Process Capability (Cp, Cpk, Pp, Ppk)

Process capability compares the natural spread of a stable process with the specification limits the customer allows. The indices Cp and Cpk use short-term (within-subgroup) variation; Pp and Ppk use overall (long-term) variation, which also includes drift and shifts between subgroups.

Cp and Pp ignore where the process is centered; Cpk and Ppk use the distance from the mean to the nearer limit, so they drop when the process drifts off target.

Process capability: a process distribution compared with its specification limits Two normal curves centered at 10.05 mm between a lower specification limit of 9.70 and an upper specification limit of 10.30. The narrower within-subgroup curve gives Cp 1.25 and Cpk 1.04; the wider overall curve gives Pp 1.00 and Ppk 0.83. The mean is closer to the upper limit, so the upper side sets Cpk. Shaft diameter: spec 10.00 ± 0.30 mm, process mean 10.05 mm LSL 9.70 USL 10.30 target 10.00 mean 10.05 9.5510.45 Within-subgroup σ = 0.08 → Cp 1.25, Cpk 1.04 Overall σ = 0.10 → Pp 1.00, Ppk 0.83
An original worked example. Both curves sit inside the limits, but the mean is 0.05 mm above target, so the upper side is closer and sets Cpk and Ppk. The overall curve is wider because it includes drift between subgroups.

How are Cp, Cpk, Pp, and Ppk calculated?

IndexFormulaSigma usedReads as
Cp(USL − LSL) ÷ 6σwithinShort-term, within subgroupsPotential: could the spread fit if it were centered?
Cpkmin[(USL − x̄) ÷ 3σwithin, (x̄ − LSL) ÷ 3σwithin]Short-termActual short-term performance, including centering
Pp(USL − LSL) ÷ 6σoverallLong-term, all dataPotential over the whole study period
Ppkmin[(USL − x̄) ÷ 3σoverall, (x̄ − LSL) ÷ 3σoverall]Long-termWhat the customer actually received

Within-subgroup sigma is usually estimated from a control chart, for example R̄ ÷ d2 on an X̄–R chart. Overall sigma is the ordinary sample standard deviation of every reading. If the two are close, the process is behaving the same way over time. If σoverall is much larger, something is shifting between subgroups and you should look for special causes before quoting any index.

Worked example: shaft diameter

A shaft is specified at 10.00 ± 0.30 mm, so LSL = 9.70 and USL = 10.30. Twenty-five subgroups of five give a mean of 10.05 mm, a within-subgroup sigma of 0.08 mm, and an overall sigma of 0.10 mm.

  • Cp = 0.60 ÷ (6 × 0.08) = 1.25
  • Cpk = min(0.25 ÷ 0.24, 0.35 ÷ 0.24) = min(1.04, 1.46) = 1.04
  • Pp = 0.60 ÷ (6 × 0.10) = 1.00
  • Ppk = min(0.25 ÷ 0.30, 0.35 ÷ 0.30) = min(0.83, 1.17) = 0.83

The gap between Cp (1.25) and Cpk (1.04) says centering costs capability: moving the mean back to 10.00 would raise Cpk to 1.25 with no change in spread. The gap between Cpk and Ppk says the process also drifts over time. With the overall sigma, a normal model predicts about 6,200 parts per million above the USL and about 230 below the LSL.

How do you read capability indices?

CpkNearest limit isExpected out of spec on that side (normal, no shift)
1.003σ from the meanabout 1,350 ppm
1.334σ from the meanabout 32 ppm
1.675σ from the meanabout 0.3 ppm
2.006σ from the meanabout 0.001 ppm

Many customers ask for Cpk of at least 1.33 on ongoing production and 1.67 for new or safety-critical characteristics, but the real requirement is whatever your customer or control plan says. A Cpk of 2.0 corresponds to the classic “six sigma” spread of six standard deviations from the mean to the nearer limit; the familiar 3.4 DPMO figure adds an assumed 1.5σ long-term shift (see what is 6 sigma).

Before you calculate

  1. Confirm the process is stable. Capability describes a predictable process. Use a control chart first; an in-control process is a prerequisite, not an extra.
  2. Check the measurement system. If gauge error is a large share of the tolerance, the indices describe the gauge as much as the part. Run a measurement system analysis first.
  3. Check the distribution. The ppm figures above assume normal data (see statistical distributions). Skewed characteristics such as flatness, runout, or cycle time need a transformation or a non-normal capability method.
  4. Collect enough data (see sample size). Indices from a few dozen readings have wide confidence intervals. A common starting point is about 100 readings in rational subgroups over enough time to see normal sources of variation.

Capability in DMAIC

Capability is a Measure-phase baseline and a Control-phase proof. In DMAIC you report the baseline Ppk, set a goal, and show the improved Ppk after the fix with the control plan in place. It translates the voice of the process into the language of the voice of the customer.

Frequently asked questions

What is the difference between Cpk and Ppk?

Both use the distance from the mean to the nearer specification limit. Cpk divides by short-term, within-subgroup sigma; Ppk divides by the overall sigma of all the data. Ppk is never higher than Cpk in a real study unless sampling noise makes it so, and a large gap points to drift or special causes.

Can Cpk be higher than Cp?

No. Cpk equals Cp when the process is exactly centered and is lower whenever it is not. A negative Cpk means the mean itself is outside a specification limit.

Is a Cpk of 1.0 good enough?

Usually not for ongoing production. At Cpk 1.0 the nearer limit is only three standard deviations away, so roughly 0.13% of output falls outside it even with no drift. Most quality agreements ask for 1.33 or more.

Can you calculate capability with only one specification limit?

Yes. With only an upper limit, report Cpu (or Ppu); with only a lower limit, report Cpl (or Ppl). Cp and Pp need both limits and are not defined for one-sided specifications.

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