Process capability (Cp, Cpk, Pp, Ppk): theory, step-by-step guide and case study

Theory: what capability really measures

A capability index compares two widths: the width the customer allows (the tolerance from LSL to USL) and the width the process actually uses (its natural spread). The natural spread is defined as six standard deviations, because a normally distributed process puts 99.73 % of its output inside ±3σ.

Cp only looks at the spread and ignores where the process is centred. It tells you what the process could achieve if it were perfectly on target. Cpk measures the distance from the mean to the nearest limit, so it falls when the process moves off centre. If Cp is high but Cpk low, centring the process is the cheapest improvement.

Two different sigmas: Cpk versus Ppk

The pairs Cp/Cpk and Pp/Ppk use the same formulas with a different standard deviation:

• σ within (short term) is estimated from the variation inside rational subgroups: R̄/d₂ for X-bar/R charts, S̄/c₄ for X-bar/S charts or MR̄/1.128 for individuals. It contains only the inherent noise of the process. Cp and Cpk use it.
• σ overall (long term) is the ordinary standard deviation of all values. It also contains every shift and drift between subgroups. Pp and Ppk use it.

For a perfectly stable process both sigmas estimate the same thing and Cpk ≈ Ppk. A clear gap (Ppk well below Cpk) means the process moves between subgroups: tool wear, warm-up, material batches, different shifts. See Cpk vs Ppk for more.

Stability comes first

A capability index is a prediction: it says how the process will behave in the future. That prediction is only valid if the process is stable, meaning only common-cause variation is present. This is why a capability study always starts with a control chart. If the chart shows special causes (points beyond the limits, runs, trends), the indices describe a mixture of different processes and cannot be trusted.

Normality and non-normal data

The link between Cpk and defect rate (Cpk 1.33 ≈ 63 PPM) assumes a normal distribution. Many characteristics are naturally skewed, for example flatness, runout, roughness or concentricity, which are bounded by zero. For those, the ISO 22514-2 percentile method uses the 0.135 % and 99.865 % points of a fitted distribution (lognormal, Weibull, gamma) instead of μ ± 3σ.

How sure are you? Confidence intervals

Cpk is calculated from a sample, so it has statistical uncertainty. With 125 values the 95 % confidence interval of a Cpk of 1.43 is roughly 1.24 to 1.62. With only 30 values the same Cpk would have an interval of roughly 1.04 to 1.82. Always look at the lower bound before you sign off.

Step by step: running a capability study

1. Check the gauge. Before you measure 125 parts, make sure the measurement system is not a large part of the variation. Run a Type 1 study and a Gage R&R.
2. Plan rational subgroups. Take 3 to 5 consecutive parts at a time, so that each subgroup contains only short-term variation, and spread the subgroups over the normal sources of variation: shifts, tool changes, material lots. 25 subgroups of 5 is the classic plan.
3. Enter the data. In the tool, set Values per subgroup to 5 and paste the table from Excel. A single column is split into subgroups automatically. Enter LSL, USL, the target and the required Cpk (1.33 or 1.67).
4. Read the control chart. The X-bar chart shows whether the process mean is stable, the R chart whether the spread is stable. Red points are Nelson rule violations. Click a point to add a note or, once the cause is known and removed, exclude it.
5. Check the histogram and normality. The Anderson-Darling p-value should be above 0.05. If not, select Best fit under Distribution.
6. Read the indices with their intervals. Compare the lower confidence bound with the requirement, and compare Cpk with Ppk.
7. Report. Fill in the report header (part, characteristic, gauge) and download the PDF. Record the characteristic, sample size and frequency in the control plan.

Case study: bearing seat Ø25 on a CNC lathe

A supplier of gearbox shafts must show capability for the bearing seat diameter before series start. The drawing says 25.000 ± 0.100 mm, the customer requires Cpk ≥ 1.33. The air gauge passed a Gage R&R. Over three shifts the quality engineer takes 25 subgroups of 5 consecutive shafts, 125 values in total. (This is the data set behind Load sample in the tool.)

1. Stability

The X-bar/R chart shows a centre line of 25.0177 mm with control limits from 24.9919 to 25.0434 mm. All 25 subgroup means and ranges are inside the limits and no Nelson rule is violated: the process is stable, so the indices may be interpreted.

2. Normality

The Anderson-Darling test gives p = 0.539. Normality is not rejected, so the standard formulas apply.

3. The indices

4. Interpretation

• Cpk 1.43 and Ppk 1.37 meet 1.33: the process is capable.
• Cp 1.74 versus Cpk 1.43: the mean sits 0.018 mm above the nominal. Moving the tool offset by −0.018 mm would raise Cpk to about Cp, 1.74, with no other change.
• Cpk ≈ Ppk: there is almost no drift between subgroups, which matches the stable chart.
• The lower confidence bound of Cpk is 1.24, below 1.33. For a demanding customer the engineer would either centre the process first or measure more parts to narrow the interval.

Common mistakes

• Calculating Cpk on an unstable process. The number looks precise but predicts nothing.
• Subgroups that are not rational. Parts from different cavities or spindles in one subgroup inflate σ within and make Cpk look worse than it is; parts collected hours apart do the opposite.
• Ignoring a skewed distribution. A normal Cpk on flatness or runout data usually underestimates the defect rate at the upper limit.
• Too few parts. 30 values give a confidence interval of about ±0.4 around Cpk.
• Rounded data. If the gauge resolution is coarse (fewer than 10 distinct values), sigma is distorted. Check the resolution in the Type 1 study.
• Confusing Cpk with Cmk. Machine capability (Cmk) is a short run on one machine and uses different limits (often 1.67 or 2.0).

Frequently asked questions