Cpk to PPM conversion: formula, table and the sigma link.
Every Cpk value maps to a defect rate in parts per million. A Cpk of 1.33 is roughly 63 PPM; a Cpk of 1.67 is under one. Here is exactly how the conversion works, the table you can read off, and the sigma-shift trap that makes two engineers quote different numbers for the same process.
How Cpk, sigma level and PPM connect
Cpk measures how many standard deviations fit between the process mean and the nearer specification limit, divided by three. Because the distance to the limit is 3 × Cpk standard deviations, and a normal distribution ties any distance-in-sigmas to a tail probability, every Cpk implies a defect rate you can express in PPM.
The clean relationship worth memorising is: Sigma level = 3 × Cpk. A Cpk of 1.00 is a 3-sigma process, 1.33 is 4-sigma, 1.67 is 5-sigma, and 2.00 is the 6-sigma gold standard. For the deeper meaning of the index itself, see Cp vs Cpk explained.
The formula
For a centred process, the defect rate comes straight from the standard normal cumulative function Φ:
- One-sided PPM = (1 − Φ(3 × Cpk)) × 1,000,000
- Two-sided PPM = 2 × (1 − Φ(3 × Cpk)) × 1,000,000
Use the one-sided form when only one limit realistically produces defects (for example a minimum wall thickness). Use the two-sided form for a symmetric tolerance where the process is centred. When the process is off-centre, Cpk already reflects the nearer limit, so the one-sided estimate against that limit is the honest number.
Cpk to PPM conversion table
Read these as centred, long-term values with no distributional shift:
| Cpk | Sigma level | PPM (two-sided) | Yield |
|---|---|---|---|
| 0.33 | 1.0σ | 317,311 | 68.27% |
| 0.50 | 1.5σ | 133,614 | 86.64% |
| 0.67 | 2.0σ | 45,500 | 95.45% |
| 1.00 | 3.0σ | 2,700 | 99.73% |
| 1.33 | 4.0σ | 63 | 99.9937% |
| 1.50 | 4.5σ | 6.8 | 99.99932% |
| 1.67 | 5.0σ | 0.57 | 99.999943% |
| 2.00 | 6.0σ | 0.002 | 99.9999998% |
This is why automotive safety-critical characteristics usually demand Cpk ≥ 1.67: at roughly half a defect per million, containment cost collapses. A plain Cpk of 1.00 — still "capable" on paper — leaks about 2,700 defects per million, which no OEM will accept on a key characteristic.
The 1.5-sigma shift
Here is where quotes diverge. Classic Six Sigma assumes a process drifts about 1.5 sigma over the long term, so a short-term 6-sigma process is quoted as 3.4 PPM rather than 0.002 PPM. The shifted table looks different:
| Sigma level | PPM with 1.5σ shift |
|---|---|
| 3.0σ | 66,807 |
| 4.0σ | 6,210 |
| 4.5σ | 1,350 |
| 5.0σ | 233 |
| 6.0σ | 3.4 |
Both tables are "correct" — they answer different questions. The centred table converts a measured Cpk today; the shifted table estimates long-term performance. Always state which one you are using. The distinction is really the difference between capability and performance, which is unpacked in Cpk vs Ppk.
Worked example
A turned shaft has a diameter spec of 20.00 +0.00 / −0.05 mm. From 50 pieces you measure a mean of 19.976 mm and a standard deviation of 0.0067 mm.
- USL = 20.000, LSL = 19.950, mean = 19.976, σ = 0.0067.
- Cpk = min[(USL − mean), (mean − LSL)] / (3σ) = min[0.024, 0.026] / 0.0201 = 1.19.
- Sigma level = 3 × 1.19 = 3.58; the nearer limit is the upper one.
- One-sided PPM = (1 − Φ(3.58)) × 1,000,000 ≈ 172 PPM.
So this "capable-looking" shaft is running near 172 rejects per million against its tight upper limit — fine for a general fit, marginal for a critical one. Centre the process (shift the mean toward 19.975) and Cpk climbs while the PPM falls sharply. You can check any of these figures in the DPMO and sigma level calculator, and compute Cpk itself from your data with the Cp/Cpk calculator.
For the defect-rate side of the same coin, the DPMO and sigma level guide shows how to go from counted defects back to a sigma score, and reusable study sheets live in the templates library.
Common mistakes
- Mixing shifted and unshifted numbers. Quoting 3.4 PPM for a 6-sigma process and 0.002 PPM interchangeably. Pick one convention and label it.
- Using two-sided PPM on an off-centre process. Once the mean drifts, almost all defects come from the near limit; double-counting the far tail flatters the number.
- Reporting PPM to three decimals from 30 data points. A tiny sample cannot justify a sub-PPM claim; the confidence interval on σ is enormous.
- Confusing Cpk with Ppk. Cpk uses within-subgroup sigma; Ppk uses overall sigma. They give different PPM for the same data.
Frequently asked questions
What PPM is a Cpk of 1.33?
About 63 PPM for a centred, two-sided process (a 4-sigma level). With the classic 1.5-sigma shift, a 4-sigma process is quoted as about 6,210 PPM.
How do you convert Cpk to sigma level?
Multiply by three: sigma level = 3 × Cpk. A Cpk of 1.67 is a 5-sigma process.
What Cpk equals 3.4 PPM?
A 6-sigma process (short-term Cpk of 2.0) is quoted as 3.4 PPM once the standard 1.5-sigma long-term shift is applied.
Does the Cpk to PPM conversion assume a normal distribution?
Yes. The tail-probability method uses the normal curve. Non-normal data needs a transformation or a distribution-specific method, or the PPM estimate will be wrong.