charlieviettq 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
- Compute process mean: μ = Σxᵢ / n
- Compute process standard deviation: σ = estimated from R-bar/d₂ or S-bar/c₄ (within-subgroup) — NOT overall std dev
- Cp = (USL - LSL) / 6σ (potential capability, ignoring centering)
- Cpk = min((USL - μ) / 3σ, (μ - LSL) / 3σ) (actual capability)
- 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