Gauge R&R explained: how to run, calculate and read a study

A gauge R&R (gage repeatability and reproducibility) study estimates how much of the variation you see in your measurements comes from the measurement system itself. It splits that variation into repeatability (the same operator measuring the same part several times) and reproducibility (different operators measuring the same parts), and compares the total with the variation between parts or with the tolerance. The usual study has 10 parts, 3 operators and 2 or 3 trials. Under the AIAG criteria, a %GRR below 10 % is generally acceptable, 10–30 % may be acceptable, above 30 % is not, and the study should resolve at least 5 distinct categories.

What gauge R&R measures

Every measurement you record is the part’s real value plus some measurement error. If the error is large compared with how much parts really differ, you can’t separate good parts from bad ones, and your SPC charts mostly show gauge noise. A gauge R&R study puts numbers on that error. In the average and range method used by the AIAG manual it reports five quantities, each as a standard deviation (AIAG MSA 4th ed., as reproduced by SPC for Excel, 2015):

Variances add; standard deviations don’t. That one fact explains most of the confusion around gauge R&R percentages, and we’ll come back to it in the worked example.

The words come from metrology, with a twist. The international vocabulary (VIM) defines repeatability as precision under the same procedure, operators, measuring system, conditions and location over a short period, and reproducibility as precision across different locations, operators and measuring systems (VIM 2.20–2.25). Precision itself is the closeness of agreement between repeated measurements, usually expressed as a standard deviation, variance or coefficient of variation, and it is not the same thing as accuracy (VIM 2.15). In an automotive gauge R&R, “reproducibility” is narrower: usually only the operator changes (SPC for Excel, 2015; Minitab, methods and formulas).

Repeatability vs reproducibility

A quick example. One inspector measures the same shaft diameter three times with a bore gauge and gets 25.012, 25.018 and 25.009 mm: that scatter is repeatability. Three inspectors each measure it three times and their averages are 25.013, 25.021 and 25.006 mm: the difference between those averages is reproducibility.

They point to different fixes:

How to run a gauge R&R study

The common design is a crossed study: every operator measures every part in every trial. The example in the AIAG manual uses 10 parts, 3 operators and 3 trials, 90 readings in total (SPC for Excel, 2015; Minitab example). Two or three trials and two or three operators are common: those are the cases covered by the published average and range constants (Ermer, 2006, table 1). The manual’s example is a design, not a rule; your customer may specify its own.

1. 1

   Calibrate and check the gauge first

   Fix bias and calibration problems before you study precision (Ermer, 2006). A gauge R&R cannot see a gauge that reads consistently wrong: shifting every reading by the same amount leaves all the variance components unchanged.

2. 2

   Choose parts that cover the process range

   Take parts from normal production that represent its expected variation, not 10 consecutive parts and not parts of different types. Number them where the operators can't see the number.

3. 3

   Choose the operators

   Use people who actually do this measurement, with the gauge, fixture and procedure used in production.

4. 4

   Randomize and measure blind

   Each operator measures all parts in a new random order in each trial, without seeing earlier readings. Record values to the full resolution of the gauge.

5. 5

   Analyze

   Use ANOVA (preferred) or the average and range method, and look at the components, not only the headline %GRR.

6. 6

   Decide and document

   Compare with the criteria your customer uses, and record parts, operators, gauge ID, date and conclusions.

   Fails: diagnose before you buy a new gauge

Two design choices matter more than any formula:

• Part selection. If the parts barely differ, part variation is small, total variation is small, and %GRR (which divides by total variation) looks bad even with an excellent gauge. The reverse is also true: parts far outside the normal range make any gauge look good. This follows directly from how the percentages are calculated (Minitab, methods and formulas).
• Randomization and blind measurement. Operators who remember a part’s earlier reading, or measure it twice in a row, make repeatability look better than it is (SPC for Excel, 2015).

In a calibration or test lab with few artifacts and operators, the NIST e-Handbook suggests more than two check standards, all operators if there are only a few (or a random sample of more than two from a large group), and measurements on more than two days if there is only one operator (NIST e-Handbook 2.4.2).

How to calculate gauge R&R: a worked example

We’ll use the data of the AIAG MSA example (4th edition) as published independently by SPC for Excel and Minitab: 10 parts, operators A, B and C, 3 trials, and a tolerance of 8 units. All results below were recomputed with our own code and match both publications to the last published decimal (apart from small rounding differences between the tables and the text of the SPC for Excel article itself). You can load the same data in the gauge R&R calculator.

Average and range method (by hand)

The average and range method (ARM) turns ranges into standard deviations with constants. In the AIAG 4th edition convention the constants give 1σ values: K1 = 0.5908 for 3 trials, K2 = 0.5231 for 3 operators and K3 = 0.3146 for 10 parts (SPC for Excel, 2015). They are simply 1/d2 and 1/d2* from the statistics of ranges of normal samples, which is why they change with the number of trials, operators and parts (Ermer, 2006).

1. Average range of the 30 operator-part cells: R̿ = 0.3417 (operators: 0.184, 0.513, 0.328).
2. Repeatability: EV = R̿ × K1 = 0.3417 × 0.5908 = 0.202.
3. Operator averages 0.190, 0.068 and −0.254, so their range is X̄diff = 0.4447.
4. Reproducibility: AV = √((X̄diff × K2)² − EV² / (parts × trials)) = √((0.4447 × 0.5231)² − 0.202² / 30) = 0.230. The subtraction removes the share of repeatability that leaks into the operator averages.
5. Gauge R&R: GRR = √(EV² + AV²) = 0.306.
6. Part variation: the part averages range over Rp = 3.511, so PV = Rp × K3 = 1.105.
7. Total variation: TV = √(GRR² + PV²) = 1.146.
8. Percentages: %EV = 17.61 %, %AV = 20.04 %, %GRR = 26.68 %, %PV = 96.4 %.

These match the published values (SPC for Excel, 2015, tables 2 and 7).

ANOVA method (preferred)

The analysis of variance (ANOVA) method uses all 90 readings instead of ranges, and it can estimate one more thing the ARM can’t: the operator × part interaction, which shows up when operators disagree on some parts but not on others (SPC for Excel, 2015; Ermer, 2006). It is the method we recommend and the default in our calculator.

For this data the interaction has F = 0.434 and p = 0.974, far above the usual 0.05 cutoff, so it is pooled into repeatability and the model is refitted without it (Minitab example). The 0.05 cutoff is a software setting, not a universal rule (Minitab lets you change it, and some tools keep the interaction in every study), so state the rule you used. Any variance component that comes out negative is set to zero (Minitab, methods and formulas; NIST e-Handbook 2.4.4).

The variance components then give:

Published values: Minitab example; %Contribution also in SPC for Excel (2015).

Same data, three different “answers”

Nothing here is wrong. The two methods estimate the same components in different ways, and 7.76 % and 27.86 % are the same result expressed on different scales: 0.2786² ≈ 0.0776. Judged on the standard-deviation percentages, as Minitab applies the AIAG guideline, the study is marginal by either method. SPC for Excel (2015) notes that the AIAG manual itself calls the ANOVA result for these data acceptable because the %GRR is below 10 %, which is exactly the mix of scales described below. The ndc difference shows how sensitive a truncated number can be: 5.09 becomes 5, and 4.86 becomes 4.

How to read the results: %GRR, %Contribution and %Tolerance

Each percentage divides the gauge R&R by something different, and each answers a different question:

Formulas: Minitab, methods and formulas (6 × SD by default). Criteria: AIAG MSA 4th ed., paraphrased from the table reproduced by SPC for Excel (2015, citing p. 78) and from Minitab (interpreting the results, which applies the 10 % guideline to %Study Var and %Tolerance).

Three practical rules:

• Use the 10 / 30 % bands on standard-deviation metrics, as Minitab does with %Study Var and %Tolerance (Minitab, interpreting the results). Applying them to %Contribution makes a gauge look far better than it is: 10 % of the variance is about 32 % of the standard deviation. That is the root of the claim, repeated online, that “10 % in ANOVA equals 30 % in average and range”: the methods agree; the columns differ.
• Pick the denominator that matches the gauge’s job. A gauge used to accept parts at final inspection is judged against the tolerance. A gauge feeding an SPC chart has to see process variation, so %Study Variation matters. In the example, the gauge uses 22.68 % of the tolerance but 27.86 % of the process variation.
• Check the multiplier. Older material and some software use 5.15σ (99 %) instead of 6σ for study variation and %Tolerance (Ermer, 2006; SAS/QC). %Study Variation doesn’t change; %Tolerance does.

Number of distinct categories (ndc)

The ndc estimates how many groups of parts the measurement system can reliably tell apart within the process variation:

ndc = 1.41 × PV / GRR, truncated to a whole number (1.41 ≈ √2; if the result is below 1, it is reported as 1, as Minitab does).

The AIAG asks for ndc ≥ 5 (Minitab example). In the example, ANOVA gives 1.41 × 1.04233 / 0.30237 = 4.86, so ndc = 4; the average and range method gives 1.41 × 1.105 / 0.306 = 5.09, so ndc = 5 (truncation rule: Minitab, number of distinct categories; the ndc of 4 is published in the Minitab example, the average and range value is our own calculation). An ndc of 1 means the gauge can’t distinguish parts at all within this process; ndc ≥ 5 corresponds to a %Study Variation of roughly 27 % or less.

What to do when a gauge R&R fails

Don’t start by buying a gauge. The components tell you where the problem is:

The NIST e-Handbook adds a useful warning: gauges can respond differently to the working environment (temperature, humidity, operators, wear), and a gauge study is how those biases come to light (NIST e-Handbook 2.4.5). Once you’ve changed something, repeat the study on the same parts if you can, so before and after are comparable.

Ready to try your own data? The gauge R&R calculator runs ANOVA and average and range side by side and shows each convention it uses.

Gauge R&R vs calibration vs measurement uncertainty

These three answer different questions, and you need all of them.

A calibration tells you how the gauge relates to the true value in a near-ideal setting, but not whether it behaves the same way in the working environment. That is the job of the gauge study (NIST e-Handbook 2.4.5). Calibrate first, then study precision: a gauge R&R can’t see a gauge that is consistently off (Ermer, 2006).

Gauge studies also feed uncertainty budgets. The standard deviations from a well-run study can serve as Type A contributions (repeatability, operator effects), but if the measurement process drifts, the study no longer describes it; NIST suggests running a gauge study before and after the measurements it is meant to describe, so you can tell whether the process changed in between (NIST e-Handbook 2.4.6). A full budget then adds what a gauge R&R never sees, such as the calibration uncertainty of the gauge and temperature effects. See measurement uncertainty explained.

Other gauge studies

A crossed gauge R&R is the most common study, not the only one:

• Type 1 study: one reference part of known value measured repeatedly, to check bias and repeatability against the tolerance before a full study. It’s part of the VDA 5 sequence and is a useful diagnostic when an R&R fails (Dietrich and Radeck, 2014).
• No operator influence: for automated systems such as a CMM running a program, reproducibility between operators doesn’t apply; studies without the operator factor are used instead (Dietrich and Radeck, 2014).
• Attribute studies: go/no-go gauges and visual inspection give pass/fail answers, not numbers, so they need an attribute agreement study instead, which checks whether appraisers agree with themselves, with each other and with a known standard (Minitab, attribute agreement analysis).
• Destructive tests: when a part can be measured only once, operators can’t re-measure the same part, and the crossed design above doesn’t apply. A nested design, where each operator measures different parts, is the usual alternative (Minitab, nested gage R&R).

ASTM also publishes a general MSA guide, ASTM E2782-24, for measurement systems used on manufactured parts. For the broader picture of bias, linearity and stability, see measurement system analysis.

Gauge R&R study checklist

Print this and keep it with the study record.