Process Capability Calculator (Cp, Cpk, PPM)

Understanding the process capability calculator — concept and principle

Process capability indices compare the natural spread of a stable process with the tolerance a customer or drawing allows. Cp asks whether the spread would fit if the process were perfectly centred; Cpk also accounts for where the mean actually sits. This calculator gives both, the expected fraction of out-of-specification parts in parts per million (PPM) assuming a normal distribution, and the distance to the nearest limit in standard deviations.

The spread would just fit (Cp 1.00), but the shift toward USL drops Cpk to 0.87.

How it is calculated

Cp = (USL − LSL) / (6σ)
Cpk = min(USL − μ, μ − LSL) / (3σ)
PPM = [ (1 − Φ((USL − μ)/σ)) + Φ((LSL − μ)/σ) ] × 10⁶, where Φ is the standard normal cumulative distribution.
Z to nearest limit = min(USL − μ, μ − LSL) / σ = 3 × Cpk; the sigma level adds the conventional 1.5σ shift: Z + 1.5.

μ is the process mean, σ the standard deviation, LSL and USL the lower and upper specification limits, all in the same unit.

The principle

For a normal distribution, about 99.73% of values fall within μ ± 3σ, so 6σ is treated as the process's natural width. Cp is simply tolerance width divided by that natural width. If the mean drifts toward one limit, the near side becomes the weak point, so Cpk uses only the nearer half-tolerance against 3σ. The PPM figure converts each limit into a z-score and reads the tail area beyond it.

Worked example

Inputs: μ = 10.02, σ = 0.05, LSL = 9.85, USL = 10.15.
Cp = (10.15 − 9.85) / (6 × 0.05) = 0.30 / 0.30 = 1.
Distances: USL − μ = 0.13, μ − LSL = 0.17; nearer side 0.13.
Cpk = 0.13 / 0.15 = 0.8667.
z-scores: upper 0.13/0.05 = 2.6, lower −0.17/0.05 = −3.4. Tail areas 0.004661 and 0.000337.
PPM = (0.004661 + 0.000337) × 10⁶ ≈ 4,998; Z to nearest limit = 2.6, so sigma level = 2.6 + 1.5 = 4.1.

Each point is the calculator re-run with only “Process mean” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Process mean	Cpk
2.5	-48.96666667
5.01	-32.26666667
7.51	-15.56666667
10	0.8666666667
12.5	-15.83333333
15	-32.53333333
20	-65.93333333

How to read the result

Where does your result fall?

Cpk below 1 means the process already produces noticeable scrap; 1.00 means the nearest limit is exactly 3σ away. Many customers require Cpk ≥ 1.33 for established processes, and 1.67 is a common expectation for new or critical characteristics. When Cp is clearly larger than Cpk, recentring the process is the quickest gain; when both are low, the variation itself must shrink.

Common mistakes

Computing capability on a process that is not in statistical control.
Mixing within-subgroup σ (for Cp/Cpk) with overall σ (for Pp/Ppk).
Assuming normality for skewed data such as flatness or runout.
Using too few measurements; estimates from 30 parts are very uncertain.

Keep in mind: PPM figures rely on the normal tails, which are rarely accurate far from the mean. The sigma level adds the Six Sigma 1.5σ shift to Z, which assumes σ is short-term; whether that shift applies to your data is a convention, so state which you use.

Frequently asked questions

What is the difference between Cpk and Ppk?

Both use the same formula, but Cpk uses short-term (within-subgroup) σ and Ppk uses the overall long-term σ.

What does Cp = 1 mean in defects?

A perfectly centred process with Cp = 1 produces about 2,700 PPM, the area outside ±3σ.

Can Cpk be negative?

Yes, when the mean lies outside a specification limit; most parts are then out of tolerance.

Sources: D. C. Montgomery, Introduction to Statistical Quality Control, Wiley · AIAG, Statistical Process Control (SPC) Reference Manual

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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