17.4 Process Capability Metrics & Outliers
Key Takeaways
- Cp measures potential capability from process spread vs. tolerance and ignores centering; Cpk measures actual capability including location relative to the nearest specification.
- Valid capability analysis requires a stable process, correct specifications, adequate data, and a distribution (often approximate normality) that matches the index assumptions—or an appropriate alternative method.
- Cpk < 1 generally indicates the process is not meeting specs consistently; higher values (e.g., ≥1.33) are common industry targets but customer/regulatory requirements govern.
- Outliers are observations unusually far from the bulk of the data; they can be special causes, data errors, or extreme common-cause tails—and they strongly affect mean, SD, and capability.
- Auditors should verify how outliers were identified (charts, residual rules, domain review), whether they were investigated, and whether exclusion from analysis was justified and documented.
17.4 Process Capability Metrics & Outliers
/practice/cqaPractice questions with detailed explanations
What “capability” means
Process capability describes how well a stable process’s natural variation and location fit within specification limits for a characteristic. It is not the same as:
- Control (stability vs. control limits—voice of the process)
- Performance sometimes labeled Pp/Ppk (often using overall long-term σ including all sources)
- Customer satisfaction (broader than one CTQ index)
Auditors meet capability claims in validation reports, supplier scorecards, PPAP/FAI packages, and management reviews. Your job is to understand the metrics and apply skepticism: correct formula, correct inputs, correct assumptions.
Specifications and process parameters
| Symbol | Meaning |
|---|---|
| LSL / USL | Lower / upper specification limits (customer or design) |
| Tolerance width | USL − LSL (two-sided) |
| μ or $\bar{x}$ | Process mean (population or sample estimate) |
| σ or s | Process standard deviation (definition depends on method) |
| Target T | Preferred value (may equal midpoint; not always) |
Critical audit check: Are LSL/USL the current approved limits for this product/revision? Many “incapable” or “capable” stories use obsolete drawings.
Cp — potential capability (spread only)
For a two-sided specification and a process spread taken as 6σ (common normal-based model spanning roughly ±3σ):
$C_p = \dfrac{USL - LSL}{6\sigma}$
- Cp asks: If the process were perfectly centered, would the spread fit the tolerance?
- Cp does not detect off-center processes. A process can have high Cp and still produce massive defects if mean sits near one limit.
Worked mini-calculation — Cp
USL = 20.0, LSL = 10.0, σ = 1.0
- Tolerance = 10.0
- 6σ = 6.0
- Cp = 10 / 6 ≈ 1.67
That looks strong—but only if the mean is near the center. If mean = 18 with the same σ, defects explode near the USL even though Cp is unchanged.
Cpk — actual capability (spread + centering)
$C_{pk} = \min\left(\dfrac{USL - \mu}{3\sigma},\ \dfrac{\mu - LSL}{3\sigma}\right)$
- Uses the nearest specification
- Falls when the process is off-center or when σ is large
- For a centered process, Cpk ≈ Cp; when off-center, Cpk < Cp
Worked mini-calculation — Cpk
USL = 20, LSL = 10, μ = 16, σ = 1.0
- (USL − μ)/(3σ) = (20 − 16)/3 = 4/3 ≈ 1.33
- (μ − LSL)/(3σ) = (16 − 10)/3 = 6/3 = 2.00
- Cpk = min(1.33, 2.00) = 1.33
- Cp was 10/(6) ≈ 1.67 → Cpk < Cp shows centering loss
Second worked example — incapable centering
USL = 10.50, LSL = 9.50, μ = 10.40, σ = 0.10
- Cp = (1.00)/(0.60) ≈ 1.67
- (USL − μ)/(3σ) = 0.10 / 0.30 ≈ 0.33
- (μ − LSL)/(3σ) = 0.90 / 0.30 = 3.00
- Cpk ≈ 0.33 → process looks “capable” by Cp but is badly off-center; audit must challenge any claim based on Cp alone.
Interpreting magnitude (rules of thumb)
Targets vary by industry and customer. Common teaching benchmarks (not universal law):
| Cpk (approx.) | Rough interpretation |
|---|---|
| < 1.0 | Process spread/location not fitting specs; nonconformances expected |
| ≈ 1.0 | Marginal; about ±3σ at the nearer limit if normal and stable |
| ≥ 1.33 | Common many-industry minimum for ongoing production |
| ≥ 1.67 | Stronger target in some automotive/high-reliability contexts |
| ≥ 2.0 | Associated with “6σ” style short-term thinking in some curricula |
Auditor stance: Compare claims to contractual/customer requirements and to stability evidence, not only to a memorized number.
Prerequisites for valid capability (Understand/Apply)
- Statistical stability — process in control; special causes addressed
- Correct specs — current, applicable, same units
- Representative data — right product, period, sampling plan
- Measurement system adequacy — high measurement error inflates apparent σ and tanks Cpk
- Distribution fit — normal-based Cp/Cpk mislead for strongly skewed data; may need transform, non-normal indices, or percent nonconforming methods
- σ definition clarity — within-subgroup vs. overall (Cp/Cpk vs. Pp/Ppk distinctions in many Six Sigma contexts)
If any prerequisite fails, treat published indices as claims needing validation, not as facts.
Centering vs. spread — improvement strategy
| Diagnosis | Evidence | Typical improvement direction |
|---|---|---|
| Wide spread, well centered | Cp and Cpk both low, similar | Reduce variation (method, equipment, materials) |
| Tight spread, off-center | Cp high, Cpk much lower | Center the mean (setup, target, bias) |
| Both problems | Cp low, Cpk even lower | Reduce variation and center |
| Unstable | Chart out of control | Restore control before capability projects |
This table is pure audit gold: it connects metrics to correct CAPA/system projects.
Outliers — definition and impact
An outlier is an observation unusually distant from the bulk of the data—far in the tail relative to the rest of the sample or to a fitted model.
Impacts:
- Pulls the mean toward the extreme
- Inflates range, variance, and SD
- Can crash Cpk (or, if wrongly deleted, inflate Cpk unrealistically)
- May be the most important signal (special cause, safety event) or a clerical error (extra zero typed)
Identification methods (common)
| Method | Idea | Caveat |
|---|---|---|
| Visual | Histogram, box plot, time-series plot | Subjective alone |
| Control chart | Point beyond limits / pattern rules | Needs proper limits |
| z-score / residual | e.g., |z| > 3 under a normal model | Model-dependent |
| IQR rule | < Q1 − 1.5·IQR or > Q3 + 1.5·IQR | Sensitive to distribution; screening only |
| Domain review | Impossible physically? Wrong unit? | Always required before deletion |
Never delete outliers solely to improve a customer scorecard. Document investigation: root cause, data error correction, or retained extreme with analysis that includes it (or uses robust methods).
Worked outlier impact sketch
Data: 10.0, 10.1, 9.9, 10.0, 10.2, and 14.0 (possible bad reading).
- With 14.0: mean jumps, s inflates, Cpk collapses
- Investigation finds a decimal error; true value 10.4 → correct the record
- Investigation finds a real process excursion → treat as special cause, contain product, fix cause; do not silently drop the point and claim high Cpk for the period without disclosure
Auditor checklist for capability & outliers
- Request control chart evidence for the capability period
- Verify LSL/USL/revision and characteristic definition
- Confirm σ method and software settings
- Compare Cp vs. Cpk to diagnose centering vs. spread
- Ask how outliers were detected, investigated, and treated
- Cross-check with actual nonconformance rates—huge mismatch suggests bad assumptions or sorting
- Align findings with system improvement (common cause / low Cpk) vs. event CAPA (outlier special cause)
Common exam traps
- Using Cp when the process is off-center (Cp looks fine; Cpk does not)
- Computing capability on unstable data
- Confusing control limits with specification limits
- Treating outlier deletion as routine “data cleaning” without cause
- Assuming Cpk ≥ 1.33 is a universal legal requirement
- Ignoring measurement system error as a driver of poor Cpk
Chapter close
For USL = 50, LSL = 20, and σ = 5, what is Cp, and what does it not tell you?
A supplier reports Cp = 1.8 and Cpk = 0.6 for the same dataset. What is the best auditor interpretation?
Which prerequisite most directly undermines a published Cpk of 1.5 for last month’s production?
An analyst removes the three highest values from a capability study solely because they make Cpk look bad, with no investigation. What should the auditor conclude?