Process capability
Compare a stable process’s variation with externally defined lower and upper specification limits.
Concepts and reasoning
Can a stable process meet its requirements?
A process may be predictable yet routinely produce items outside specification. Conversely, a short favorable batch may meet specifications while the process is unstable. Capability compares process variation and location with externally defined requirements. It becomes interpretable only after considering whether the process is sufficiently stable and representative.
Specification limits come from product, engineering, or clinical requirements. They are not calculated from the observed sample to make a process look acceptable. Control limits answer a different question: what variation is expected from the process under a statistical model?
Cp measures how the specification width compares with six within-process standard deviations. Cpk also accounts for centering. Pp and Ppk use overall variation, which can include shifts between subgroups. The terms “within” and “overall” depend on a meaningful sampling plan, not just which formula a software package offers.
Rational subgroups
A rational subgroup groups observations collected under conditions intended to share the same short-term process state. Variation within those groups estimates one component of variability; changes between groups reveal another. Arbitrary grouping can conceal drift or inflate the within estimate.
For example, taking five consecutive measurements every hour can help separate within-hour variation from shifts across hours. Mixing one measurement from each of five different days into a subgroup answers a different question. Describe the sampling process before interpreting the indices.
The NIST capability chapter presents these indices alongside their assumptions. A capability index is a summary under a model, not a universal certificate of quality.
Mathematical foundations
Width and centering
With lower and upper specification limits LSL and USL, process mean μ, and within-process SD σw,
Replace σw with the overall SD to obtain Pp and Ppk. Use estimates from data in practice and recognize that the resulting indices also have sampling uncertainty.
For a centered normal process, Cp = 1 places the limits three SDs from the mean. Under that exact stable normal model, about 0.27% of individual observations lie beyond those limits. This tail calculation is sensitive to nonnormality and shifts; it does not justify applying an assumed “sigma shift” to every process.
When the mean lies outside a specification limit, Cpk can be negative. That is an informative result, not a calculation error to be replaced by zero. One-sided specifications require the relevant one-sided index rather than an invented second limit.
Worked example
Worked example: the same spread, different centering
Suppose specifications are 0 to 12 and the estimated within-process SD is 1.
| Mean | Cp | Upper-side index | Lower-side index | Cpk |
|---|---|---|---|---|
| 6 | 2.000 | 2.000 | 2.000 | 2.000 |
| 9 | 2.000 | 1.000 | 3.000 | 1.000 |
| 13 | 2.000 | −0.333 | 4.333 | −0.333 |
At mean 9, Cp still looks large because the spread has not changed. Cpk identifies that the nearer upper limit is only three SDs away. At mean 13, the process average is already outside the allowed range.
Now return to mean 6 but suppose the overall SD is 2 because the process shifts between batches. Cp and Cpk remain 2 using the within estimate, while Pp and Ppk equal 12/(6 × 2) = 1. The gap is a reason to examine the time structure and subgrouping.
Under a centered normal model with SD 2, specifications at 0 and 12 are ±3 SDs from the mean. With SD 1 they are ±6 SDs. Those imply very different tail risks. But if the process is drifting, neither a single normal tail calculation nor a pooled index may adequately describe future production.
A small sample may estimate SD poorly, causing all these ratios to vary considerably. Reporting three decimal places should never be interpreted as having three-decimal scientific precision.
Interpreting the results
Separate process width from centering
Cp compares the specification width with six within-subgroup standard deviations; Cpk also accounts for how far the mean lies from the nearer specification. A large gap between Cp and Cpk indicates that centering is important. A negative Cpk means the mean itself lies beyond a specification boundary.
Pp and Ppk use overall variation, which can include between-subgroup shifts. Differences between within and overall indices may reveal changing conditions rather than a contradiction in the formulas. They should prompt investigation of process stability and subgroup construction.
Observed out-of-spec proportion is the fraction of supplied measurements outside the limits. A normal-model prediction extrapolates tail probabilities from the estimated mean and spread. Those quantities need not agree, especially in small samples or nonnormal data. A capability index is not the probability a future item will meet specification.
Assumptions and limitations
Specifications and control limits answer different questions
Specification limits are externally defined requirements; control limits describe expected process variation. Estimating specifications from the observed data removes the independent standard against which capability is assessed. A stable process can still be incapable, while an unstable process can temporarily produce acceptable items.
Within-subgroup variation is meaningful only when subgroups represent observations collected under suitably similar conditions. Pooling arbitrary batches or mixing different products can create a misleading denominator. Overall variation needs observations representative of future production conditions.
Normal-tail predictions assume a relevant normal distribution and a stable process. Skewness, mixtures, autocorrelation, measurement error, or changing means can invalidate the translation from indices to defect probabilities. NIST's capability reference distinguishes spread and centering in these indices.
Diagnostics and sensitivity analysis
Establish stability and measurement quality first
Review measurements in time order and use an appropriate control chart before summarizing capability. Investigate shifts between subgroups and whether the subgroup definition isolates short-term variation as intended. A single pooled histogram can conceal a drifting process.
Inspect distribution shape and the measurement system's resolution and repeatability. If rounding or instrument error dominates the within-subgroup spread, an apparently favorable index can be misleading. Compare observed defects with normal-model predictions and explain major differences.
Assess sensitivity to representative operating periods and subgroup definitions, using justified exclusions rather than removing unfavorable values. Report the specification source, sample sizes, within and overall variation estimators, and process conditions. Do not generalize a short stable trial to future production without considering the conditions it omitted.
References and other material
Read the methodological references and supporting tutorials below. The examples in this lesson are original teaching calculations, not findings from these sources.
- NIST handbook: process capability — Capability indices, assumptions and specification limits.