Can Cp be larger than Cpk?

Yes. Cp ignores off-center location, while Cpk uses the nearer specification boundary; they are equal when the mean is at the midpoint under the same spread estimate.

Is Cpk meaningful for a one-sided requirement?

Use the applicable one-sided capability measure and make the requirement explicit rather than inventing a second limit.

Can Cpk change without a change in variation?

Yes. Moving the mean relative to the limits changes it.

AI Glossary / Statistics and probability / Intermediate

Capability index, Cpk

Definition

The capability index Cpk compares the distance from a process mean to its nearer specification limit with three process standard deviations. It accounts for both process spread and off-center operation. A meaningful capability interpretation requires a suitably stable process, appropriate measurement and a defensible distributional analysis; the index is not a guarantee that every future item will conform.

What it means

A process can have a narrow spread yet operate too close to an unacceptable boundary. Cpk captures that problem by looking at the less favorable side. Specifications describe acceptable output, usually from engineering or customer requirements. They are different from control limits, which describe the behavior expected from a process under statistical monitoring. An in-control process can consistently produce unacceptable output, while a short sample of acceptable output does not prove stable capability.

How it works

For two-sided limits, calculate the upper distance USL minus the mean and the lower distance mean minus LSL. Divide each by three times the relevant process standard deviation and take the smaller result. In symbols, Cpk = min[(USL - mu)/(3 sigma), (mu - LSL)/(3 sigma)]. Sample estimates replace the unknown population quantities. In common industrial software, Cpk uses within-subgroup variation, whereas Ppk uses overall variation; the estimation convention must be reported. The familiar translation from these distances to tail probabilities assumes an appropriate normal model. Nonnormal processes require suitable alternative analysis rather than an automatic normal-tail interpretation.

Three practical examples

Manufacturing: assess whether a stable machining process keeps a measured diameter sufficiently far from both tolerance boundaries.
Supplier review: compare capability evidence collected under aligned specifications, measurement methods and sampling conditions, instead of ranking suppliers by unsupported headline numbers.
Process improvement: distinguish a centering problem from excessive variation and verify whether a proposed adjustment improves the limiting side.

Think of it this way

Think of driving a vehicle through a lane. The lane width and the vehicle's sideways variation matter, but so does whether it travels near one edge. Cpk pays attention to the closest edge; a wide lane alone does not make an off-center path comfortable.

Worked example

Suppose a hypothetical stable process has specification limits 9.4 and 10.6 millimeters, mean 10.2 millimeters and within-process standard deviation 0.1 millimeters. The upper index is (10.6 - 10.2)/(3 × 0.1) = 0.4/0.3, approximately 1.333. The lower index is (10.2 - 9.4)/0.3, approximately 2.667. Cpk is therefore 1.333. By comparison, Cp = (10.6 - 9.4)/(6 × 0.1) = 2, showing the spread-based potential if appropriately centered. Moving the mean to 10.0 while preserving the same standard deviation makes both side indices equal to 2. If instead the mean rises to 10.7, Cpk becomes (10.6 - 10.7)/0.3, approximately -0.333: the mean is already beyond the upper specification. The arithmetic does not establish that a real process can be recentered without changing its variation, nor does it estimate the uncertainty of these hypothetical parameters.

Illustrative case study

Fictional case: Meridian Components receives a high Cpk report from a short production run. Reviewing time-ordered data reveals that the sample covered only a freshly adjusted machine and omitted later shifts. The quality team collects representative subgroups over the intended operating conditions, checks the measurement system and examines stability before reporting capability. It separates within-process and overall summaries and includes uncertainty. This fictional case illustrates why sampling context matters; it does not assert that the supplier's true process capability has a particular value.

Common misconceptions and mistakes

Cpk is not the percentage of products meeting specification, and multiplying it by 100 does not create such a percentage. It does not directly measure closeness to an engineering target that differs from the specification midpoint. A negative value is possible when the process mean lies outside a limit. Widening specifications increases the index without improving the process and is unacceptable unless the requirements themselves are legitimately changed. A normal-looking histogram does not prove temporal stability. A high point estimate from limited or selectively collected data should not substitute for uncertainty assessment.

Business use

Businesses can use Cpk as one element of process qualification and improvement, alongside observed nonconformance, control charts and measurement-system evidence. Agree on the standard-deviation estimator, subgroup strategy, operating period and acceptance requirement before comparing reports. Avoid universal pass thresholds detached from product risk and contractual requirements. In AI-monitored production, verify that automated measurements are calibrated and that drift is detected. Capability calculations based on model outputs inherit errors or changes in those outputs, even when the final formula is implemented correctly.

Check your understanding

Q: Why is the upper boundary limiting in the worked example?

Answer: A: The mean is only 0.4 millimeters below the upper limit but 0.8 millimeters above the lower limit. With the same spread denominator, the smaller distance produces the smaller index.

Related terms

Specification limits
Control limits
Rational subgroups
process capability
Cp
Ppk
specification limit
control limit
rational subgroup
process stability and measurement uncertainty

Learn next

plot several normal distributions with the same spread but different means against fixed limits. Then compare within-subgroup and overall variation in time-ordered process data, and examine how confidence intervals change the interpretation of an estimated capability index.

Sources

Sources: NIST Engineering Statistics Handbook, process capability definitions and limitations: https://www.itl.nist.gov/div898/handbook/pmc/section1/pmc16.htm ; Minitab, Cpk versus Ppk estimation conventions: https://blog.minitab.com/en/blog/process-capability-statistics-cpk-vs-ppk ; Minitab, normal capability analysis interpretation: https://support.minitab.com/minitab/help-and-how-to/quality-and-process-improvement/capability-analysis/how-to/capability-analysis/normal-capability-analysis/interpret-the-results/key-results/ .

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