Cpk, PPM and sigma level: theory, conversion guide and examples

Theory: three ways to describe the same tail

Cpk, PPM and sigma level all describe how far the process is from the nearest specification limit. For a normal distribution they can be converted exactly.

Φ is the standard normal cumulative distribution. A centred process with Cpk 1.00 (Z = 3) has 2,700 PPM outside both limits; Cpk 1.33 (Z = 4) has 66 PPM; Cpk 1.67 (Z = 5) has about 0.6 PPM.

The 1.5 sigma shift

Six Sigma practice assumes that over months the process mean shifts by up to 1.5 σ. The “sigma level” therefore refers to short-term capability, and the quoted defect rate is calculated at Z − 1.5. This is why a 6 σ process is said to produce 3.4 DPMO.

Step by step

1. Pick the input: Cpk / Ppk, PPM or sigma level.
2. Pick the specification type: two-sided centred, or one-sided.
3. Set the shift to 0 for automotive Cpk reporting or 1.5 for Six Sigma DPMO.
4. Read the result and check it against the table.

Case study: translating a customer target

A customer requires “no more than 50 PPM” for a two-sided characteristic. The engineer converts 50 PPM: for a centred process that corresponds to Z ≈ 4.06 per side, i.e. Cpk ≈ 1.35. The internal target of Cpk 1.33 is therefore just at the edge; the team sets 1.50 to keep a margin, which corresponds to about 7 PPM.

The same customer’s Six Sigma team asks for “4.5 sigma”. With the 1.5 σ shift this is 3.0 σ long term, about 1,350 PPM: a much weaker requirement. The conversion shows the two targets are not the same and must be clarified in the supplier agreement.

Common mistakes

• Mixing values with and without the 1.5 σ shift.
• Converting Cpk of a non-normal process into PPM.
• Using the two-sided formula for a one-sided characteristic.

Frequently asked questions