data collected for the quality control chart, then one can compute two different indices of variability in the data. One is the regular standard deviation for all observations, ignoring the fact that the data consist of multiple samples; the other is to estimate the process's inherent variation from the within-sample variability. When the total process variability is used in the standard capability computations, the resulting indices are usually referred to as process performance indices (as they describe the actual performance of the process; common indices are Pp, Pr, and Ppk), while indices computed from the inherent variation (within-sample sigma) are referred to as capability indices (since they describe the inherent capability of the process; common indices are Cp, Cr, and Cpk).