The relation in one formula
Cpk is the distance from the mean to the nearest limit in units of 3σ. So the nearest limit sits at z = 3 × Cpk standard deviations. If the data is normal, the expected rate beyond that limit is the normal tail above z, times 1,000,000 for ppm. For a two-sided tolerance you add the tail beyond the other limit, which is small unless the process is close to centered.
Conversion table (normal model, no shift)
| Cpk | z to nearest limit | ppm, nearest side only | ppm, centered two-sided |
|---|---|---|---|
| 0.67 | 2.0 | 22,750 | 45,500 |
| 1.00 | 3.0 | 1,350 | 2,700 |
| 1.33 | 4.0 | 32 | 63 |
| 1.50 | 4.5 | 3.4 | 6.8 |
| 1.67 | 5.0 | 0.29 | 0.57 |
| 2.00 | 6.0 | 0.001 | 0.002 |
Rows use z = 3 × Cpk with the exact fractions (1.33 = 4/3, 1.67 = 5/3). Values rounded.
Example 1: same Cpk, different ppm
Process A is centered with Cp = Cpk = 1.33. Each limit is 4σ away, so the expected rate is about 32 ppm per side, 63 ppm in total. Process B has a much narrower spread, Cp = 2.00, but its mean is shifted so that Cpk = 1.33. The near limit is 4σ away (about 32 ppm) and the far limit is 8σ away (practically zero). Total: about 32 ppm.
Same Cpk, about half the expected ppm, and a very different fix: B needs re-centering, A needs less variation. Always report Cp (or Pp) next to Cpk (or Ppk).
Example 2: from complaint ppm back to Cpk
A customer reports 3 rejected parts in 60,000 delivered: 3 / 60,000 × 1,000,000 = 50 ppm. If you assume every reject is the same characteristic, beyond one limit, from a normal process, the equivalent z is about 3.89 and the equivalent Cpk about 1.30.
Be skeptical about that 1.30. Three parts is a tiny count; the next 60,000 could show 1 or 7 just by chance. The rejects may be three different failure modes. And the customer only counts what their own checks catch. Use the equivalent Cpk to start a conversation, then run a real study on the characteristic in question with the Cpk / Ppk calculator.
Sigma level and the 1.5σ shift
Six Sigma tables quote 3.4 ppm for “6 sigma”. That figure assumes the long-term mean drifts 1.5σ toward a limit, so the real distance is 4.5σ: Cpk 1.5. Without the shift, Cpk 2.0 means about 0.001 ppm per side. If a customer or a scorecard asks for a sigma level, state which convention you used. The calculator converts ppm to sigma level and sigma level back to ppm with the shift applied.
What to write in an 8D
- State the measured ppm with its counts (rejects and quantity delivered), not just the rate.
- Say which characteristic the capability index refers to, which σ (within or overall), and the date range.
- Do not use a pre-complaint PPAP Cpk as proof the process could not have produced the defect. Check it against recent data.
- For D6, compare like with like: ppm before and after over a similar volume, or Ppk before and after on the same characteristic. See D6 verification of effectiveness.
For the difference between the within and overall indices, see Cpk vs Ppk.
FAQ
- What PPM does a Cpk of 1.33 correspond to?
- Under a normal model, Cpk 1.33 (taken as 4/3) puts the nearest limit 4 sigma from the mean: about 32 ppm beyond that limit. If the process is centered on a two-sided tolerance, add the same again on the other side: about 63 ppm in total.
- Is 3.4 ppm six sigma?
- Only with the 1.5 sigma shift convention. 3.4 ppm is the one-sided tail at z = 4.5, which is Cpk 1.5. Without the shift, a six sigma process (Cpk 2.0) gives about 0.001 ppm per side.
- Can I compute Cpk from my complaint PPM?
- You can compute an equivalent z and divide by 3, but treat it as a rough indicator. Field PPM depends on detection, sorting, how the customer counts and which characteristic failed. It is not a capability study.
- Why does my real PPM not match the Cpk prediction?
- The usual causes: the data is not normal (tails are heavier), the study used sigma within while the process shifts over time, the gauge is poor, or defects come from a different characteristic or failure mode than the one studied.