Does the 1.5σ shift apply?

Do not silently apply it. The direct nearest-tail Cpk conversion uses Z = 3 × Cpk with no shift. A 1.5σ shift is a separate Six Sigma reporting convention and must be labeled explicitly.

Capability Conversion

Cpk to PPM Conversion

4 min read Last updated July 2026

PPM at a given Cpk is computed from the normal distribution using z = 3 × Cpk. The result is the predicted single-sided defect rate.

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The engineering question this page answers

What defect rate does my Cpk actually predict — and how far can I trust that prediction?

Decision logic

Confirm normality and stability on the source dataset ↓ Decide the required output: nearest-tail (one-sided) or total two-sided PPM ↓ Nearest tail: p = 1 − Φ(3 × Cpk); PPM = p × 1e6 ↓ Two-sided centered: PPM_total ≈ 2 × p_nearest (only valid when Cp = Cpk) ↓ For non-centered processes compute each tail separately using Cpu and Cpl

Capability study readiness

Typical PPM and sigma equivalent

Typical PPM: Reference (nearest tail, no shift): 1.00 → 1350 · 1.33 → 32 · 1.50 → 3.4 · 1.67 → 0.29 · 2.00 → 0.0009.

Formula

Nearest-tail PPM = (1 − Φ(3 × Cpk)) × 1e6 Two-sided (Cp = Cpk): PPM_total = 2 × (1 − Φ(3 × Cpk)) × 1e6 Non-centered: PPM_total = (1 − Φ(3 × Cpu)) × 1e6 + (1 − Φ(3 × Cpl)) × 1e6

Examples

Example. Machined bore, USL = 10.10, LSL = 9.90, mean = 10.02, σ_within = 0.025.

Cp = (10.10 − 9.90) / (6 × 0.025) = 1.33
Cpu = (10.10 − 10.02) / (3 × 0.025) = 1.07
Cpl = (10.02 − 9.90) / (3 × 0.025) = 1.60
Cpk = 1.07 → nearest-tail PPM ≈ 830
Two-sided total PPM ≈ 830 + 1 ≈ 831 (dominated by upper tail)
Engineering conclusion: variation is fine (Cp = 1.33) but centering loses ~830 PPM at USL — a setpoint/tool-offset problem.

Conversion conditions

Distribution modelNormal distribution, justified from stable study data.

Specification modelcontext-dependent

Tail conventioncontext-dependent

Centering assumptionNearest-tail PPM uses Cpk directly; centered total two-sided PPM additionally assumes equal tail distances.

Sigma basiswithin-subgroup

Sigma-shift conventionnone

Output typepredicted

Required inputs

Cpk
tail convention
within versus overall basis
normality and stability justification

Invertibility. Nearest-tail conversion is invertible under the same normal no-shift model; total two-sided conversion requires centering information.

Limitations

No sigma shift is applied unless explicitly selected.
Predicted PPM must not be labelled as observed defects.

Worked example and engineering limits

Worked example. At Cpk 1.33 with no shift, nearest-tail PPM is about 33; the centered two-sided prediction is about 66.

Failure case. For a non-centred process, Cpk alone cannot determine actual total two-sided PPM without both tail distances or Cp and mean.

What cannot be inferred. Actual scrap, asymmetric tails, or long-term performance from Cpk alone.

Engineering next step. Report tail convention and compare the prediction with observed defects by tail.

Common mistakes

Reporting two-sided PPM from Cpk alone — Cpk carries no information about the far tail unless the process is centered.
Silently injecting the 1.5σ shift, producing PPM 3.4 at "6 sigma" without labeling it as long-term.
Using Ppk instead of Cpk (or vice-versa) without noting which sigma was used.
Comparing predicted PPM against observed defective parts that include measurement error.
Extrapolating to Cpk > 2 where sample sizes are insufficient to validate the tail.

Engineering insight

The tail is where the model is weakest and the customer is most exposed. Treat Cpk-derived PPM as a design-review indicator, not as a shipping quality metric.

When NOT to use this metric

Any non-normal distribution — the tail probability from Φ is meaningless.
Highly skewed processes near a single spec limit (concentricity, flash, hardness) — use non-normal methods.
Very short-run or DoE data — predicted PPM has no long-term meaning.

Relationship to other capability metrics

Cpk vs PPM: A monotonic mapping only under normality; the mapping breaks whenever normality does.
Cpk vs Ppk PPM: Ppk-derived PPM approximates long-term defect rate; Cpk-derived PPM approximates short-term entitlement.
Predicted vs observed PPM: Observed excess normally indicates instability or MSA — not a "worse" 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

predicted versus observed defect behavior — Model prediction must be compared with production behavior.

Convert in the other direction with PPM → Cpk, translate through Cpk → Sigma, and compare Cpk with long-term Ppk under Cpk vs Ppk. If PPM is dominated by one tail, work through Process Not Centered.

Verification checklist

Normality confirmed on the same dataset used for Cpk
One- vs two-sided PPM stated explicitly
No 1.5σ shift unless explicitly labelled
Sigma type (within / overall) recorded
Observed defect rate reviewed against predicted for divergence

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.

Sources and engineering references

External engineering references used for this page. Qhubio applies these references to the practical guidance above.

External engineering referenceNIST/SEMATECH
What is Process Capability?
Definitions and interpretation of capability indices.
External engineering referenceNIST/SEMATECH
Normal Distribution
Normal-distribution assumptions used in PPM conversions.