How the capability indices are calculated
Capability indices compare the specification width with the spread of the process. The potential indices Cp and Pp ignore where the process is centered. The performance indices Cpk and Ppk use the distance from the mean to the nearer limit:
Pp and Ppk use the same expressions with the overall standard deviation \(\sigma_o\) in place of \(\sigma_w\). With a one-sided specification, only CPU (or CPL) exists, and Cpk equals it. Cp and Pp are undefined.
The two standard deviations come from different estimators:
Here \(\bar R\) is the average subgroup range and \(\overline{MR}\) the average moving range of consecutive points. \(d_2\) is the bias-correction constant for the expected range of \(n\) normal values (2.326 for \(n = 5\)). The Taguchi index \(C_{pm} = (USL - LSL)/\bigl(6\sqrt{\sigma_o^2 + (\bar x - T)^2}\bigr)\) also penalizes distance from the target.
Expected nonconforming parts per million come from the normal model, \(\text{PPM}_{
Cpk versus Ppk
The AIAG SPC manual separates inherent variation (within subgroups, collected over a short time) from total variation (every measurement, including drift and shifts between subgroups). Cpk uses within-subgroup variation and describes what the process could do if special causes were eliminated. Ppk uses total variation and describes what the customer actually received during the study.
For a stable process the two are nearly equal. When Ppk is clearly below Cpk, there are shifts between subgroups: tool wear, lot changes, warm-up, or set-up differences. That is a stability problem to fix before quoting capability.
What counts as capable
| Cpk | Distance to nearest limit | Expected PPM (centered, two-sided) | Typical interpretation |
|---|---|---|---|
| 1.00 | 3σ | 2700 | Barely capable; any drift produces scrap |
| 1.33 | 4σ | 63 | Common minimum for ongoing production |
| 1.67 | 5σ | 0.57 | Common requirement for new or safety/critical characteristics |
| 2.00 | 6σ | 0.002 | Six Sigma capability (short-term) |
For initial process studies, the AIAG PPAP manual's acceptance criteria are commonly applied as follows. An index above 1.67 meets the criteria. Values from 1.33 to 1.67 may be acceptable with customer approval. Values below 1.33 do not meet them. Customer-specific requirements always take precedence.
Worked example
The default data are 100 measurements of a laser-scribe line width, in 20 subgroups of 5, with specification 50 ± 6 µm. The mean is 50.93 µm. The average range is \(\bar R = 2.992\) µm, so \(\sigma_w = 2.992/2.326 = 1.286\) µm. The overall standard deviation is 1.293 µm.
The spread alone would support Cpk ≈ 1.56, but the mean sits 0.93 µm above target, so Cpk = 1.31 is just short of 1.33. About 40 PPM is expected above the USL. Re-centering the process at 50.0 µm, for example with a laser-power or focus offset, would raise Cpk to 1.56 and cut the expected nonconforming rate to about 3 PPM, without any reduction in variation. Ppk (1.31) is close to Cpk, which suggests the process was stable during the study. The 95 % interval for Ppk, roughly 1.11–1.50, is a reminder that 100 values pin the index down only to about ±0.2.
Assumptions and limits
- Stability comes first. A capability index predicts future performance only if the process is in statistical control. Plot a control chart before trusting the numbers.
- Normality. PPM and Z.bench assume a normal distribution, so tail estimates in the low-PPM range are only as good as that assumption. Check the histogram and a normal probability plot. For skewed characteristics such as flatness, runout, or burr height, use a transformation or a percentile-based method (ISO 22514-2) instead.
- Sample size. Capability estimates are noisy. Use at least 100 values (for example 25 subgroups of 4–5) and report the confidence interval, not just the point estimate.
- Measurement system. Observed variance is the sum of process and gauge variance, \(\sigma^2_\text{obs} = \sigma^2_\text{process} + \sigma^2_\text{gauge}\). A gauge with a high %GRR deflates Cpk, and poor resolution makes \(\bar R\) unreliable. AIAG MSA guidance treats %GRR under 10 % as acceptable, 10–30 % as conditionally acceptable, and over 30 % as unacceptable, with at least five distinct categories (ndc ≥ 5).
- Rational subgroups. σ within is only meaningful if each subgroup was collected close together in time under the same conditions. Subgroups that span a shift change inflate \(\bar R\), which lowers Cpk toward Ppk and hides the between-subgroup instability that the comparison is meant to reveal.
References
- AIAG, Statistical Process Control (SPC) Reference Manual, 2nd ed. (2005).
- AIAG, Measurement Systems Analysis (MSA) Reference Manual, 4th ed. (2010).
- AIAG, Production Part Approval Process (PPAP), 4th ed. (2006).
- D. C. Montgomery, Introduction to Statistical Quality Control, 8th ed., Wiley.
- A. F. Bissell, “How reliable is your capability index?” Journal of the Royal Statistical Society, Series C (Applied Statistics) 39(3), 331–340 (1990).