What Pp and Ppk are
Pp and Ppk are the process performance indices: they measure what the process actually delivered against the tolerance, using the overall standard deviation of all parts, including variation between subgroups and drift of the mean. Cp and Cpk use only within-subgroup variation (short term) and show what the process would do if it were stable.
The difference is just one thing: sigma. When Cpk and Ppk are close, the process is stable. When Ppk sits well below Cpk, there is between-subgroup variation that the short-term view hides. For the short-term indices, see Cp and Cpk.
Visual map of Pp and Ppk
The map compares the two index families and carries the example of 25 parts in 5 subgroups, with Cpk = 1.94 and Ppk = 0.82.
Pp and Ppk formulas, and how they differ from Cp and Cpk
Pp = (USL − LSL) ÷ (6 · s) Ppk = min[(USL − x̿), (x̿ − LSL)] ÷ (3 · s)
Here s is the sample standard deviation of all readings (divisor n − 1). In Cp and Cpk, sigma is estimated within subgroups:
σshort term = R̄ ÷ d2 (d2 = 2.326 for subgroups of 5)
- Pp compares the tolerance with the total spread, ignoring centering. Ppk includes the off-center mean.
- Cp and Cpk use short-term sigma: the potential of the process under control.
- Pp and Ppk use overall sigma: actual performance over the study period.
- Ppk ≤ Pp and Cpk ≤ Cp always hold. With a stable, centered process, all four are close.
Worked example: 25 parts, tolerance 9.90 to 10.40
Illustrative example: a diameter with LSL = 9.90 mm and USL = 10.40 mm. Five subgroups of 5 consecutive parts were measured at regular intervals.
| Subgroup | x1 | x2 | x3 | x4 | x5 | Mean | Range |
|---|---|---|---|---|---|---|---|
| 1 | 10.02 | 9.98 | 10.05 | 10.00 | 9.96 | 10.002 | 0.09 |
| 2 | 10.04 | 10.08 | 10.01 | 10.10 | 10.03 | 10.052 | 0.09 |
| 3 | 10.12 | 10.07 | 10.15 | 10.09 | 10.11 | 10.108 | 0.08 |
| 4 | 10.16 | 10.21 | 10.13 | 10.18 | 10.20 | 10.176 | 0.08 |
| 5 | 10.22 | 10.17 | 10.25 | 10.19 | 10.24 | 10.214 | 0.08 |
The mean of each subgroup climbs from 10.002 to 10.214: the process drifted. The average range is R̄ = (0.09 + 0.09 + 0.08 + 0.08 + 0.08) ÷ 5 = 0.084. With a grand mean of 10.110, the distances to the limits are 10.40 − 10.110 = 0.290 and 10.110 − 9.90 = 0.210 (the lower side is the critical one).
| Item | Calculation | Result |
|---|---|---|
| Grand mean | (sum of the 25 readings) ÷ 25 | 10.110 |
| Tolerance | USL − LSL = 10.40 − 9.90 | 0.50 |
| Short-term σ | R̄ ÷ d2 = 0.084 ÷ 2.326 | 0.0361 |
| Overall σ | standard deviation (n−1) of the 25 readings | 0.0851 |
| Cp | 0.50 ÷ (6 × 0.0361) | 2.31 |
| Cpk | min(0.290, 0.210) ÷ (3 × 0.0361) | 1.94 |
| Pp | 0.50 ÷ (6 × 0.0851) | 0.98 |
| Ppk | 0.210 ÷ (3 × 0.0851) | 0.82 |
Reading: the Cpk of 1.94 says that, looking only at the spread inside each subgroup, the process is comfortably capable. The Ppk of 0.82 shows that, taken together, the drift used up the margin: assuming a normal distribution, the fraction out of tolerance is about 7,000 ppm (roughly 6,700 below 9.90 and 330 above 10.40). The X-bar and R chart of this process would show the rising trend of the means. The example uses few parts for teaching purposes only.
How to read the gap between Cpk and Ppk
| Situation | What it indicates | Action |
|---|---|---|
| Cpk ≈ Ppk | Stable process; little between-subgroup variation | Keep up SPC |
| Cpk much larger than Ppk | Mean drift or variation between lots, shifts, tools or raw material | Find the cause of the drift; stabilize before any change to limits |
| Low Ppk and high Pp | Spread fits the tolerance, but the mean is off-center | Center the process |
| Low Pp and Ppk | Total variation too large for the tolerance | Reduce variation (see Six Sigma and DMAIC) |
When to use Pp and Ppk
- Initial performance study, before there is enough data to prove stability. This is the typical use in PPAP, which usually asks for about 100 consecutive parts and a minimum Ppk set by the customer (often 1.67 for the initial study). Confirm the requirement in the customer manual.
- Short production runs, where there are not enough subgroups to estimate short-term sigma.
- Comparing actual with potential: the ratio of Ppk to Cpk quantifies what instability costs.
- Auditing old data with no subgroup structure, using only the overall standard deviation.
The indices assume reliable measurement: validate the system first with MSA and Gauge R&R and check the distribution shape in a histogram. To detect small drifts of the mean over time, use CUSUM and EWMA charts.
Common mistakes and limitations
- Calculating Ppk for an unstable process and treating it as a forecast: the index describes only the period measured.
- Using overall sigma in Cpk, or R̄ ÷ d2 in Ppk. The names depend on the sigma used.
- Calculating from too few parts and reporting the index with no confidence interval.
- Ignoring the normality assumption. With a skewed distribution, the ppm estimated from the normal is misleading.
- Comparing indices from different bases (subgroup size, period) without saying so.
- Adjusting the mean after every part to “improve Cpk”, which increases variation.
Frequently asked questions
What is the difference between Pp and Cp?
Pp uses the overall standard deviation of all parts; Cp uses the within-subgroup standard deviation. Both compare the tolerance with 6 sigma.
What is the difference between Cpk and Ppk?
Both account for the centering of the mean, but Ppk uses overall sigma and Cpk uses short-term sigma.
Why is Ppk lower than Cpk?
Because overall sigma includes between-subgroup variation and mean drift, which short-term sigma does not see.
What is a good Ppk?
It depends on the customer and the characteristic. In the PPAP initial study, 1.67 is a common reference value; below 1.33 is usually rejected.
How many parts are needed to calculate Ppk?
The more the better. In PPAP it is usually about 100 consecutive parts; few parts give an unstable index.
Can I calculate Pp and Ppk without subgroups?
Yes. The standard deviation of all readings is enough. Cp and Cpk require subgroups to estimate short-term sigma.
Sources
- AIAG. Statistical Process Control (SPC) Reference Manual. 2nd ed. Southfield: AIAG, 2005.
- AIAG. Production Part Approval Process (PPAP). 4th ed. Southfield: AIAG, 2006.
- MONTGOMERY, D. C. Introduction to Statistical Quality Control. Wiley.
- KOTZ, S.; JOHNSON, N. L. Process Capability Indices. London: Chapman & Hall, 1993.
Keep studying: trusted sources and links
- NIST e-Handbook: Process Capability
- ASQ: Quality Resources
- Wikipedia: Process capability
- Wikipedia: Production part approval process
Save this address: Manufatura Enxuta is your school for lean manufacturing. Come back whenever you need to study, with a complete article, a visual map and trusted sources. Follow one of the learning paths or check the A–Z glossary.
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About the author
Lean Manufacturing & Behavioral Management Specialist
Over 20 years in the automotive and metalworking industries (GM and Dana), Lean Manufacturing practitioner since 2006. SENAI instructor and mentor in Brazil’s Brasil Mais Produtivo program, delivering consulting, training and audits for 50+ companies, combining quality, productivity and people development.
