Cp and Cpk in two lines
Cp is the ratio between the width of the tolerance and six standard deviations of the process: a Cp of 1 means the process takes up exactly the tolerance, a Cp of 2 that it takes up half of it. Cp does not look at where the mean is, so a process shifted entirely out of tolerance can have an excellent Cp.
Cpk measures the distance between the mean and the nearest limit, in units of three standard deviations. It is always less than or equal to Cp, and equals Cp only if the process is perfectly centered. The difference between the two tells you how much capability you recover by centering the process, without touching its variation.
Formulas
Cp = (USL − LSL) ÷ (6σ)
Cpu = (USL − μ) ÷ (3σ)
Cpl = (μ − LSL) ÷ (3σ)
Cpk = min(Cpu, Cpl)
σ (within subgroups) = R̄ ÷ d2 or s̄ ÷ c4
- USL and LSL: upper and lower specification limits, from the drawing or the customer.
- μ: process mean, usually the grand mean of the control chart.
- σ: short-term standard deviation, estimated within subgroups: for subgroups of five parts d2 is 2.326.
- With a single specification, for example only a maximum, you calculate only the side that exists: Cpk equals Cpu or Cpl and Cp is not defined.
Example: a dimension with a ±0.05 tolerance
A hole has a nominal dimension of 10.00 mm with a tolerance of ±0.05 mm: LSL = 9.95 mm and USL = 10.05 mm. The control chart, with subgroups of five parts, is stable with a mean of 10.012 mm and an average range R̄ = 0.0233 mm, so σ = 0.0233 ÷ 2.326 = 0.010 mm.
| Index | Calculation | Result |
|---|---|---|
| Cp | 0.10 ÷ (6 × 0.010) | 1.67 |
| Cpu | (10.05 − 10.012) ÷ 0.030 | 1.27 |
| Cpl | (10.012 − 9.95) ÷ 0.030 | 2.07 |
| Cpk | min(1.27, 2.07) | 1.27 |
| Expected out of tolerance | upper side 3.8σ from the mean | about 72 ppm |
The process has the right variation (Cp 1.67) but is shifted 12 microns toward the upper limit, and the Cpk of 1.27 falls short of the usual target of 1.33. The fix here is not a more precise machine: correcting the tool offset is enough. If Cp had been low instead, centering would not have been enough.
What Cpk value you need
The minimum value is set by the customer or the industry standard. The most common references, with the expected share of out-of-tolerance parts for a centered, normally distributed process, are these.
| Cpk | Distance from the limit | Expected out of tolerance | Typical use |
|---|---|---|---|
| 1.00 | 3σ | about 2,700 ppm | Barely capable process: any shift produces scrap |
| 1.33 | 4σ | about 63 ppm | Usual minimum for processes in production |
| 1.67 | 5σ | about 0.6 ppm | Critical or safety characteristics, new processes |
| 2.00 | 6σ | about 0.002 ppm | Short-term target of Six Sigma programs |
Cpk vs Ppk
Cpk and Ppk have the same formula but use a different σ. Cpk uses short-term variation, within subgroups: it tells you what the process is capable of. Ppk uses the overall standard deviation of all parts, which also includes the shifts from one subgroup to the next: it tells you what the process actually did over the period.
In the example, if the overall standard deviation were 0.013 mm, Pp would be 0.10 ÷ 0.078 = 1.28 and Ppk would be 0.038 ÷ 0.039 = 0.97. A Ppk much lower than Cpk is the sign that the mean shifts over time: tool changes, batches, shifts. That is where to look for the cause, even before looking at the machine.
Common mistakes: unstable process, small sample
- Calculating Cpk on an out-of-control process: if the control chart shows points beyond the limits or drifts, σ and the mean do not describe one process but a mixture of processes. Cpk predicts nothing until the process is stable.
- Using too few parts: with 30 parts and an estimated Cpk of 1.33, the 95% confidence interval runs from about 0.97 to 1.69. With 125 parts it narrows to about 1.15–1.51. A Cpk on a small sample should be stated together with the number of parts.
- Ignoring the shape of the distribution: the link between Cpk and ppm holds for normal data. Flatness, roughness or concentricity, bounded by zero on one side, are often skewed and call for percentile-based methods.
- Forgetting the measurement system: if the gauge has a variation close to that of the process, Cpk measures the gauge. Before Cpk you need a measurement system analysis (gauge R&R).
- Rounding the tolerance or the mean: with tolerances in hundredths, one decimal place less changes Cpk by tenths.
Cpk really pays off when you watch it over time: per die, per shift, per week. A Cpk that falls while every part is still good is the signal to act before the scrap. This is the path of the use case The critical dimension that drifts before the first scrap, where a Python function written by your team calculates the control chart and Cpk per die and per week on Muvia Industrial.
Calculator
Enter the specification limits, the mean and the standard deviation of the process: the calculator returns Cp, Cpu, Cpl, Cpk and the expected out-of-tolerance parts in parts per million. The starting values are those of the example.
Cp and Cpk calculator
It starts from the example of the 10.00 ± 0.05 mm dimension. The standard deviation is the within-subgroup one (R̄/d2).
- Cp
- 1.67
- Cpu (toward USL)
- 1.27
- Cpl (toward LSL)
- 2.07
- Cpk
- 1.27
- Expected out of tolerance (ppm)
- 72.3
Expected ppm hold for a normal distribution and a stable process: with a control chart out of control, Cpk has no meaning.
Frequently asked questions
Cp compares the tolerance with the process variation and does not look at where the mean is. Cpk measures the distance of the mean from the nearest limit, so it also accounts for centering. Cpk is always less than or equal to Cp.
The most common reference is 1.33 for processes in production, equal to four standard deviations between the mean and the nearest limit. For critical characteristics and new processes 1.67 is often required. The binding value is the one set by the customer.
Cpk = the minimum of (USL − mean) ÷ 3σ and (mean − LSL) ÷ 3σ, where σ is the within-subgroup standard deviation. With LSL 9.95, USL 10.05, mean 10.012 and σ 0.010, Cpk is 1.27.
Cpk uses short-term variation, within subgroups; Ppk the overall standard deviation of all the parts in the period. A Ppk much lower than Cpk indicates that the process mean shifts over time.
Yes: with only one limit you calculate only the side that exists, Cpu for a maximum or Cpl for a minimum, and Cpk equals that. Cp, on the other hand, requires both limits.