Cp and Cpk Calculator

The difference between Cp and Cpk

Cp asks whether the process could fit inside the specification limits. It compares the width of the specification with the width of the process, and knows nothing about where the process sits.

Cpk asks whether it does. It measures the distance from the process center to the nearer specification limit, so it falls as the process drifts off center even when its spread has not changed at all.

A process can be capable and off center, which shows as a good Cp and a poor Cpk. The gap between the two numbers is the centering, and closing it is usually an adjustment rather than an improvement project.

Cp and Cpk against Pp and Ppk

Cp and Cpk use the within-subgroup variation — the short-term spread of the process when it is behaving. Pp and Ppk use the total variation across everything measured, which includes drift, shift changes and whatever else happened over the study period.

A large gap between Cpk and Ppk says the process is not stable. The short-term capability is good and something is moving it around. That is a different problem from a process that is simply too wide, and it has a different fix.

Capability on an unstable process means nothing

A capability index describes a distribution. If the process is not in statistical control there is no single distribution to describe, and the number is an average of things that were not the same as each other.

Establish stability with a control chart first. A capability study run on an out-of-control process produces a figure that will not reproduce next month, when predicting next month is the whole point of calculating it.

One-sided tolerances

Where only one specification limit exists, Cp cannot be calculated at all — there is no specification width to compare against. Cpk can, using the one limit that exists, and is sometimes written Cpu or Cpl to make clear which side it refers to.

A customer asking for Cp on a one-sided characteristic is asking for something that does not exist, and the useful reply names Cpu or Cpl rather than inventing a lower limit.

Cpm and Cr

Cpm, the Taguchi capability index, charges the process for distance from the target as well as for spread: Cpm = (USL − LSL) ÷ 6√(s² + (x̄ − T)²). A process centered on its target has Cpm equal to Pp; one that sits off target loses Cpm even when every part is inside the limits. Leave the target blank to use the midpoint of the specification.

Cr, the capability ratio, is the reciprocal of Cp: the fraction of the specification the process spread uses. A Cr of 0.75 means the process uses three quarters of the tolerance, the same statement as Cp = 1.33.

Non-normal data and the Box-Cox transform

The PPM and sigma level are read from the normal curve, so they are only as good as the normal assumption. Flatness, runout, concentricity, leak rate and cycle time are bounded at zero and usually skewed to the right, and a normal curve fitted to them misplaces the tail that the specification sits in.

The Box-Cox option finds the power λ that makes the data look most normal (by maximum likelihood), transforms the readings and both limits by y = (x^λ − 1) ÷ λ (or ln x when λ = 0), and computes the indices on the transformed scale. Check the transformed data on a probability plot before quoting the result. Every reading and the lower limit must be above zero; where they are not, a distribution fit (Weibull, lognormal) or a percentile method is the alternative.

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