Process Capability (Cp, Cpk)

Cp = (USL − LSL) / (6σ); Cpk = min[(USL − μ)/(3σ), (μ − LSL)/(3σ)]

Study notes

• Capability index estimates are valid only if the sample size is large enough, generally about 50 independent data values.
• Cpk can be expressed as Cp(1 − k), where k = |m − μ| / ((USL − LSL)/2) and m = (USL + LSL)/2.
• For two-sided specifications, Cpk = min(Cpl, Cpu) and Cp = (Cpu + Cpl)/2, where Cpu = (USL − μ)/(3σ) and Cpl = (μ − LSL)/(3σ).
• Assuming a centered normal distribution, Cp = 1.00 corresponds to 0.27% rejects, Cp = 1.33 to 64 ppm, Cp = 1.66 to 0.6 ppm, and Cp = 2.00 to 2 ppb.
• For non-normal data, remedies include a Box-Cox transformation or nonparametric indices Cnp, Cnpk, and Cnpm based on percentiles.
• The exact variance of the Cpk estimator is given by Zhang et al. (1990) with conditions n ≥ 25, 0.75 ≤ Cpk ≤ 4, |c| ≤ 100, and d ≤ 24.
• An approximate 100(1−α)% confidence interval for Cpk is provided for n ≥ 100.
• Cpm incorporates a target value T: Cpm = (USL − LSL) / (6√(σ² + (μ − T)²)).
• One-sided capability indices are Cpu = (USL − μ)/(3σ) for upper specification and Cpl = (μ − LSL)/(3σ) for lower specification.