# Process Capability (Cp, Cpk)

Operations · `Process Capability (Cp, Cpk)` · verification: source checked

Process capability indices compare the output of an in-control process to specification limits. Cp compares the specification width to the process width (6σ). Cpk accounts for process centering by taking the minimum of the upper and lower capability ratios.

## Specification

| Property | Value |
|---|---|
| Kind | metric |

## Formula

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

- `USL`: Upper specification limit
- `LSL`: Lower specification limit
- `σ`: Process standard deviation
- `μ`: Process mean

## Safety

- Capability index estimates are valid only if the sample size used is large enough, generally about 50 independent data values.
- The exact variance formula for Cpk is stated for n ≥ 25, 0.75 ≤ Cpk ≤ 4, |c| ≤ 100, and d ≤ 24.
- The approximate confidence interval for Cpk is given for n ≥ 100.

## 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.

## Sources

1. [6.1.6. What is Process Capability?](https://www.itl.nist.gov/div898/handbook/pmc/section1/pmc16.htm) (accessed 2026-10-25)

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