process-capability-and-spc

name	process-capability-and-spc
description	Baseline a process: compute DPMO/sigma level and Cp/Cpk, select and read the right control chart (I-MR / Xbar-R / p / c / u), and apply Western Electric/Nelson rules to separate common-cause from special-cause variation. Routes deeper capability inference and hypothesis testing to applied-statistics.

Skill: process-capability-and-spc

Invoked by: process-improvement/lean-six-sigma-blackbelt (primary — owns the baseline verdict and control-phase monitoring design). Also used by process-analyst for data-collection planning and initial control-chart selection.

When to invoke: DMAIC Measure phase (establishing the baseline); Control phase (setting up ongoing monitoring); any time someone asks "how capable is this process?" or "is this process stable?"; before and after an Improve-phase change to quantify the gain.

Output: a stated sigma level / DPMO / yield (with the 1.5σ-shift convention explicitly noted); Cp/Cpk (or Pp/Ppk for long-term) with a capability verdict; a control chart type selected from the decision tree; out-of-control signals identified against stated Western Electric / Nelson rules; a clear separation of common-cause from special-cause.

The two questions this skill answers

1. Is the process stable? (Control chart — separates common-cause from special-cause variation)
2. Is the stable process capable? (Capability indices — compares variation to the customer specification)

Answer #1 before #2. Capability on an unstable process is meaningless — the indices will change as special causes come and go.

Part 1 — DPMO, Sigma Level, and Yield

Operational definitions (required before any count)

Before counting defects, define:

• Unit — the entity being produced (one invoice, one support ticket, one hire, one deployment)
• Defect — any output that fails a customer CTQ (not an internal preference)
• Opportunity — the number of ways a single unit can fail (be specific; inflating opportunities inflates sigma artificially)

DPMO and sigma level

DPMO (Defects Per Million Opportunities):

DPMO = (number of defects / (units × opportunities per unit)) × 1,000,000

Example — claims processing: 47 defects in 800 claims, each with 3 CTQ opportunities:

DPMO = (47 / (800 × 3)) × 1,000,000 = 19,583

Sigma level from DPMO (long-term, with the 1.5σ shift baked in — the industry-standard convention):

Long-term sigma	DPMO	Yield
2σ	308,537	69.1%
3σ	66,807	93.3%
4σ	6,210	99.4%
5σ	233	99.977%
6σ	3.4	99.9997%

Always state the convention. "This process is at 3.7 sigma (long-term, with 1.5σ shift)." Never quote a sigma level without stating which convention. The 1.5σ shift is an industry assumption about long-term process drift, not a mathematical derivation; omitting it creates confusion when comparing across sources.

For deeper capability inference (confidence intervals on sigma, capability with small samples) — route to applied-statistics.

Part 2 — Cp, Cpk, Pp, Ppk

Use these indices only for continuous, measurable CTQs (cycle time in hours, error rate, invoice amount variance), not for defect counts.

Index	What it measures	When to use
Cp	Process spread vs. spec width (centering ignored)	Short-term potential; subgroup data
Cpk	Process spread vs. spec, with centering penalty	Short-term actual; most common single-number summary
Pp	Long-term spread vs. spec width	Overall, all sources of variation
Ppk	Long-term spread vs. spec, with centering	Long-term actual; use when reporting to customers

Formulas:

Cp  = (USL - LSL) / (6 × σ_within)
Cpk = min[(USL - x̄) / (3 × σ_within),  (x̄ - LSL) / (3 × σ_within)]

(Pp / Ppk: same formula but σ_overall replaces σ_within)

Capability thresholds:

Cpk value	Verdict
< 1.00	Incapable — the process is producing defects against the spec
1.00–1.33	Marginally capable — meets spec when centered; little room for drift
1.33–1.67	Capable — industry standard for established processes
≥ 1.67	Highly capable — Six Sigma target (Cpk ≥ 1.50 short-term → 6σ long-term)

Confusing Cpk and Ppk is an anti-pattern. Cpk uses within-subgroup variation (σ_within); it overestimates long-term performance if between-subgroup variation is significant. Report both when available; explain which is which.

For capability confidence intervals, sample-size planning for a capability study, and Gage R&R analysis (measurement system validation before calculating capability) — route to applied-statistics.

Part 3 — Control chart selection

Use this decision tree before drawing a chart:

Is the data continuous (measured) or attribute (counted)?
├── Continuous
│   ├── Individual measurements (subgroup size = 1, e.g., daily cycle time)
│   │   └── I-MR chart (Individuals + Moving Range)
│   └── Subgroups (multiple measurements per time period)
│       ├── Subgroup size 2–8  →  Xbar-R chart
│       └── Subgroup size ≥ 9  →  Xbar-S chart
└── Attribute (defects / defectives)
    ├── Tracking defectives (pass/fail units)?
    │   ├── Constant subgroup size  →  p chart (proportion defective)
    │   └── Varying subgroup size   →  p chart (also handles variable n)
    │       OR np chart (fixed n only, counts defectives not proportion)
    └── Tracking defects (multiple defects per unit)?
        ├── Constant area of opportunity  →  c chart (count of defects)
        └── Varying area of opportunity   →  u chart (defects per unit)

Most common choices for office/operational processes:

• I-MR — single daily/weekly measurement (cycle time per order, claims per day, deployment failures per week)
• p chart — proportion defective with varying batch sizes (proportion of tickets re-opened, proportion of invoices requiring rework)
• c chart — count of defects per unit with fixed opportunity (errors per audit report)

Reading a control chart

Every control chart has:

• Center line (CL) — the process average
• Upper Control Limit (UCL) and Lower Control Limit (LCL) — ±3σ from the center line, calculated from the data

Control limits are not specification limits. Control limits describe what the process actually does. Specification limits describe what the customer needs. Do not draw spec limits on a control chart and call a point "out of control" because it missed the spec.

Part 4 — Western Electric / Nelson rules

Apply these rules to detect special-cause signals. State which ruleset you are using before applying it.

Western Electric Rules (the standard four):

Rule	Signal
Rule 1	1 point beyond ±3σ (outside UCL or LCL)
Rule 2	2 of 3 consecutive points beyond ±2σ (same side)
Rule 3	4 of 5 consecutive points beyond ±1σ (same side)
Rule 4	8 consecutive points on the same side of the center line

Nelson Rules add:

Rule	Signal
Rule 5	6 consecutive points trending steadily up or down
Rule 6	14 consecutive points alternating up/down
Rule 7	15 consecutive points within ±1σ (hugging the center)
Rule 8	8 consecutive points beyond ±1σ (both sides, none within ±1σ)

Using multiple rules increases false-alarm rate. For a baseline study, Rules 1–4 are sufficient. Apply additional rules only when the process is known to exhibit specific patterns (trending, cycling).

Interpreting signals

Signal type	Meaning	Action
Common cause — all points within control limits, no rule violations	Natural variation inherent to the current process design	Investigate and redesign the process (Improve phase); reacting to individual points is tampering
Special cause — any rule violation	An assignable cause entered or left the process	Identify and eliminate the special cause; do not calculate capability until the process is stable

See best-practice: separate-common-cause-from-special-cause.md — reacting to common-cause variation as if it were special cause (tampering) increases process variation.

Capability study checklist

• Unit, defect, and opportunity operationally defined before counting
• Measurement system validated (Gage R&R or equivalent) before collecting capability data — route to applied-statistics if formal MSA is needed
• Process stability confirmed first — control chart shows no special-cause signals
• Correct chart type selected from the decision tree (not assumed)
• Western Electric / Nelson ruleset stated explicitly
• Sigma level quoted with convention (long-term with 1.5σ shift, or short-term — state which)
• Cpk and Ppk both calculated and labeled correctly (short-term vs. long-term)
• Capability verdict stated against the threshold table
• Post-Improve remeasure planned: same metric, same definition, comparable sample size

Anti-patterns this skill flags

• Calculating Cpk before confirming stability — an out-of-control process yields a misleading Cpk
• Using control limits as specification limits — these are different things with different purposes
• Quoting sigma level without stating the 1.5σ-shift convention — creates apples-to-oranges comparisons
• Confusing Cpk (short-term) with Ppk (long-term) — inflates reported capability
• Drawing a control chart without stating the ruleset — any one point outside 3σ might be Rule 1; calling a run of 7 "out of control" when Rule 4 requires 8 is wrong
• Reacting to a single point without confirming it is a special-cause signal — the most common form of tampering
• Inflating the number of opportunities per unit to inflate the sigma level — sigma is only comparable across processes when opportunities are defined consistently
• Skipping MSA — calculating Cpk with a measurement system that contributes > 10% of process variance inflates variation and understates capability

See also

• Skill: ../root-cause-analysis/SKILL.md — uses the control chart to establish whether the baseline is stable before root-cause analysis
• Skill: ../control-plan-and-sustain/SKILL.md — uses the control chart type and UCL/LCL in the Control phase
• Best-practice: ../../best-practices/measure-the-baseline-before-you-change-anything.md
• Best-practice: ../../best-practices/separate-common-cause-from-special-cause.md
• applied-statistics/agents/applied-statistician.md — Gage R&R / MSA, capability CIs, sample-size planning, hypothesis tests for pre/post comparison
• applied-statistics/skills/statistical-qa-of-metrics/SKILL.md — validates metric definitions and measurement reliability before the capability study begins
• Knowledge: ../../knowledge/six-sigma-statistics-and-spc.md — the sigma↔DPMO table, control-chart rules, and Cp/Cpk thresholds with retrieval dates

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Last reviewed: 2026-06-03 by claude