How to calculate Cp and Cpk
Process capability compares two widths: the tolerance the customer allows and the process’s natural spread, which is 6 standard deviations (3 on each side of the mean).
Cp = (USL − LSL) ÷ 6σ
Cpk = the smaller of (USL − mean) ÷ 3σ and (mean − LSL) ÷ 3σ
The σ in Cp and Cpk is the within-subgroup (short-term) standard deviation:
- with subgroups of size n: σ = R̄ ÷ d2, where R̄ is the average subgroup range and d2 comes from the table (2.326 for n = 5)
- with individual values: σ = MR̄ ÷ 1.128, where MR̄ is the average moving range between consecutive values
Pp and Ppk use the same formulas with the standard deviation of all measurements together (the ordinary sample standard deviation, long term).
Worked example
The data in the calculator is 20 subgroups of 5 parts of a shaft specified at 10.00 ± 0.05 mm (LSL = 9.95, USL = 10.05).
- Grand mean: 10.00985 mm
- Average subgroup range: R̄ = 0.0283 mm
- Within σ = 0.0283 ÷ 2.326 = 0.01217 mm
- Cp = (10.05 − 9.95) ÷ (6 × 0.01217) = 0.10 ÷ 0.0730 = 1.37
- Upper side: (10.05 − 10.00985) ÷ (3 × 0.01217) = 1.10. Lower side: (10.00985 − 9.95) ÷ (3 × 0.01217) = 1.64
- Cpk = the smaller one = 1.10
The standard deviation of all 100 values is 0.01187 mm, giving Pp = 1.40 and Ppk = 1.13. Ppk is close to Cpk, so the process is stable between subgroups, and the control chart of the same data agrees: no points outside the limits.
In plain words: the process spread fits the tolerance with room to spare (Cp 1.37), but the mean sits 0.01 mm above nominal, close to the upper limit. Just centering the process would take Cpk from 1.10 to about 1.37 without reducing variation at all.
How to read the result
| Cpk | Status | What to do |
|---|---|---|
| 1.33 or more | capable | keep it in control with a control chart |
| 1.00 to 1.33 | marginal | center it and reduce variation; tighter inspection meanwhile |
| below 1.00 | not capable | it already makes bad parts; contain and act now |
Always compare all four indices:
- High Cp, low Cpk: the problem is where the mean sits. Move the process to the center.
- Low Cp: the spread is too wide for the tolerance. Reduce variation (machine, method, material) or review the tolerance with engineering.
- Ppk well below Cpk: the process drifts over time (lot changes, wear, shifts). Look for the special cause on the control chart.
Expected ppm is estimated from the normal curve with the overall standard deviation, and the sigma level uses the convention of adding 1.5 to Z. Both only hold if the data is roughly normal: check the histogram’s shape.
Common mistakes
- Computing Cpk on an unstable process. If the control chart shows points outside the limits, Cpk describes the past and does not predict the future. Stabilize first.
- Using the overall standard deviation for Cpk. That is Ppk. Many ready-made templates mix them up and report a “Cpk” that is not Cpk.
- Too few measurements. With 20 or 30 values, Cpk swings a lot between samples. Use at least 100.
- A poor gauge. If measurement uncertainty is large relative to the tolerance, it adds to the spread and drags Cpk down. See the measurement uncertainty calculator.
- Ignoring the shape of the data. Strongly skewed data (flatness, roughness, times) is not normal, and the estimated ppm will be wrong.