Process Capability

Process Capability Index Cpk

Measures how well a process fits within specification limits, accounting for centering.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Quality Tools > Capability Analysis > Normal

Description

Cpk is the minimum of CPU and CPL. It quantifies the capability of a process relative to both upper and lower specification limits, penalizing off-center processes. A Cpk ≥ 1.33 is generally considered capable. Unlike Cp, which only measures spread, Cpk also accounts for how well the process is centered between the specification limits.

Formulas

Cpk

C_{p k} = min (\frac{U S L - x ˉ}{3 s}, \frac{x ˉ - L S L}{3 s})

Minimum of upper and lower capability indices

C_{p} = \frac{U S L - L S L}{6 s}

Process capability regardless of centering

CPU

C_{P U} = \frac{U S L - x ˉ}{3 s}

Upper capability index

CPL

C_{P L} = \frac{x ˉ - L S L}{3 s}

Lower capability index

CI(Cp)

C I (C_{p}) = C_{p} \frac{χ _{α /2, n - 1}^{2}}{n - 1}, C_{p} \frac{χ _{1 - α /2, n - 1}^{2}}{n - 1}

Exact confidence interval for Cp based on the chi-squared distribution (α = 1 − confidence level)

CI(Cpk)

C I (C_{p k}) = C_{p k} \pm z_{1 - α /2} \cdot \frac{1}{9 n} + \frac{C _{p k}^{2}}{2 ( n - 1 )}

Approximate confidence interval for Cpk per Kushler & Hurley (normal approximation)

χ² approx.

χ_{p, ν}^{2} \approx ν (1 - \frac{2}{9 ν} + z_{p} \frac{2}{9 ν})^{3}

Wilson-Hilferty approximation of the chi-squared quantile function (z_p = normal quantile)

Assumptions

Data is approximately normally distributed
Process is in statistical control
Measurements are independent

Limitations

Not suitable for non-normal distributions without transformation
Requires sufficient sample size (n ≥ 30 recommended)
Does not account for between-subgroup variation (use Ppk for that)

References

Montgomery, Introduction to Statistical Quality Control, 8th Ed. — Chapter 7
Kushler, R. & Hurley, P. (1992), Confidence Bounds for Capability Indices, Journal of Quality Technology, 24(4), 188–195
Wilson, E.B. & Hilferty, M.M. (1931), The Distribution of Chi-Square, PNAS, 17(12), 684–688