A process capability index compares the spread of a process with the width of its specification. Cp, Cpk, Pp and Ppk are the four indices used in process capability analysis, and they all answer versions of one question: will this process reliably produce output inside the limits? They differ in two ways. Whether they account for centring. And whether they use the short-term or the long-term spread. Confusing them is how a process with an impressive index can still ship non-conforming units.
Cp, also called the capability ratio, is the specification width divided by the process spread: the tolerance band over six standard deviations.
Cp = (USL − LSL) / 6σ
It measures potential. Whether the process is narrow enough to fit inside the specification if it were perfectly centred. A Cp of 1.0 means the spread exactly fills the tolerance. 1.33 leaves a comfortable margin. Below 1.0 the process is simply too wide, wherever you put it. What Cp cannot see is where the process actually sits.
Cpk adds what Cp leaves out. It measures the distance from the process mean to the nearer specification limit, in units of three standard deviations.
Cpk = min[ (USL − mean) / 3σ , (mean − LSL) / 3σ ]
Cpk falls as the process drifts off-centre, even though the spread has not changed.
When the process is perfectly centred, Cpk equals Cp. As it shifts towards one limit, Cpk drops below Cp. The gap between them is exactly the penalty for being off-centre. A process can have Cp 2.0 and Cpk 0.8. Plenty narrow, but sitting so close to a limit that it fails anyway.
The arithmetic is simple. With limits at 7 and 11 and a standard deviation of 0.5, a process centred at 9 has Cp = (11 − 7) / (6 × 0.5) = 1.33, and Cpk matches at 1.33. Let the mean drift to 10 with the spread unchanged and Cp stays 1.33. But Cpk = (11 − 10) / (3 × 0.5) = 0.67. The process is now failing on the upper side while its Cp still looks acceptable.
The Cp/Cpk pair uses the within-subgroup standard deviation. That is the short-term spread, estimated from variation inside rational subgroups. It represents the process at its best. The Pp/Ppk pair uses the overall standard deviation of all the data. That also captures the drift and shifts that occur between subgroups over time.
Cp/Cpk describe capability: what the process could do if the between-subgroup variation were removed. Pp/Ppk describe performance: what it actually delivered. A large gap between them is itself a finding. It says the process is capable in principle but is not being held stable over time.
Cp, Cpk and their long-term counterparts all measure against the specification limits. None of them refers to the target. Cpm adds that. It penalises the process for sitting away from a nominal target value, not only for approaching a limit. Where hitting a target matters rather than only staying inside the tolerance, Cpm falls as the centre moves off target even while the process stays comfortably within the limits. A drift Cpk alone would not flag.
An index is a point estimate from a finite sample. It carries a confidence interval. A Ppk of 1.30 with an interval reaching down to 1.05 is a different message from one holding above 1.25.
A capability index only means anything if the process was in control when the data were taken. Capability answers “does a stable process meet the specification?” Establish stability on a control chart first. An index computed on an out-of-control process describes nothing repeatable. It also assumes the distribution is roughly normal. Where the data are skewed, see capability for non-normal data before reading the index at face value.
Download the process capability example (.xlsx) — copper plating with a stability check, histogram against specification limits, and Pp/Ppk indices with confidence intervals, ready to open in the Analyse-it trial.
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Quoting Cp alone. Cp ignores centring, so a high Cp can sit over a process that is failing on one side. Report Cpk beside it.
Confusing capability with performance. Cp/Cpk use short-term spread; Pp/Ppk use long-term. Comparing across the pair, or reporting only the more favourable one, misstates what the process actually delivers.
Computing capability on an unstable process. Without control first, the index estimates nothing repeatable. Confirm stability, then measure capability.
Dropping the confidence interval. A capability index from a small sample is uncertain. Report the interval, especially near a pass/fail threshold like 1.33.
Analyse-it reports the indices with their uncertainty, inside Excel:
Every feature from all five editions for 15 days, with no sign-up and no licence key. Capability analysis is in the Quality Control & Improvement and Ultimate editions, from US$ 290 a year. Validated against published reference datasets and thousands of internal test cases. See sigma level and PPM for turning capability into a defect rate, or the process capability reference guide.