six-sigma-methods

name	six-sigma-methods
description	Six Sigma methodology — DMAIC (Define, Measure, Analyze, Improve, Control), process capability indices (Cp, Cpk, Ppk), Gage R&R measurement system analysis (% study variation, number of distinct categories), hypothesis testing (t-test, ANOVA, chi-square), DOE (Design of Experiments — 2k factorial, response surface), control charts (X-bar/R, I-MR, p-chart, u-chart, CUSUM), DPMO and sigma level conversion, DFSS (Design for Six Sigma), and ASQ/AIAG standards.
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Six Sigma Methods — Complete Skill

DMAIC Framework

Phase Overview

Define: Problem statement: quantified, time-bounded, no solution prescribed Project charter: business case, scope, team, milestones, financial benefit Voice of Customer (VoC): CTQ (Critical to Quality) tree; Y = output metric defined SIPOC diagram: Supplier-Input-Process-Output-Customer; defines process boundaries

Measure: Baseline Y: collect data; calculate DPU/DPMO; calculate sigma level; establish baseline Cpk Measurement system analysis (MSA/Gage R&R): verify measurement system acceptable before collecting data Data collection plan: sample size, sampling method, operational definitions

Analyze: Root cause identification: fishbone (Ishikawa), 5-Why, cause-and-effect matrix Multi-vari analysis: graphical exploration of variation patterns Hypothesis testing: confirm root causes statistically (t-test, ANOVA, regression) Process map: identify value-added vs. non-value-added steps; where variation enters

Improve: DOE (Design of Experiments): identify significant factors; optimize settings Simulation and pilot: verify improvements; prove results before full implementation Change management: stakeholder buy-in; pilot team training

Control: Control plan: document controls for each key input X → maintain improved output Y Control charts: SPC implementation; response plan for out-of-control conditions Mistake-proofing (poka-yoke): prevent defect from occurring Handoff to process owner; close project

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Process Capability

Cp and Cpk

Cp (process potential — centered assumption): Cp = (USL − LSL) / (6σ) [USL = upper spec limit; LSL = lower spec limit; σ = process standard deviation] Cp = tolerance width / process spread; Cp ≥ 1.33 for capable process (±4σ room)

Cpk (actual process capability — accounts for centering): Cpk = min(CPU, CPL) CPU = (USL − x̄) / (3σ) [upper; distance from mean to USL in σ units] CPL = (x̄ − LSL) / (3σ) [lower; distance from mean to LSL in σ units] Cpk = 1.0 → 3σ level (2700 ppm); Cpk = 1.33 → 4σ level (63 ppm); Cpk = 1.67 → 5σ level (0.57 ppm)

Six Sigma goal: Cpk = 1.50 → process shifted by 1.5σ → 3.4 DPMO (from tables) Note: 6σ target is 3.4 DPMO including 1.5σ long-term mean shift

Ppk (process performance — includes between-subgroup variation): Ppk = min((USL − x̄)/(3×S_overall), (x̄ − LSL)/(3×S_overall)) [S_overall = total sample std dev; not Rbar/d2] Ppk ≤ Cpk always (Ppk includes more variation); gap = Ppk/Cpk indicates between-subgroup problems

Example: Dimension spec: 100 ± 1.0 mm; x̄ = 100.2 mm; σ = 0.25 mm CPU = (101.0 − 100.2) / (3 × 0.25) = 0.8 / 0.75 = 1.067 CPL = (100.2 − 99.0) / (3 × 0.25) = 1.2 / 0.75 = 1.600 Cpk = min(1.067, 1.600) = 1.067 (marginally capable; mean shifted toward USL)

DPMO and Sigma Level

DPMO (Defects Per Million Opportunities): DPMO = (number of defects / (number of units × number of opportunities per unit)) × 10⁶

Sigma level to DPMO conversion (with 1.5σ shift):

Sigma Level	DPMO	Cpk (short-term)	Yield [%]
2σ	308,537	0.50	69.1
3σ	66,807	1.00	93.3
4σ	6,210	1.33	99.4
5σ	233	1.67	99.98
6σ	3.4	2.00	99.9997

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Measurement System Analysis (Gage R&R)

AIAG MSA Study

Gage R&R components: Repeatability (Equipment Variation, EV): variation within one operator using same gage on same part Reproducibility (Appraiser Variation, AV): variation between appraisers using same gage on same part Part variation (PV): actual part-to-part variation being studied Total gage R&R (GRR) = √(EV² + AV²)

%GRR (% Study Variation): %GRR = (GRR / TV) × 100% [TV = Total Variation = √(GRR² + PV²) = process σ × 6 from study] Standard: %GRR < 10% → acceptable; 10–30% → marginal (may be acceptable for specific application); > 30% → unacceptable

Number of Distinct Categories (NDC): NDC = 1.41 × PV / GRR [rounded down to integer] NDC ≥ 5 required for acceptable measurement system (can detect 5+ distinct levels of product variation) NDC < 2 → gage cannot differentiate products

Typical Gage R&R study: 10 parts × 2–3 appraisers × 2–3 replicates = 40–90 measurements Analyze with ANOVA method (better than Range/Average method for separating AV and EV)

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Statistical Process Control (SPC)

Control Charts

X-bar and R chart (subgroup size n = 3–8): Center line: x̄̄ (grand mean); UCL_X = x̄̄ + A₂ × R̄; LCL_X = x̄̄ − A₂ × R̄ UCL_R = D₄ × R̄; LCL_R = D₃ × R̄ [D₃ = 0 for n < 7] Control factors A₂, D₃, D₄: from table (n = 5: A₂ = 0.577; D₄ = 2.115; D₃ = 0)

I-MR chart (individual observations; n = 1): Moving range: MR_i = |X_i − X_{i-1}|; MR̄ = average moving range UCL_X = X̄ + 3 × MR̄/1.128; UCL_MR = 3.267 × MR̄ [constant 1.128 = d₂ for n=2] Used for: slow processes (1 measurement per period); chemical batches; financial data

Western Electric Rules (WECO) — out-of-control signals:

1. One point beyond 3σ limits (±3σ)
2. Two of three consecutive points beyond 2σ
3. Four of five consecutive points beyond 1σ
4. Eight consecutive points on same side of center line Triggers investigation; identify assignable cause; remove from data if valid special cause

Attribute charts: p-chart: fraction defective (variable subgroup size n) np-chart: number defective (fixed n); UCL = n×p̄ + 3√(n×p̄×(1−p̄)) u-chart: defects per unit (variable n); c-chart: count of defects (fixed n)

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Hypothesis Testing

Common Tests

One-sample t-test: H₀: μ = μ₀; H₁: μ ≠ μ₀ t = (x̄ − μ₀) / (s/√n) [degrees of freedom ν = n−1] Reject H₀ if |t| > t_{α/2, n−1} [from t-table at significance level α]

Two-sample t-test (compare two means): t = (x̄₁ − x̄₂) / √(s_p² × (1/n₁ + 1/n₂)) [s_p = pooled std dev; assumes equal variance] Check equal variance first with F-test or Levene's test

One-way ANOVA (compare ≥3 means): F = MSB/MSW [MSB = Mean Square Between; MSW = Mean Square Within] Reject H₀ (all means equal) if F > F_{α, k−1, N−k} [k = groups; N = total sample size]

Sample size calculation (t-test): n = 2 × [(z_{α/2} + z_β)² × σ²] / δ² [δ = minimum detectable difference; z = z-scores for Type I/II error] For α = 0.05, β = 0.10 (80% power): (1.96 + 1.28)² = 10.5 → n ≈ 21σ²/δ²

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Design of Experiments (DOE)

2k Factorial Design

Full factorial 2k: 2 levels (high/low) for each of k factors; 2k runs Effect estimate: E_main = (Y_high − Y_low) / (2k/2) [contrast ÷ half the runs] Interaction: E_AB = (Y_AB+) − Y_AB−) / (2k/2) [when A and B interact; plotted as crossing lines]

Half-fraction (2^(k-1)): resolution III (main effects confounded with 2-way interactions) or resolution IV/V Used when k > 4 and full factorial too expensive

Response Surface Methodology (RSM): Central Composite Design (CCD): 2k factorial + 2k axial points + center points → fits quadratic model Optimize response: y = β₀ + Σβᵢxᵢ + Σβᵢᵢxᵢ² + Σβᵢⱼxᵢxⱼ + ε Find optimum: set ∂y/∂xᵢ = 0 for each factor → stationary point → maximum/minimum/saddle

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Standards and References

Standard	Scope
AIAG MSA-4	Measurement Systems Analysis reference manual
AIAG SPC-2	Statistical Process Control reference manual
ASQ Body of Knowledge	Certified Six Sigma Black Belt (CSSBB) reference
IATF 16949	Automotive QMS (requires SPC, MSA, PPAP)
ISO 13053-1/2	Six Sigma in business improvement
Montgomery "Design and Analysis of Experiments"	DOE standard textbook

Output

Provide: project charter (Y metric; baseline value; goal; financial benefit; timeline), process capability (data: x̄ ± σ [units]; USL/LSL [units]; Cp = (USL−LSL)/(6σ); Cpk = min(CPU,CPL); sigma level; DPMO from Cpk table), Gage R&R (EV [%]; AV [%]; GRR% = total R&R/TV [%]; NDC [integer]; acceptable? %GRR < 10% ideal), root cause analysis (Pareto chart top causes; fishbone categories; statistical validation: t-test/ANOVA p-values; root causes confirmed by data), DOE (factors studied: k; design: full 2k or fractional; main effects [significance p-value]; interactions; optimal settings; predicted Y_new [value]), control plan (key input Xs; SPC chart type; UCL/LCL; sampling plan; response plan for OOC; poka-yoke implementation), final capability (Cpk_new; DPMO_new; sigma level; vs. baseline; financial benefit realized [$]), and applicable standard (AIAG MSA-4 for GRR; AIAG SPC-2 for control charts; ISO 13053 for DMAIC framework).