# Cp process capability

Tolerance-to-spread ratio, estimation assumptions, within versus overall variation and limits on interpreting capability.

Reviewed: 2026-11-01

Tags: inspection, metrology, process-capability, statistics

## Definition and scope

**Cp** compares a two-sided specification width with six process standard deviations:

**Cp = (USL − LSL) / (6σ)**.

USL and LSL are the specification limits. NIST presents Cp for a stable process and a normal population. The index is dimensionless and uses spread alone. [NIST/SEMATECH, process capability](https://itl.nist.gov/div898/handbook/pmc/section1/pmc16.htm)

The worked study uses limits 9.90–10.10 mm and within SD 0.020 mm, giving **Cp = 1.667**. A mean of 10.04 mm leaves only 0.06 mm to the upper limit, so **Cpk = 1.000**. Overall SD 0.025 mm gives **Ppk = 0.800**. These are assumed inputs.

State the SD estimator and data basis. Cp = 1 means the tolerance width equals the estimated six-SD width. Apply the part acceptance rule separately.

Use the product specification limits for capability. Control limits assess process behavior from data.

## Cp does not establish centering

For the same σ estimate, the centering-sensitive index is:

**Cpk = min[(USL − μ)/(3σ), (μ − LSL)/(3σ)]**,

where μ is the process mean, estimated from data in a study. NIST explains that Cp omits off-centering while Cpk uses the smaller distance from mean to specification limit. Thus, reducing variation and adjusting the mean are different improvement actions. [NIST/SEMATECH, assessing capability](https://www.itl.nist.gov/div898/handbook/ppc/section4/ppc46.htm)

A high Cp can describe a narrow distribution positioned close to, or even beyond, a limit. Read Cp alongside centering, the time-ordered data and actual observations. A common display of “Cp ≥ 1.33” is not a universal acceptance requirement; the required threshold and confidence basis come from the customer or study agreement.

Cp needs **two meaningful specification limits**. For a characteristic having only an upper or lower requirement, use an appropriate one-sided assessment rather than inventing a second limit. NIST defines Cpu and Cpl for the respective upper and lower sides. [NIST/SEMATECH, one-sided indices](https://itl.nist.gov/div898/handbook/pmc/section1/pmc16.htm)

## Which standard deviation?

In Minitab's normal-capability convention, **Cp/Cpk use within-subgroup standard deviation**, representing potential capability if between-subgroup shifts and drifts were eliminated. They do not automatically describe every source of variation encountered over an extended production period. [Minitab, within capability](https://support.minitab.com/en-us/minitab/help-and-how-to/quality-and-process-improvement/capability-analysis/how-to/capability-sixpack/normal-capability-sixpack/interpret-the-results/all-statistics-and-graphs/within-capability/)

Minitab calculates **Pp/Ppk using overall standard deviation**, estimated from all observations. A substantial difference between within and overall estimates may indicate instability or additional sources of variation. Report both the convention and estimator: labels alone can obscure how software calculated an index. Overall data also represent only the period and conditions actually sampled, not every future operating condition. [Minitab, process-data statistics](https://support.minitab.com/en-us/minitab/help-and-how-to/quality-and-process-improvement/capability-analysis/how-to/capability-analysis/normal-capability-analysis/interpret-the-results/all-statistics-and-graphs/process-data/)

## Reconciled worked example

**Original illustrative study assumptions:** a diameter has LSL **9.90 mm** and USL **10.10 mm**. A suitable study estimates mean **10.04 mm**, within σ **0.020 mm** and overall standard deviation **0.025 mm**. These values are assumed inputs, not results from a supplied data set. No customer threshold is implied.

| Calculation | Result |
| --- | --- |
| Specification width | 10.10 − 9.90 = **0.20 mm** |
| Cp | 0.20 / (6 × 0.020) = **1.667** |
| Cpu | (10.10 − 10.04) / (3 × 0.020) = **1.000** |
| Cpl | (10.04 − 9.90) / (3 × 0.020) = **2.333** |
| Cpk | min(1.000, 2.333) = **1.000** |
| Pp, using overall spread | 0.20 / (6 × 0.025) = **1.333** |
| Ppk, using overall spread | min(0.06/0.075, 0.14/0.075) = **0.800** |

Cp appears strong because the within spread is narrow. The upper side limits Cpk because the mean is off-centre. The larger overall spread reduces Pp and Ppk further. If the mean alone were moved to **10.00 mm**, with both spreads unchanged, Cp and Pp would remain unchanged; Cpk would become **1.667** and Ppk **1.333**. This is arithmetic under the stated assumptions, not proof that a mean adjustment would leave a real process's variation unchanged.

The within interval μ ±3σ is **9.98–10.10 mm** in the original case. This explains the upper-side constraint. It does not establish the number of defects in a finite sample, and the difference between within and overall spread warrants investigation before making a future-capability claim.

## Collect representative, time-ordered evidence

Establish a stable process before using capability to predict ongoing output. Minitab recommends evaluating stability and collecting enough continuous data over enough time to represent sources of variation. Its guidance suggests at least 100 observations, but this is guidance rather than a universal contractual sample size. A large quantity collected in a single brief run may still omit relevant shifts, material lots or operating conditions. [Minitab, data considerations](https://support.minitab.com/en-us/minitab/help-and-how-to/quality-and-process-improvement/capability-analysis/how-to/capability-analysis/normal-capability-analysis/before-you-start/data-considerations/)

Use **rational subgroups** when appropriate: similar items produced close together under comparable conditions, while preserving representative sampling across time. Minitab notes that subgrouping affects within-variation estimates even though overall standard deviation does not change merely because the same observations are regrouped. Assigning arbitrary subgroup IDs can therefore change the apparent potential capability without improving the process. [Minitab, subgroup selection](https://support.minitab.com/en-us/minitab/help-and-how-to/quality-and-process-improvement/capability-analysis/supporting-topics/data-and-data-assumptions/using-subgroups-to-assess-capability/)

Keep machine, cavity, tool, shift and batch information so separate populations can be recognized. JMP warns that insufficient data or omitted natural variation can underestimate σ, while special causes and instability can inflate it. Decide whether combining streams answers the intended study question; do not pool unlike processes simply to obtain more observations. [JMP, estimating sigma](https://www.jmp.com/en/statistics-knowledge-portal/quality-and-reliability-methods/process-capability)

## Model and measurement limits

The usual six-sigma interpretation relies on the assumed distribution. ASQ emphasizes sensitivity of Cp/Cpk to normality and notes that capability estimates can be accompanied by confidence intervals. A skewed or mixed population can make a normal-model tail estimate misleading. Use an appropriate validated nonnormal analysis or other assessment when required; do not force a bell curve onto the data solely to obtain a familiar index. [ASQ, capability estimates and studies](https://asq.org/quality-resources/process-capability)

Evaluate the measurement process as well. NIST describes measurement variability arising with time, temperature and operators. Such effects can enter the observed data and alter the estimated spread. Preserve sufficient resolution and evaluate repeatability, bias and relevant reproducibility before interpreting small differences between capability estimates. Repeated readings of one item are not equivalent to measurements of independent production items. [NIST/SEMATECH, measurement variability](https://www.itl.nist.gov/div898/handbook/mpc/section1/mpc114.htm)

## Reporting and decisions

Report the characteristic and units, specification revision, sampled process/period, observation and subgroup counts, subgroup rationale, stability and distribution assessment, mean, standard-deviation estimates and methods, indices and any confidence intervals. Preserve actual nonconforming counts separately from model-predicted tails. Include software settings and exclusions so another analyst can reproduce the result.

An index threshold is a study decision rule, not a replacement for individual product acceptance. Avoid releasing a known out-of-tolerance part because the process has a high Cp, or declaring future performance from an unrepresentative snapshot. Use the study to identify the relevant action: reduce variation, correct centering, address instability, improve measurement or revise the analysis model.

## Related reference

[Inspection and quality documents](../../sourcing/documents/index.md)