Cpk Process Capability Index

Overview

Cpk measures how well a process fits within specification limits, accounting for both variation (spread) and centering. Cpk = min((USL - μ) / 3σ, (μ - LSL) / 3σ). Cpk ≥ 1.33 is typically required; Cpk ≥ 1.67 for critical characteristics. Unlike Cp, Cpk penalizes off-center processes.

When to Use

Trigger conditions:

• Assessing whether a manufacturing process can meet customer specifications
• Comparing capability across processes, machines, or time periods
• Qualifying a process for production readiness

When NOT to use:

• When the process is not in statistical control (stabilize first with SPC)
• For non-normal distributions without transformation

Algorithm

IRON LAW: Cpk Is Only Valid for a STABLE, IN-CONTROL Process
Computing Cpk on an unstable process gives a meaningless number.
The process MUST be in statistical control (per SPC charts) before
capability analysis. An unstable process with Cpk=2.0 today may
produce defects tomorrow when it shifts.

Phase 1: Input Validation

Collect: 100+ measurements from a stable process. Determine: USL, LSL (customer specifications). Verify process is in control (SPC charts show stability). Gate: Process in control, specifications defined, 100+ data points.

Phase 2: Core Algorithm

1. Compute process mean: μ = Σxᵢ / n
2. Compute process standard deviation: σ = estimated from R-bar/d₂ or S-bar/c₄ (within-subgroup) — NOT overall std dev
3. Cp = (USL - LSL) / 6σ (potential capability, ignoring centering)
4. Cpk = min((USL - μ) / 3σ, (μ - LSL) / 3σ) (actual capability)
5. Estimate PPM defective from Cpk (e.g., Cpk=1.33 → ~63 PPM)

Phase 3: Verification

Check: Cp vs Cpk difference indicates centering issue (Cp >> Cpk = off-center). Distribution is approximately normal (histogram, normality test). Gate: Capability computed, centering assessed, normality verified.

Phase 4: Output

Return capability indices with defect rate estimates.

Output Format

{
  "capability": {"cp": 1.8, "cpk": 1.45, "ppm_defective": 27},
  "centering": {"mean": 50.2, "target": 50.0, "offset_pct": 0.4},
  "specs": {"usl": 55, "lsl": 45, "target": 50},
  "metadata": {"samples": 200, "sigma_method": "rbar_d2", "normality_p": 0.35}
}

Examples

Sample I/O

Input: USL=55, LSL=45, μ=50.2, σ=1.5 Expected: Cp = (55-45)/(6×1.5) = 1.11. Cpk = min((55-50.2)/4.5, (50.2-45)/4.5) = min(1.07, 1.16) = 1.07. Below 1.33 target.

Edge Cases

Input	Expected	Why
μ exactly at target	Cp = Cpk	Perfectly centered
μ outside specs	Cpk < 0	Process mean beyond specification limit
One-sided spec only	Use Cpk for that side only	e.g., surface finish has only USL

Gotchas

• σ estimation method: Use within-subgroup σ (R̄/d₂), NOT overall σ. Overall σ includes between-subgroup variation that inflates σ and understates Cpk.
• Non-normal data: Cpk assumes normality. For skewed data (surface finish, concentricity), use Box-Cox transformation or non-parametric capability indices.
• Short-term vs long-term: Cp/Cpk are short-term (within subgroup variation). Pp/Ppk use overall variation (long-term). Customers often want Ppk.
• Sample size confidence: Cpk from 30 samples has wide confidence intervals. Report confidence intervals alongside point estimates.
• Cpk ≠ defect-free: Even Cpk=2.0 has a theoretical defect rate (~0.002 PPM). For ultra-critical applications, higher Cpk or process validation is required.

Scripts

Script	Description	Usage
scripts/cpk.py	Compute Cp, Cpk, Cpm, and PPM defective from process data	python scripts/cpk.py --help

Run python scripts/cpk.py --verify to execute built-in sanity tests.

References

• For Cp/Cpk/Pp/Ppk comparison, see references/capability-indices.md
• For non-normal capability analysis, see references/non-normal-capability.md