What a Gage R&R study tells you
Every measured value carries two kinds of variation: the real differences between parts and the variation added by the measurement itself. A Gage R&R study separates them. Typically 3 operators measure the same 10 parts 2 or 3 times each, in random order and without seeing their earlier readings.
- Repeatability (EV, equipment variation): the spread when the same operator measures the same part again with the same gauge.
- Reproducibility (AV, appraiser variation): the difference between operators measuring the same parts.
- Gage R&R (GRR): both together, GRR = √(EV² + AV²).
- Part variation (PV): the real spread between the parts. Total variation is TV = √(GRR² + PV²).
%GRR = 100 · GRR / TV %Tolerance = 100 · 6 GRR / (USL − LSL) ndc = 1.41 · PV / GRR
Is my measurement system acceptable?
| %GRR | Usual decision (AIAG MSA) |
|---|---|
| below 10 % | Acceptable. |
| 10 % to 30 % | May be acceptable depending on the importance of the application, the cost of the gauge and the cost of repair. |
| above 30 % | Not acceptable. Find and remove the causes of measurement variation. |
The number of distinct categories, ndc, should be 5 or more. It says how many groups of parts the gauge can reliably tell apart within the process spread. Judge against study variation when the gauge is used for process control and SPC; judge against tolerance when it is used to accept or reject parts. If the 10 parts do not cover the real process range, enter the process standard deviation from your SPC data: it replaces the part variation from the study and gives a fairer %GRR.
ANOVA or average and range?
The average-and-range method is the classic hand calculation from the AIAG MSA manual. It uses the average range of the trials for repeatability and the spread of the operator averages for reproducibility, with the constants K₁, K₂ and K₃. It works for 2 or 3 operators and 2 or 3 trials.
The ANOVA method uses all the information in the data, works for any number of operators and trials, and can detect a part × operator interaction: operators that agree on some parts and disagree on others. When the interaction is not significant (p > 0.05) it is pooled into repeatability, as the MSA manual does. ANOVA is the preferred method when a computer is available, so it is the default headline here. Both are shown so you can compare with older reports.
Worked example: the AIAG MSA reference study
Three appraisers, A, B and C, measure ten parts three times each. This is the example data set of the AIAG MSA 4th edition manual. Press Load worked example to see it.
| Result | Average and range | ANOVA |
|---|---|---|
| Repeatability EV (σ) | 0.2019 | 0.1999 |
| Reproducibility AV (σ) | 0.2296 | 0.2268 |
| Gage R&R (σ) | 0.3058 | 0.3024 |
| Part variation PV (σ) | 1.1046 | 1.0423 |
| %GRR of study variation | 26.7 % | 27.9 % |
| Number of distinct categories | 5 | 4 |
%GRR is between 10 % and 30 %, so the gauge is marginal. Reproducibility is slightly larger than repeatability, and the range chart shows that appraiser B has the largest ranges. Training appraiser B and agreeing on how the part is held would be the first steps. The interaction is far from significant (p ≈ 0.97), so the ANOVA pools it into repeatability.
Common mistakes
Choosing parts that are too similar
If all ten parts come from one batch, the part variation is small and %GRR looks bad even with a good gauge. Pick parts that cover the process range, or enter the process σ.
Letting operators remember their readings
Randomise the order in each trial and hide the part numbers. Otherwise operators repeat their first reading and repeatability looks better than it is.
Too coarse a resolution
If the gauge reads in steps larger than about one tenth of the process spread, many ranges are zero and ndc falls. The range chart shows this as values stuck on a few levels.
Judging a sorting gauge against study variation only
A gauge used to accept parts must be judged against the tolerance. A gauge used for SPC must be judged against the process variation. Report the one that matches the use.
Questions
How must the data be laid out?
One row per part. The columns hold all trials of the first operator, then all trials of the second, and so on. With 3 operators and 3 trials that is 9 columns. Choose the number of operators; the number of trials follows from the number of columns. A first column with part numbers and a header row are detected and skipped.
Why do the two methods give different numbers?
They estimate the same quantities in different ways. The average-and-range method uses ranges and tabulated constants; ANOVA uses sums of squares. Differences of a few percent are normal. If they differ a lot, look for an interaction or for an outlier in the range chart.
Why 6σ and not 5.15σ for the tolerance?
The current AIAG MSA manual (4th edition) uses 6 standard deviations, covering 99.73 % of a normal distribution. Older reports used 5.15σ (99 %). The %study variation and ndc do not depend on this choice.
How do I know the calculations are right?
The calculator reproduces the AIAG MSA reference example by both methods. The ANOVA, its p-values and the variance components were also checked against an independent implementation in Python (statsmodels) on several random studies, with and without interaction, with differences below one part in a billion.
Is my data uploaded?
No. Everything is calculated in your browser. The study is kept only in this browser’s local storage, and a share link carries the data inside the link itself.
Next steps: check the process with our capability calculator, or build an uncertainty budget, where the repeatability from this study is one of the contributions.
References
- AIAG, Measurement Systems Analysis (MSA) reference manual, 4th edition, 2010.
- NIST/SEMATECH e-Handbook of Statistical Methods, section 2.4 Gauge R & R studies.
- D. C. Montgomery, Introduction to Statistical Quality Control, chapter on gauge and measurement system capability studies.