Which formula should I use?

Use short-term (Z/3) when comparing Cpk to a within-subgroup sigma; use the shifted form only inside a Six Sigma program.

Capability Conversion

Sigma Level to Cpk Conversion

3 min read Last updated July 2026

Sigma level and Cpk measure the same property on different scales. The mapping is simple but depends on whether the 1.5σ shift is included.

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The engineering question this page answers

Which sigma convention is being quoted, and how do I translate it into a Cpk my customer will accept?

Decision logic

Ask whether the sigma level is short-term (Z_ST) or long-term (Z_LT) ↓ Short-term direct: Cpk = Z_ST / 3 ↓ Six Sigma long-term convention: Cpk_LT = (Z_ST − 1.5) / 3 ↓ Confirm the sigma level was derived from a normal, stable, centered process ↓ State the convention explicitly on every deliverable

Formula

Cpk (no shift) = Z / 3 Cpk (1.5σ shift) = (Z − 1.5) / 3 Sigma level Z = NORMSINV(1 − DPMO/1e6)

Examples

Example. Supplier reports "4.5 sigma" on a critical dimension.

Without shift: Cpk = 4.5 / 3 = 1.50
With Six Sigma 1.5σ shift: Cpk_LT = (4.5 − 1.5) / 3 = 1.00
The same "4.5 sigma" claim maps to either PPAP-acceptable or borderline-unacceptable depending on convention.
Engineering conclusion: request the raw σ, spec limits and dataset before accepting.

Conversion conditions

Distribution modelNormal distribution, justified from stable study data.

Specification modelone-sided

Tail conventionnearest-tail

Centering assumptionSigma level means distance to the nearest specification limit.

Sigma basisuser-specified

Sigma-shift conventionnone

Output typeindex-comparison

Required inputs

sigma definition
whether a shift convention applies
study basis

Invertibility. Cpk equals nearest-specification sigma distance divided by three under the stated no-shift model.

Limitations

A marketing Six Sigma convention using a 1.5σ shift is a different definition.
Do not apply a shift automatically.

Worked example and engineering limits

Worked example. Six nearest-specification sigmas with no shift gives Cpk = 6 / 3 = 2.00.

Failure case. A reported Six Sigma level with a 1.5σ shift cannot be treated as six nearest-specification sigmas.

What cannot be inferred. Centering, observed defects, or long-term drift.

Engineering next step. State Z_ST or Z_LT and the shift convention next to the Cpk value.

Common mistakes

Assuming the Six Sigma shift is universal — most PPAP/AIAG references do not include it.
Applying Z / 3 to a DPMO derived from multi-CTQ opportunity counting — the Z is not a process Z.
Interpreting rolled throughput yield (RTY) as if it were a single-parameter capability.
Mixing sigma level from attribute yield with a Cpk that requires variables data.

Engineering insight

"Sigma level" is a communication metric. Cpk is an engineering metric. Never accept a sigma claim without the underlying dataset and the σ definition.

When NOT to use this metric

Attribute or discrete yield metrics where no continuous σ exists.
Non-normal continuous processes.
Multi-CTQ product-level sigma — capability is per-characteristic.

Relationship to other capability metrics

Z vs Cpk: Cpk = Z_nearest_tail / 3; nothing more.
Short-term vs long-term: Differ by the assumed 1.5σ shift — a modelling choice, not a measurement.
Sigma vs Ppk: Long-term sigma corresponds to Ppk, not Cpk.

Recommended actions

Use the free Process Capability Calculator to compute Cp, Cpk, Pp, Ppk, sigma level, and predicted PPM from your measurement data — no sign-up required.

Continue the investigation

Inverse conversion in Cpk → Sigma; connect to defect rate via PPM → Cpk and Cpk → PPM. For deeper interpretation see Process Capability Index.

Verification checklist

Short-term vs long-term sigma stated
1.5σ shift applied or not, explicitly
Sigma derived from variables data, not RTY
Normality validated where continuous data is used

Assumptions and applicability

Process condition: statistical stability is required.
Distribution assumption: approximately normal data.
PPM convention: nearest specification-limit tail.
Sigma-shift convention: no sigma shift applied.
Cpk alone identifies distance to the nearest specification limit; it does not determine total two-sided defects unless centering or both tail distances are known.
Predicted PPM is a model estimate, not a replacement for observed defect data.