Reference
Process sigma
Every figure here is computed, not looked up. The sigma level is Φ⁻¹(1 − DPMO/10⁶) + 1.5, evaluated with Wichura's AS241 inverse normal.
| Sigma level | DPMO | Yield | Long-term Z |
|---|---|---|---|
| 1σ | 691,462 | 30.8538% | -0.500 |
| 2σ | 308,538 | 69.1462% | 0.500 |
| 3σ | 66,807 | 93.3193% | 1.500 |
| 4σ | 6,210 | 99.3790% | 2.500 |
| 5σ | 233 | 99.9767% | 3.500 |
| 6σ | 3.4 | 99.9997% | 4.500 |
The 1.5σ shift is a convention — Motorola's observation that processes drift over time — not a derivation. The long-term Z column is the unshifted value, shown so the difference is never hidden.
Why 3.4 DPMO is six sigma
At six sigma the long-term Z is 4.5, and the area beyond it is Φ(−4.5) = 3.398 × 10⁻⁶ — 3.4 defects per million. A lookup table that stores this as a boundary and compares with > gets it wrong, because 1 − 3.4/10⁶ equals the threshold exactly rather than exceeding it.
Control chart constants, n = 5
- d₂
- 2.326
- A₂
- 0.577
- D₃
- 0.000
- D₄
- 2.114
From a single sourced table (Montgomery, ISQC 7e, Appendix VI) whose values are verified in CI against their defining identities — including D₃ + D₄ = 2, which holds exactly and caught two transcription errors.