Kitstead Labs
Process statistics
Cpk vs Ppk: Formulas, Differences, and Practical Examples
A capability report can show a reassuring Cpk while the underlying sequence changes level. This guide follows one small synthetic dataset from raw measurements to all four capability and performance indices. It explains which variability estimate each index uses and why neither an index nor an in-spec sample replaces a stability review.
Quick answer
Cpk uses a within-process variability estimate; Ppk uses overall sample variability for the chosen dataset. Both compare the mean with the nearer specification limit. State the estimators and sampling window before comparing the numbers, and examine stability before interpreting them as capability.
Capability and performance answer related questions
A specification describes an engineering requirement. Variation describes what the process actually did. Cp and Pp compare the full specification width with six estimated standard deviations; Cpk and Ppk also account for an off-center mean by using the nearer specification limit. All four indices are dimensionless.
The distinction here is the variability estimator. “Within” uses local differences between successive individual measurements. Overall sample standard deviation uses deviations from the mean of the entire selected dataset, so shifts across the sequence contribute. Neither label defines a calendar duration: Ppk is not automatically a long-term study.
Formulas and assumptions
Let LSL and USL be lower and upper specification limits, x̄ the sample mean, and n the number of observations. For this individual-observation method, MRi = |xi − xi−1| and MR̄ averages the n−1 moving ranges. Use measurement order, not sorted magnitude.
s = √[Σ(xᵢ − x̄)² / (n − 1)]
Cpk = min(USL − x̄, x̄ − LSL) / (3σ̂within)
Pp = (USL − LSL) / (6s)
Ppk = min(USL − x̄, x̄ − LSL) / (3s)
The moving-range estimator is appropriate to individual observations with a coherent sampling sequence and sufficiently independent local errors. Rational subgroup methods use other within estimators; do not substitute this formula without checking the sampling plan. These equations can be calculated on unstable or nonnormal data, but their capability and tail-probability interpretation then needs particular caution. Zero estimated variation does not justify an infinite capability claim.
Worked example: a shift hidden by a high Cpk
All values below are synthetic thickness measurements, not customer data. Set LSL = 9.50 µm and USL = 10.50 µm. The short sequence deliberately changes level halfway through, making it useful for demonstrating why a capability number needs a chart. Ten values are not a production qualification study.
| Observation | Thickness (µm) | Moving range (µm) |
|---|---|---|
| 1 | 10.00 | — |
| 2 | 10.04 | 0.04 |
| 3 | 10.02 | 0.02 |
| 4 | 10.06 | 0.04 |
| 5 | 10.04 | 0.02 |
| 6 | 10.20 | 0.16 |
| 7 | 10.24 | 0.04 |
| 8 | 10.22 | 0.02 |
| 9 | 10.26 | 0.04 |
| 10 | 10.24 | 0.02 |
The values sum to 101.32 µm, giving x̄ = 10.132 µm. The nine moving ranges sum to 0.40 µm: MR̄ = 0.40/9 = 0.044444 µm. The sum of squared deviations from the mean is 0.10416 µm².
s = √(0.10416 / 9) = 0.107579 µm
Nearest limit distance = 10.50 − 10.132 = 0.368 µm
| Index | Calculation | Result |
|---|---|---|
| Cp | 1.00 / (6 × 0.039401) | 4.230 |
| Cpk | 0.368 / (3 × 0.039401) | 3.113 |
| Pp | 1.00 / (6 × 0.107579) | 1.549 |
| Ppk | 0.368 / (3 × 0.107579) | 1.140 |
The calculations use unrounded estimates internally. Cpk exceeds Ppk because most adjacent differences are small relative to the overall spread across the two levels. Every observation is within specification, yet the process sequence contains a deliberate shift. A Cpk of 3.113 therefore cannot be read as evidence that this process is stable or qualified.
Interpret the indices together with the sequence
A large Cpk–Ppk gap is a reason to inspect time order, equipment, material and sampling differences. It does not identify a cause. Similar values also do not prove stability: both estimators can be influenced by outliers, measurement resolution or mixed populations. Use the I-MR chart worked example to see control limits for these same measurements.
A process mean outside either specification can yield a negative Cpk or Ppk. That is meaningful evidence of poor centering relative to the stated requirement, not a software error to clamp to zero. A target can be useful for centering decisions, but it is not a substitute for LSL and USL in these formulas.
Common errors in a capability review
- Using a population standard deviation divisor n when the stated overall estimator uses n−1.
- Sorting values before calculating moving ranges, or joining unrelated tools or phases as if they were a single continuous process.
- Converting a capability estimate directly into a guaranteed yield without checking stability, distribution shape and measurement adequacy.
- Treating a generic threshold such as 1.33 as a universal qualification rule. Acceptance depends on the agreed engineering requirement and uncertainty.
When the question is whether experimental conditions differ, use an experimental comparison instead of a capability index. The split-lot analysis guide explains that distinction.
Apply the method in desktop Excel
The released SPC & Capability Analyzer v0.1 computes Cp/Cpk with MR̄/1.128 and Pp/Ppk with sample standard deviation. It supports ordered measurements, I-MR charts and descriptive Tool, Chamber and Lot summaries. Its single-point Individuals control-limit checks do not include advanced run rules or subgroup capability.
Sources and further reading
- NIST: Process capability — index definitions and interpretation assumptions.
- NIST: Individuals control charts — the moving-range variability estimator.