Measurement System Analysis

Gage R&R — ANOVA-Based Measurement System Analysis Type 2

Decomposes measurement variation into repeatability (EV), reproducibility (AV), and part-to-part variation using two-factor crossed ANOVA with interaction.

Statusvalidated

Version1.1.0

Minitab equivalentStat > Quality Tools > Gage Study > Gage R&R Study (Crossed)

Description

The Gage R&R study (Type 2) evaluates a measurement system by having multiple operators measure multiple parts with replications. The two-factor crossed ANOVA (Parts × Operators) with interaction decomposes total variation into components: σ²_Repeatability (equipment variation EV), σ²_Reproducibility (appraiser variation AV = σ²_Operator + σ²_Interaction), σ²_Part (part-to-part variation), and σ²_GRR = σ²_EV + σ²_AV. If the interaction is not significant (p ≥ α), it is pooled with repeatability. Study variation = k·σ with k = 6 (default, AIAG MSA 4th ed., Minitab, Bosch booklet 10) or 5.15 (AIAG 3rd ed.). The decision criterion is %GRR on the verdict basis: %Tolerance when LSL and USL are given (automatic), otherwise %Study Var.; < 10% = acceptable, 10–30% = conditionally acceptable, ≥ 30% = not acceptable. An optional historical process σ adds %Process = σ/σ_hist·100 (informational only). Additionally, the number of distinct categories (ndc = ⌊1.41 · σ_Part / σ_GRR⌋) is computed; ndc ≥ 5 is considered acceptable.

Formulas

SS_Part

S S_{Part} = o \cdot r \cdot i = 1 \sum p (\overset{ˉ}{Y}_{i ..} - \overset{ˉ}{Y}_{...})^{2}

Sum of squares for part variation

SS_Operator

S S_{Operator} = p \cdot r \cdot j = 1 \sum o (\overset{ˉ}{Y}_{. j .} - \overset{ˉ}{Y}_{...})^{2}

Sum of squares for operator variation

SS_Interact

S S_{Interact} = r \cdot i = 1 \sum p j = 1 \sum o (\overset{ˉ}{Y}_{ij .} - \overset{ˉ}{Y}_{i ..} - \overset{ˉ}{Y}_{. j .} + \overset{ˉ}{Y}_{...})^{2}

Sum of squares for Part × Operator interaction

σ²_GRR

σ_{GRR}^{2} = σ_{EV}^{2} + σ_{AV}^{2} = σ_{Equip}^{2} + σ_{Operator}^{2} + σ_{Interact}^{2}

Total measurement system variation (Repeatability + Reproducibility)

%GRR

% GRR_{StudyVar} = \frac{σ _{GRR}}{σ _{Total}} \times 100

Measurement system variation as percentage of total variation (decision criterion without tolerance or with verdict basis study variation)

ndc

ndc = ⌊ 1.41 \cdot \frac{σ _{Part}}{σ _{GRR}} ⌋

Number of distinct categories (≥ 5 acceptable)

%Tolerance

% Tol = \frac{k \cdot σ _{GRR}}{T} \times 100 (T = U S L - L S L)

Measurement system variation as percentage of tolerance (only if LSL/USL provided); decision criterion with the automatic verdict basis

%Process

% Process = \frac{σ}{σ _{hist}} \times 100

Share of the historical process σ (only if given; informational, may exceed 100%, no verdict)

Verdict

% GRR_{verdict} = {% Tol % GRR_{StudyVar} basis \neq = studyVar \land T given otherwise

Verdict on pctGRRVerdict: < 10% pass, < 30% warn, otherwise fail

Assumptions

Crossed study design: each operator measures each part equally often (balanced design)
At least 2 parts, 2 operators, 2 replicates per cell
Recommended: 10 parts × 3 operators × 2–3 replicates (AIAG MSA-4)
Measurements are approximately normally distributed
Parts represent the typical process range

Limitations

Negative variance estimates are clamped to 0 — this can distort results in very small studies
Interaction is automatically pooled when p ≥ α — pooling changes degrees of freedom and variance components
Only for crossed studies — nested designs require a different ANOVA structure

References

AIAG — Measurement Systems Analysis Reference Manual, 4th Edition (MSA-4), Chapter 4
VDA Band 5 — Prüfprozesseignung: Eignung von Messsystemen, Mess- und Prüfprozessen, 3. Auflage
Montgomery, D. C. — Design and Analysis of Experiments, 10th Edition, Chapter 14
Wheeler, D. J. — EMP III: Evaluating the Measurement Process, SPC Press