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Short answer: Cpk = min[(USL − μ) / 3σ, (μ − LSL) / 3σ], where USL and LSL are the upper and lower specification limits, μ is the process mean and σ is the process standard deviation. It is the distance from the mean to the nearer specification limit, measured in units of three standard deviations. A higher Cpk means the output sits further inside its limits.
This page covers the formula, a fully worked off-centre calculation, the assumptions that must hold before the number means anything, and how capability fits into regulated laboratory and manufacturing quality work. In this context Cpk is the process capability index used in quality control; it has nothing to do with creatine phosphokinase (CPK), a blood enzyme test.
Definition and scope
The NIST/SEMATECH e-Handbook of Statistical Methods describes process capability as comparing “the output of an in-control process to the specification limits by using capability indices.” Cpk is one of those indices. It answers a narrower question than the standard deviation alone: not “how variable is the process?” but “how much of that variability fits inside what the specification allows, given where the process is centred?”
Related indices share the same ingredients. For background on the spread term, see the guide to standard deviation.
The Cpk formula
NIST defines the indices as follows:
- Cpu = (USL − μ) / 3σ, the upper-side capability.
- Cpl = (μ − LSL) / 3σ, the lower-side capability.
- Cpk = min(Cpl, Cpu), whichever side is closer to its limit.
- Cp = (USL − LSL) / 6σ, the spread-only index that ignores centring.
With real data, the sample mean x̄ replaces μ and the sample standard deviation s replaces σ. NIST also gives an equivalent form, Cpk = Cp(1 − k), where k = |m − μ| / ((USL − LSL)/2) and m = (USL + LSL)/2 is the mid-point of the specification. Because 0 ≤ k ≤ 1, Cpk can never exceed Cp, and the two are equal only when the process mean sits exactly at the mid-point.
Worked Cpk calculation
The following example is the one NIST uses. A characteristic has USL = 20 and LSL = 8. A sample gives x̄ = 16 and s = 2.
| Quantity | Calculation | Result |
|---|---|---|
| Cp | (20 − 8) / (6 × 2) | 1.000 |
| Cpu | (20 − 16) / (3 × 2) | 0.667 |
| Cpl | (16 − 8) / (3 × 2) | 1.333 |
| Cpk | min(0.667, 1.333) | 0.667 |
| k | |14 − 16| / 6, with m = 14 | 0.333 |
| Cpk via Cp(1 − k) | 1.000 × (1 − 0.333) | 0.667 |
The point of the example is the gap between Cp and Cpk. Cp = 1.0 suggests the spread just fits the specification, but the mean is shifted toward the upper limit, so the upper side is the constraint. NIST’s own reading is that “we would like to have Cpk at least 1.0, so this is not a good process,” and that the remedy is to reduce variability, to re-centre the process, or both. Here, moving the mean from 16 to 14 would lift Cpk to equal Cp.
Cp versus Cpk
Use Cp to ask whether the process is capable in principle: could its spread fit the specification if it were centred? Use Cpk to ask whether it is capable as it currently runs. A large difference between the two points to a centring problem rather than a variability problem, which points to re-centring rather than reducing variability. NIST’s reference for a centred process links Cp to expected rejects:
| Cp (centred process) | Expected rejects |
|---|---|
| 1.00 | 0.27% |
| 1.33 | 64 ppm |
| 1.66 | 0.6 ppm |
| 2.00 | 2 ppb |
Those figures assume a centred, normally distributed process. An off-centre process with the same Cp will reject more, which is exactly what the lower Cpk signals.
Assumptions to check before trusting a Cpk
A Cpk value is only as good as the data behind it. NIST states that the Cp, Cpk and Cpm statistics “assume that the population of data values is normally distributed.” Work through this checklist first:
- Stability. Capability compares an in-control process with its limits. If the process is drifting or showing special-cause signals, the standard deviation is not a stable property and the index is not meaningful. A control chart is the usual way to check this.
- Normality. Check the distribution of the data; heavily skewed or bounded measurements break the 3σ interpretation.
- Sample size. NIST says the estimates are valid only if the sample is large enough, that n ≥ 25 is needed before its confidence-limit approximations hold, and that in general n ≥ 100 is needed for capability studies.
- Uncertainty. Report an interval, not just a point value. NIST’s approximation is Cpk ± z1−α/2 × √(1/(9n) + Cpk²/(2(n−1))).
NIST does not publish a pass mark such as 1.33 or 1.5; those thresholds come from individual customer or industry requirements rather than from the handbook, so state the source of whatever target you apply.
Where it matters for labs and regulated manufacturing
The FDA’s guidance Process Validation: General Principles and Practices (January 2011) says that in Stage 3, “production data should be collected to evaluate process stability and capability,” and that the quality unit should review this information. It also recommends that “a statistician or person with adequate training in statistical process control techniques” develop the data collection plan and the statistical methods. The guidance lists ASTM E2281, Standard Practice for Process and Measurement Capability Indices, as a useful reference, and it does not prescribe a numeric Cpk target.
In practice that places Cpk inside the ongoing monitoring that follows qualification. See process performance qualification for the stage before it, and continued process verification for the monitoring that keeps capability under review. The wider lifecycle is covered under FDA process validation.
Common misunderstandings
- “Cpk is a pass/fail rule.” It is an index. The FDA text asks for capability to be evaluated, not for a specific value.
- “Cp and Cpk are interchangeable.” They are equal only for a perfectly centred process; otherwise Cpk is lower.
- “A high Cpk proves a good process.” Not if the data are non-normal, too few, or collected while the process was unstable.
- “CPK is the same as the blood test CPK.” The abbreviation is shared; the topics are unrelated.
FAQ
What is the formula for Cpk?
Cpk = min[(USL − μ) / 3σ, (μ − LSL) / 3σ]. In sample terms, replace μ with the sample mean and σ with the sample standard deviation.
How do you do a Cpk calculation by hand?
Compute the mean and standard deviation of your measurements, work out Cpu and Cpl from the two specification limits, and take the smaller of the two. The worked example above shows each step.
What is the difference between Cp and Cpk?
Cp uses only the width of the specification and the spread, so it ignores where the mean sits. Cpk also accounts for centring, so Cpk ≤ Cp always.
How many data points do you need?
NIST gives n ≥ 25 as the minimum for its confidence-limit approximations and n ≥ 100 as the general requirement for capability studies.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods, section 6.1.6 “What is process capability?” itl.nist.gov/div898/handbook/pmc/section1/pmc16.htm, checked 2026-10-29.
- U.S. Food and Drug Administration, Process Validation: General Principles and Practices, January 2011. fda.gov/media/71021/download, checked 2026-10-29.
