Learn · updated 2026-10-21
How to calculate Cp and Cpk
Cp and Cpk compare how wide your process runs with how wide the specification allows. Cp ignores where the process is centred; Cpk counts the side closest to a spec limit, so it is the one that predicts out-of-spec parts.
The short answer. With spec limits LSL and USL, process mean μ and within-process standard deviation σ:
Cp = (USL − LSL) / (6σ)Cpk = the smaller of (USL − μ) / (3σ) and (μ − LSL) / (3σ)
Worked example (NIST)
The NIST/SEMATECH e-Handbook uses a process with USL = 20, LSL = 8, mean μ = 16 and σ = 2.
| Step | Calculation | Result |
|---|---|---|
| Spec width over process width | (20 − 8) / (6 × 2) = 12 / 12 | Cp = 1.00 |
| Upper side | (20 − 16) / (3 × 2) = 4 / 6 | CPU = 0.67 |
| Lower side | (16 − 8) / (3 × 2) = 8 / 6 | CPL = 1.33 |
| Take the smaller side | min(0.67, 1.33) | Cpk = 0.67 |
Cp = 1.00 says the process would just fit inside the specification if it were centred. Cpk = 0.67 says it isn't: the mean sits 4 units from the upper limit but 8 from the lower, so the upper tail spills over. Centring the process at 14 would lift Cpk to 1.00 without reducing variation at all.
Where σ comes from: Cpk vs Ppk
The formulas are the same; only σ changes.
- Cp and Cpk use the within-process σ, the short-term variation. For one measurement at a time it is the average moving range divided by 1.128 (MR̄ / d2).
- Pp and Ppk use the overall sample standard deviation s of all the data, which also picks up drift and shifts between periods.
If Ppk is much lower than Cpk, the process moves around over time; fix that before chasing short-term variation.
Example with real data
NIST's ten flow-rate measurements (the same data as our I-MR chart guide), with spec limits 45 to 57:
| Statistic | Value |
|---|---|
| Mean | 50.81 |
| σ within = MR̄ / 1.128 = 1.8778 / 1.128 | 1.6647 |
| s overall | 2.0339 |
| Cp | 1.20 |
| Cpk | 1.16 |
| Pp | 0.98 |
| Ppk | 0.95 |
Ten points is a small sample. The 95% confidence interval for this Cpk runs from about 0.59 to 1.74 (Bissell's approximation), so the honest summary is "somewhere between poor and good: collect more data." LimitLine shows this interval in its verdict.
What is a good Cpk?
| Cpk | What it means |
|---|---|
| below 1.00 | The process spreads past at least one spec limit: expect out-of-spec parts. |
| 1.00 to 1.33 | Fits, with little room. A small shift produces defects. |
| 1.33 | A common minimum target for an established process (LimitLine's default Cpk goal). |
| 1.67 or more | Often asked for on new or safety-critical characteristics. |
Targets are set by your customer or your own quality plan; these are common conventions, not universal rules.
Three checks before you trust a Cpk
- The process is stable. Plot a control chart first. A process with special causes has no single σ, and its Cpk describes nothing in particular.
- The data are roughly normal. Cpk turns σ into a defect rate through the normal curve. Strongly skewed data need a different method.
- There is enough data. With fewer than about 25 to 30 points the confidence interval is wide, as above.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods, §6.1.6 What is Process Capability?
- NIST/SEMATECH e-Handbook, §6.3.2.2 Individuals Control Charts (flow-rate data)
- A. F. Bissell (1990), “How Reliable Is Your Capability Index?”, Applied Statistics 39(3), 331–340 (Cpk confidence interval)