Six Sigma & Measurement Systems Analysis
AIAG Gage R&R %StudyVar, P/T Ratio & ndc Formula
Combine Equipment Repeatability (EV), Appraiser Reproducibility (AV), and Part Variation (PV) via root-sum-of-squares to audit %StudyVar, 6-sigma Precision-to-Tolerance (P/T), and ndc = INT(1.41 * PV / GRR).
Never add Gage Repeatability and Reproducibility linearly — combine variances in quadrature!
Before trusting Cpk or SPC charts, Automotive Industry Action Group (AIAG MSA 4th Ed.) Gage R&R audits whether measurement noise swamps actual part-to-part differences! Given Equipment Variation SD sigma_EV (Repeatability within operator), Appraiser Variation SD sigma_AV (Reproducibility across operators), and Part-to-Part SD sigma_PV, independent variances add in quadrature: Gage R&R SD sigma_GRR = SQRT(sigma_EV^2 + sigma_AV^2) and Total Study SD sigma_TV = SQRT(sigma_GRR^2 + sigma_PV^2)! AIAG grades a gage on three strict criteria: (1) %StudyVar = sigma_GRR / sigma_TV (< 10% Acceptable, 10%–30% Conditional, > 30% Unacceptable), (2) 6-Sigma Precision-to-Tolerance Ratio P/T = (6 * sigma_GRR) / (USL - LSL), and (3) Number of Distinct Categories ndc = MAX(1, INT(1.41 * (sigma_PV / sigma_GRR))) (requiring ndc >= 5 non-overlapping confidence buckets to resolve part variation)!
Verify on exact double-Pythagorean numbers (3-4-5 and 5-12-13!): Repeatability_SD sigma_EV = 0.30 mm (B2), Reproducibility_SD sigma_AV = 0.40 mm (B3 -> sigma_GRR = SQRT(0.09 + 0.16) = 0.50 mm!), Part_Variation_SD sigma_PV = 1.20 mm (B4 -> sigma_TV = SQRT(0.25 + 1.44) = 1.30 mm!), and Spec_Tolerance_Width (USL - LSL) = 10.00 mm (B5): %StudyVar is 0.50 / 1.30 = 0.3846 (38.46% > 30% Unacceptable for process control!), %Contribution (variance share) is (0.50 / 1.30)^2 = 0.1479 (14.79%), 6-Sigma P/T Ratio is (6 * 0.50) / 10.00 = 0.3000 (30.00% of spec width), and Distinct Categories ndc = INT(1.41 * (1.20 / 0.50)) = INT(3.384) = 3 (< 5 -> insufficient resolution)! Upgrading fixture repeatability/reproducibility to EV = 0.06, AV = 0.08 (GRR = 0.10) drops %StudyVar to 8.30% and lifts ndc to 16 (>= 5 Acceptable!)!
| AIAG MSA Metric (EV=0.30, AV=0.40, PV=1.20, Tol=10.0) | Formula | Verified Output |
|---|---|---|
| Combined Gage R&R SD (sigma_GRR) | SQRT(EV^2 + AV^2) | 0.5000 mm (3-4-5 Triple) |
| %StudyVar (sigma_GRR / sigma_TV) | 0.50 / SQRT(0.50^2 + 1.20^2) | 0.3846 (38.46% — Fails <30%!) |
| 6-Sigma Precision-to-Tolerance (P/T) | (6 * sigma_GRR) / Tolerance | 0.3000 (30.00% of Spec) |
| Number of Distinct Categories (ndc) | MAX(1, INT(1.41 * PV / GRR)) | 3 categories (Needs >= 5!) |
AIAG %StudyVar (sigma_GRR / sigma_TV) Formula
Validate non-negative component standard deviations with sigma_GRR > 0 and sigma_PV > 0.
=LET(s_ev, B2, s_av, B3, s_pv, B4, IF(OR(s_ev < 0, s_av < 0, s_pv <= 0, s_ev + s_av <= 0), "Invalid MSA Input", LET(s_gr, SQRT((s_ev ^ 2) + (s_av ^ 2)), s_tv, SQRT((s_gr ^ 2) + (s_pv ^ 2)), ROUND(s_gr / s_tv, 4))))AIAG Number of Distinct Categories (ndc) Formula
AIAG truncates 1.41 * (sigma_PV / sigma_GRR) downward via INT(...) (clamped at minimum 1).
=LET(s_ev, B2, s_av, B3, s_pv, B4, IF(OR(s_ev < 0, s_av < 0, s_pv <= 0, s_ev + s_av <= 0), "Invalid MSA Input", LET(s_gr, SQRT((s_ev ^ 2) + (s_av ^ 2)), MAX(1, INT(1.41 * (s_pv / s_gr))))))Try the example
Use the original values or change them to test your understanding.