Cpk process capability
Centering-sensitive capability, within versus overall variation, one-sided limits, confidence bounds and interpretation.
Tags: inspection, metrology, process-capability, statistics
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Definition and meaning
Cpk compares the distance from the process mean to the nearer specification limit with three process standard deviations:
Cpu = (USL − μ)/(3σ)
Cpl = (μ − LSL)/(3σ)
Cpk = min(Cpu, Cpl).
In the worked normal-model example, a mean of 20.04 mm lies 0.06 mm below the upper limit. Within SD of 0.015 mm gives Cpk = 1.333. Overall SD of 0.024 mm gives Ppk = 0.833. The upper limit controls both.
USL and LSL are specification limits, μ is the process mean, and σ is the standard deviation being assessed. In a sample study, mean and standard deviation are estimated. NIST defines these indices for a normal population and places capability analysis in the context of a stable process. NIST/SEMATECH, process capability
Use consistent units for the limits, mean and SD. Report the study window, measurement process and estimator with the dimensionless index.
Cp compares total tolerance width with six σ and does not consider centering. Cpk adds the mean's position relative to the limits. A gap between Cp and Cpk therefore helps distinguish an off-centre process from a spread problem. It does not establish which machine setting should be changed or that centering can be corrected without affecting variation. NIST/SEMATECH, assessing capability
Interpret the index without losing the requirement
Under the stated formula, Cpk is zero when the mean equals a specification limit and negative when the mean lies outside the allowed interval. Do not silently replace a negative result with zero. Such a replacement discards information about the direction and extent of the mean shift. This follows directly from the signed distances in the formula.
For a genuinely one-sided specification, assess the applicable Cpu or Cpl rather than inventing another limit. A physical boundary, such as a quantity that cannot be negative, is not automatically a lower product requirement. NIST treats upper-only and lower-only capability separately. NIST/SEMATECH, one-sided specifications
Cpk is also not a general target-loss index. If the engineering target differs from the tolerance midpoint, maximizing the smaller distance to the limits need not place the mean at that target. JMP distinguishes Cpm, which explicitly includes deviation from target, from Cp/Cpk. Select the assessment for the design intent rather than changing a target merely to improve a statistic. JMP, capability indices and targets
Within variation versus overall performance
In the convention described by JMP, Cpk uses a within-subgroup estimate of σ, while Ppk uses overall standard deviation. Both also depend on process centering. A large difference warrants understanding the sources of variation; it is not permission to report only the larger number. JMP, Cpk and Ppk
Software terminology and settings need attention. SAS explains that some industry conventions reserve Cpk for subgroup-based estimates and Ppk for the ordinary sample standard deviation, while its procedures can use different estimators. Record the actual method rather than assuming identical index labels guarantee identical calculations. For an in-control process containing only common-cause variation, the estimates address the same underlying σ, although finite samples can differ. SAS/QC, assumptions and terminology
Reconciled worked example
Original illustrative inputs: a diameter has LSL 19.90 mm, USL 20.10 mm, estimated mean 20.04 mm, within σ 0.015 mm, and overall standard deviation 0.024 mm. Assume the required process/model assessments support the calculations. These inputs are invented, not a supplied production data set.
| Calculation | Result |
|---|---|
| Cp | (20.10 − 19.90)/(6 × 0.015) = 2.222 |
| Cpu | (20.10 − 20.04)/(3 × 0.015) = 1.333 |
| Cpl | (20.04 − 19.90)/(3 × 0.015) = 3.111 |
| Cpk | min(1.333, 3.111) = 1.333 |
| Ppk | min(0.06/0.072, 0.14/0.072) = 0.833 |
The upper side controls Cpk. The high Cp shows that within-subgroup spread is small relative to tolerance, while Ppk reflects the larger overall spread. The within mean ±3σ interval is 19.995–20.085 mm; the corresponding overall interval is 19.968–20.112 mm. Neither interval is a guaranteed range containing every future item.
If only the mean changed to 20.00 mm, with the assumed spreads unchanged, Cpk would be 2.222, while Ppk would be 1.389. If the mean instead shifted to 20.12 mm, Cpk would be −0.444, because the upper distance becomes −0.02 mm. These scenarios isolate centering algebra; they do not predict the response of a real adjustment.
A threshold such as 1.33 must come from the study agreement. Comparing a rounded display value with a threshold can also hide a small difference: retain sufficient calculation precision and apply the prescribed rounding and decision rule.
Build an interpretable study
Statistical control matters before forecasting ongoing capability. SAS states that without it, a predictable process distribution cannot be meaningfully characterized by the usual capability indices. A histogram alone discards time order and can conceal changes in the process. Investigate instability rather than treating a favourable fitted distribution as proof of control. SAS/QC, statistical-control assumption
Minitab recommends continuous measurements, adequate data over sufficient time and an appropriate distribution. Its normal-analysis guidance suggests at least 100 observations, but that recommendation is not a universal contractual sample size. A large burst of measurements may still omit relevant material lots, tool life, shifts or environmental conditions. Minitab, data considerations
Use rational subgroups when possible: comparable items produced close together under common conditions, with independent observations. Minitab explains that within-subgroup variation can miss between-batch variation when each subgroup represents a different batch. Define the process being claimed and retain machine, cavity, batch and time identifiers so the chosen analysis represents it. Minitab, subgrouping and variation
Distribution, measurement and uncertainty
ASQ emphasizes that ordinary Cp/Cpk estimates are sensitive to normality. For a stable nonnormal population, a justified transformation or alternative distribution may be needed. A mixture of different machines or populations also needs process investigation; fitting a curve does not explain why the mixture exists. Report the model and assess whether it supports the intended tail interpretation. ASQ, practical capability concerns
Measurement variation affects the data entering the study. NIST distinguishes short-term repeatability from changes across days, environments and handling. Evaluate the measurement process at conditions relevant to production; repeated readings of one part do not replace sampling independent production parts. A calibrated instrument alone does not establish negligible measurement variation. NIST/SEMATECH, measurement variability
Cpk is a point estimate. Minitab describes confidence intervals and one-sided confidence bounds for capability indices. Specify whether a threshold applies to the estimate or to a lower confidence bound, and retain the confidence level and calculation method. Those decision rules can produce different outcomes from the same measurements. Minitab, capability confidence bounds
As a bounded published example, JMP's Clips2 analysis reports Cpk 0.966 with an interval 0.805–1.128, despite 104 observations. That example illustrates sampling uncertainty, not a prescribed interval for other studies. JMP 19.2 example, published January 13, 2026
Reporting and acceptance boundaries
Retain specification revision and units, process scope and period, sample/subgroup counts and rationale, stability/model assessment, measurement-system basis, mean, sigma estimators, Cpu/Cpl/Cpk, corresponding overall indices and confidence bounds. Report observed nonconforming counts separately from model-predicted rates. Cpk alone does not uniquely identify the two-tail defect probability; both sides and the distribution matter.
Keep individual product acceptance separate from process-study acceptance. A high Cpk does not authorize release of a known out-of-tolerance item, and zero observed defects in a sample does not establish unlimited future capability. Use the evidence to distinguish centering, variation, instability and measurement problems before selecting corrective action.