Process Capability Cp, Cpk, Pp, Ppk & Specification Fit

The formula

This is the expression the tool evaluates. Every term is named underneath, with the unit it must be supplied in.

How the result is derived

Step by step, from the values you type to the figure on screen.

1. The 8 inputs are read from the form on every keystroke: Upper Specification Limit, Lower Specification Limit, Grand Mean of All Subgroups, Mean Subgroup Range (Rbar), Subgroup Size, Number of Subgroups, Overall Standard Deviation and Confidence Level.
2. Each value is checked against the accepted range in the input table below. A value outside its range stops the calculation rather than producing a misleading figure — the results blank out and a message appears.
3. The validated values are substituted into the expression above, which resolves Cpk, Within-Subgroup together with every supporting figure in one pass — no value is carried over from a previous entry.
4. The supporting outputs — Ppk, Overall Performance, Capability Gap (Cpk - Ppk), Cp, Potential Capability, Pp, Overall Potential, Within-Subgroup Sigma (Rbar / d2), Stability Ratio (overall / within), Lower Confidence Bound on Cpk and Expected Nonconforming, Long Term — come from the same pass, so they always describe the same case as the headline figure.
5. Results are rounded for display only. The full-precision value is used throughout the chain, so reading a rounded intermediate figure back into the tool by hand can shift the last digit.

What each input means

Where to read each value on the floor, the unit it must be in, and the range the tool accepts.

What the tool returns

The headline figure and every supporting value it is built from.

Worked example

Substituting

Every one of the four indices tells a different story about the same 125 packages. Cp says the tolerance is very nearly wide enough. Cpk says the process is off centre. Ppk says one package in thirty is out of specification. The mill that reports Cpk 0.85 has not lied, but it has described its best hour rather than its month, and the customer measures the month.

How to use it

1. Work through the input groups in order — Specification & Centre, Within-Subgroup Spread and Overall Spread & Confidence. The defaults are a realistic case, so you can change one value at a time and watch what moves.
2. There is no calculate button. Every figure recalculates as you type or drag, which is what makes this usable for a what-if sweep rather than a single answer.
3. Read Cpk, Within-Subgroup in the dark results panel — that is the headline figure, expressed in Cpk.
4. Check the supporting rows underneath (Ppk, Overall Performance, Capability Gap (Cpk - Ppk), Cp, Potential Capability, Pp, Overall Potential, Within-Subgroup Sigma (Rbar / d2), Stability Ratio (overall / within), Lower Confidence Bound on Cpk and Expected Nonconforming, Long Term) before acting on the headline — they are where an implausible input usually shows itself first.
5. Reset to defaults returns every field to the reference case, which is the quickest way to check whether a surprising result came from the tool or from an input you had changed earlier.

Where this is used

• Process planning — establishing Cpk, Within-Subgroup before a trial is booked, so machine time and material in Quality Systems, Traceability, Utilities & Factory Decisions are committed against a calculated figure rather than an estimate.
• Costing and quotation — Cpk, Within-Subgroup is an input to the cost sheet, and quoting from a worked number rather than a remembered one is what keeps a margin intact.
• Troubleshooting — when the floor result drifts from plan, entering the measured values (starting with Upper Specification Limit) shows how much of the gap in Cpk, Within-Subgroup each variable explains.
• Teaching and study — the accepted ranges bracket normal Quality Systems, Traceability, Utilities & Factory Decisions practice, so moving one variable at a time shows the shape of the relationship rather than a single answer.

Reading the result

Typical bands and what each one is telling you.

Assumptions and limits

• The within-subgroup sigma is estimated by the range method, Rbar / d2, which is the AIAG default and is accurate for subgroup sizes up to about ten; beyond that the pooled standard deviation or sbar / c4 is the better estimator and this tool will read slightly low. The overall sigma is taken as the sample standard deviation of the individual readings without a c4 bias correction, matching AIAG practice; the correction is under half a percent for a hundred readings and can be ignored at these sample sizes. Everything here assumes the characteristic is measured on a continuous scale and is at least roughly normal - capability indices computed on a skewed characteristic such as neps per gram or a bounded one such as shade difference are not comparable with those computed on a symmetric one, and ISO 22514-2 describes the non-normal distribution families to use instead. The indices say nothing about measurement error: if the gauge repeatability and reproducibility consumes a meaningful share of the tolerance, part of the spread being penalised here belongs to the test method rather than the process, and a measurement systems analysis should be run before any capability improvement project is launched. Finally, capability is only interpretable on a process in statistical control; a large capability gap is direct evidence that this precondition has failed, and the correct response is to find and remove the special cause rather than to report the index.
• Every input is bounded to the range normal practice occupies (Upper Specification Limit -1000000 to 1000000 units, Lower Specification Limit -1000000 to 1000000 units and Grand Mean of All Subgroups -1000000 to 1000000 units, and so on for the rest). Those bounds are guard rails against typing errors, not a claim that the formula fails one unit outside them.
• The calculation is deterministic: the same inputs always give the same result. It carries no allowance for machine condition, operator skill, ambient conditions or lot-to-lot material variation unless an input above explicitly represents one.
• Nothing is sent anywhere. The maths runs in your browser, so the numbers you type never leave the page.

Standards and further reading

• ISO 22514-1:2014 - Statistical methods in process management, Capability and performance, Part 1: General principles and concepts.
• ISO 22514-2:2017 - Part 2: Process capability and performance of time-dependent process models, which formalises the within-subgroup versus overall distinction used here.
• AIAG Statistical Process Control (SPC) Reference Manual, 2nd edition, for Cp and Cpk from Rbar / d2, Pp and Ppk from the total standard deviation, and the d2 table in Appendix E.
• ASTM E2587 - Standard Practice for Use of Control Charts in Statistical Process Control, for subgroup construction and the range-to-sigma factors.

Questions people ask