Quality managementReading time 12 min

Process capability indices: what separates Cp, Cpk and Ppk

“Our Cpk is 1.5” sounds good. If the process is not yet in statistical control, that number is just the result of an arithmetic exercise.

Cpk

01What separates the four indices

Index	Which variation	Accounts for offset	Answers
Cp	Within-subgroup (short term)	No	Is the spread of the process small enough
Cpk	Within-subgroup (short term)	Yes	How much margin is left once the offset is taken off
Pp	Overall (long term)	No	Spread including between-lot variation
Ppk	Overall (long term)	Yes	What you actually shipped

The C indices use a within-subgroup estimate of sigma; the P indices use the overall sample standard deviation.

Cp only compares the tolerance band with the width of the process distribution and ignores where that distribution sits. Cpk also accounts for how far the mean has moved from the centre of the specification, which is why Cpk is always less than or equal to Cp.

02Indices and defect rates

Cpk	Equivalent sigma level	Theoretical defect rate (two-sided)	Where it is usually required
1.00	3σ	2700 ppm	No longer accepted in most industries
1.33	4σ	63 ppm	The basic threshold in general industry
1.67	5σ	0.57 ppm	Automotive, critical characteristics
2.00	6σ	0.002 ppm	The six sigma target

These figures assume a normal distribution and a centred process. In practice a 1.5σ long-term shift is usually added.

Worth noting: these defect rates are theoretical and assume normally distributed data. Many processes, roundness, runout and parallelism, which have a physical lower bound on one side, are inherently not normal, and applying the figures directly badly understates the defect rate.

03The prerequisite: the process must be in statistical control

This is the step most often skipped. A capability index describes the capability of a stable process. If the process is still drifting, or special-cause variation is present, the Cpk you calculate describes a stretch of history and predicts nothing.

How to confirm it

Plot a control chart (X̄-R or X̄-s) and confirm no points fall outside the control limits.
Look for non-random patterns: seven consecutive points on one side, seven consecutive rising or falling, cyclic behaviour, or data hugging the centre line (which usually means the subgrouping is wrong).
Confirm the sources of variation have been identified and controlled: tool changes, material changes, shift changes, temperature.
Only once all of that holds should you calculate a capability index.

04How many samples to take

With too few samples the confidence interval on the index is very wide. The usual practice is at least 25 subgroups of 5 (125 pieces), spread over enough time to capture between-lot variation.

Sample size n	95% lower confidence limit on a point estimate of Cpk 1.33	What it asks
30	About 1.02	A very wide interval; easy to misjudge
50	About 1.10	Still wide
100	About 1.18	Acceptable
200	About 1.23	More reliable

Approximate values. With too few samples, “Cpk 1.33” and “Cpk 1.0” are statistically indistinguishable.

05Back to the source: where the data comes from

All of this assumes you have the actual measured values, not just a pass or fail verdict. Many shops inspect a great deal and record nothing but a tick. That kind of data supports no capability index at all, and it never shows whether the process is centred or hugging the edge of the tolerance.

The second assumption is that the specification itself is correct. If the tolerances in the inspection sheet were copied off the drawing by hand, the Cpk for a mistyped one is meaningless, and it is usually meaningless in the flattering direction.