Gage R&R for Surface and Tribology Measurements
Separate repeatability, operator reproducibility, interaction, and part-to-part variation in a balanced crossed measurement study.
Study data
Variation components
ANOVA summary
| Source | SS | df | MS |
|---|---|---|---|
| Part | — | — | — |
| Operator | — | — | — |
| Part × operator | — | — | — |
| Repeatability | — | — | — |
Zero-truncated method-of-moments components prevent negative variance estimates from being presented as physical variation.
How to use
- Confirm a balanced crossed study: every operator measures every part for the same number of trials, preferably in randomized run order.
- Enter operators b, parts a, trials n, and the common measurement unit.
- Paste bn rows with a numeric values each. Group rows by operator, then trial; keep part columns in one consistent order.
- Optionally enter positive tolerance width T = USL − LSL and/or positive six-sigma process variation PV₆σ. Leave either blank rather than inventing a denominator.
- Select Analyze Gage R&R to update the single result panel. Select Reset example to restore the example and recalculate.
- Read %Study GRR and ndc, then review repeatability, reproducibility, interaction, ANOVA, and the chart. Correct any red warning.
- Do not use this model for nested/destructive, unbalanced, missing-data, drifting, categorical, or nonrepresentative studies. Apply the governing quality plan and engineering judgment.
Equations used
For part i = 1…a, operator j = 1…b, and trial k = 1…n, xᵢⱼₖ is a measurement. Overbars denote cell, part, operator, or grand arithmetic means.
SSpart = bn Σᵢ(x̄ᵢ·· − x̄···)²
SSoperator = an Σⱼ(x̄·ⱼ· − x̄···)²
SSinteraction = n ΣᵢΣⱼ(x̄ᵢⱼ· − x̄ᵢ·· − x̄·ⱼ· + x̄···)²
SSrepeat = ΣᵢΣⱼΣₖ(xᵢⱼₖ − x̄ᵢⱼ·)²
Each MS = SS/df; degrees of freedom are a−1, b−1, (a−1)(b−1), and ab(n−1).
σ²repeat = MSrepeat
σ²operator = max[(MSoperator − MSinteraction)/(an), 0]
σ²interaction = max[(MSinteraction − MSrepeat)/n, 0]
σ²part = max[(MSpart − MSinteraction)/(bn), 0]
σ²reprod = σ²operator + σ²interaction
σ²GRR = σ²repeat + σ²operator + σ²interaction
σ²total = σ²GRR + σ²part
%Study GRR = 100 σGRR / √(σ²GRR + σ²part)
%Tolerance = 100(6σGRR)/T
%Process = 100(6σGRR)/PV₆σ
ndc = floor[1.41 σpart / σGRR]
Standard deviations and six-sigma spreads use the measurement unit; variances, SS, and MS use its square. The model reports positive variation components; raw values and means retain their sign. Assumptions: crossed, balanced, random-effects data; independent residuals; stable conditions; representative parts and operators. Negative component estimates are truncated to zero. Interaction contribution and F = MSinteraction/MSrepeat are diagnostics, not an automated significance claim.
References: NIST/SEMATECH Gauge R&R studies; AIAG MSA Reference Manual, 4th edition.
Related: Friction Test Statistics and Process Capability.