Process capability

A process can be consistent and still produce out of tolerance, if it is off-centre. That is the whole difference between the two indices: Cp looks only at the spread, Cpk takes the offset into account. A high Cp together with a low Cpk is the signal of a setting to correct — good news, it is the easiest case to deal with.

The second pair, Pp and Ppk, is calculated the same way but with the overall standard deviation of all the data, whereas Cp and Cpk use the spread within subgroups. The gap between the two families measures how much the process drifts between samples. When Cpk is good and Ppk poor, the process is capable in the short term and unstable over time.

In French, industrial usage says "capabilité"; the normative vocabulary retains "aptitude". Both designate the same thing, and it is better to know both before an audit.

A capability index answers a decision question: can this process hold this tolerance in series production, or does sorting have to be done? 100 % unit inspection is expensive and lets defects through; knowing in advance that a process will not hold avoids discovering the problem in a customer complaint.

Three moments when the calculation is useful. When qualifying a piece of equipment, before series launch. When defining the monitoring plan: a characteristic that is comfortably capable justifies a lighter inspection, a characteristic that is only just capable requires a control chart. In the dialogue with the customer, where the index serves as a common language — hence its presence in contracts far more often than in a standard.

The index also carries information that the scrap rate does not give: it is predictive. A scrap rate describes what happened; a capability calculated on a stable process says what the next parts will produce. That is why the stability condition is not a formality.

Finally, the gap between the indices directs the action. Low Cp: spread, so work on the process. Good Cp and low Cpk: an offset, so an adjustment. Good Cpk and low Ppk: a drift, so a stability problem. Three diagnoses, three teams.

Dimension to hold: 12.00 mm at ± 0.05 mm. The lower limit is therefore at 11.95, the upper at 12.05, and the tolerance interval is 0.10 mm. Twenty-five subgroups of five parts are sampled over the course of production.

The standard deviation calculated within subgroups comes out at 0.010 mm. Cp = 0.10 / (6 × 0.010) = 1.67. The machine is more than consistent enough for the tolerance requested: on its own, that value would lead to the conclusion that all is well.

The mean, however, stands at 12.022 mm. Cpk retains the least favourable side: on the upper side, (12.05 − 12.022) / (3 × 0.010) = 0.93; on the lower side, (12.022 − 11.95) / (3 × 0.010) = 2.40. Cpk = 0.93. The gap between 1.67 and 0.93 is not spread, it is an offset of 22 microns — nearly a quarter of the tolerance, consumed without producing anything useful.

Translated into parts, the offset is easier to see. The upper limit is 2.8 standard deviations from the mean: in the order of 2,600 parts per million beyond it, in short-term operation. Recentring on 12.000 puts both limits at 5 standard deviations, brings that figure under one part per million and brings Cpk back to Cp, at 1.67. The gain lies in an offset correction.

The second family remains. The standard deviation calculated over all the data is 0.016 mm and not 0.010. Pp = 1.04 and Ppk = 0.028 / (3 × 0.016) = 0.58. The gap between 0.93 and 0.58 measures how much the process drifts between two samples: consistent at a given moment, it wanders over time.

The order of treatment reads directly in those four numbers: first stabilise the drift that widens the gap between 0.010 and 0.016, then recentre, and conclude on capability only afterwards. Doing the opposite means adjusting a machine that will move again the following week.