Measure the same part three times and the readings differ; have someone else measure it and they differ a little more. How much of that difference comes from the parts themselves, and how much from the gauge and the people measuring? GR&R (gauge repeatability and reproducibility, also written GRR or Gage R&R) is the method that answers this: first calculate the variation caused by the measurement system itself, then compare it with the real differences between parts, or with the tolerance, to judge whether the numbers from this gauge can be used for process control and pass/fail decisions.
GR&R is one of the most common studies in measurement systems analysis (MSA). This page follows the method in the AIAG MSA Reference Manual, 4th edition: it first defines repeatability and reproducibility, then covers the conditions to confirm before starting, the two calculation methods (average and range, and analysis of variance, ANOVA), how to interpret the results, and common mistakes on the floor.
1. Repeatability and reproducibility: the two sources of measurement variation
Total measured variation can be split into two large parts, “caused by the measurement system” and “the parts themselves,” and the measurement system part splits further into repeatability and reproducibility:
| Item | English and symbol | What it looks at | Main sources |
|---|---|---|---|
| Repeatability | Repeatability, EV (equipment variation) | The difference when the same person, gauge, part and location are measured several times | The gauge itself: resolution, clamping and locating, wear, vibration and temperature |
| Reproducibility | Reproducibility, AV (appraiser variation) | The difference between the averages of different people measuring the same parts with the same gauge | People: technique, how readings are taken, measurement location, force applied |
| GR&R | GRR | Repeatability and reproducibility combined | GRR² = EV² + AV² |
| Part variation | Part variation, PV | The real differences between parts | The process |
| Total variation | Total variation, TV | All measured variation | TV² = GRR² + PV² |
Note that variations add as squares, not directly. For example, if EV and AV each account for 5% of total variation, the combined %GRR is about 7%, not 10%; the percentages on a report will not add up to exactly 100% either.
Where GR&R sits in measurement systems analysis
| Study | What it looks at | Belongs to |
|---|---|---|
| Bias | How far the measured average is from the reference value of a standard | Accuracy |
| Linearity | Whether bias is the same at different points of the measuring range | Accuracy |
| Stability | Whether the result drifts when the same standard is measured again some time later | Accuracy (over time) |
| Repeatability and reproducibility (GR&R) | The spread when measuring repeatedly and with different people | Precision |
GR&R looks just at precision (whether measurements are consistent), not accuracy (whether they are right). A gauge that reads 0.05 mm off across the board can still give a good GR&R as long as it is off by the same amount every time, so GR&R cannot replace calibration; bias, linearity and stability must be checked separately.
2. Before you start: confirm four things
1. Is the gauge resolution good enough?
The MSA manual’s rule of thumb is “10 to 1”: the gauge’s smallest reading (discrimination) should divide the tolerance or process variation into at least 10 parts. For example, with a tolerance of ±0.05 mm (full width 0.10 mm), the gauge should read to at least 0.01 mm. When resolution is not enough, repeated readings are often identical, the range is 0, and repeatability looks “very good” — when in fact the gauge cannot tell the difference.
2. Calibrate the gauge first, and check bias and stability
Before a GR&R, confirm the gauge is within its calibration period, and if needed measure a standard once to check bias. For a temperature measurement chain (sensor and thermocouple wire plus display), you can first spot-check it on site with a calibrator to confirm the whole chain is not reading off, then evaluate repeatability and reproducibility.
3. Parts must cover the actual range of process variation
Usually take 10 parts, chosen from different lots and times, so their sizes cover the range the process actually produces, ideally including parts near the upper and lower spec limits. If the parts are chosen too narrowly, part variation PV is underestimated and %GRR (to total variation) is inflated, so a perfectly adequate gauge can be judged unacceptable.
4. Appraisers, trials and blind measurement
Usually use 3 people who normally operate this gauge, each measuring every part 2 to 3 times; the MSA manual’s example is 10 parts × 3 appraisers × 3 trials, 90 readings in all. Cover the part numbers and shuffle the order each round, so appraisers do not know which part they are measuring and cannot see their previous readings; mark the measurement location on each part so that shape differences within the part are not counted as measurement variation.
| Item | Common practice | Used to estimate |
|---|---|---|
| Number of parts | 10 parts, covering the actual process range | Part variation PV |
| Appraisers | 2 to 3 people who normally operate the gauge | Reproducibility AV |
| Trials per person per part | 2 to 3 | Repeatability EV |
| Measurement order | Random each round, blind | Avoids memory and order effects |
| Measurement location | Mark the measurement point on the part | Avoids measuring shape differences within the part |
3. Method 1: Average and range
The average and range method estimates variation from the range (the maximum minus the minimum of a set of readings). It can be calculated in Excel and is a common method on the floor. The steps:
- For each appraiser and each part, calculate the range R of the repeated readings; calculate each appraiser’s average range, then average across all appraisers to get R̿ (read “R double bar,” the average of the average ranges).
- Check against the control limit: UCLR = R̿ × D4 (D4 = 3.27 for 2 trials per part, 2.58 for 3 trials). For any set of readings above the limit, find the cause first, then remeasure or remove it before calculating.
- Calculate the overall average of all readings for each appraiser, and subtract the smallest from the largest to get X̄DIFF.
- Calculate the average of each part (all appraisers, all trials), and subtract the smallest from the largest to get RP.
Formulas and constants (AIAG MSA, 4th edition)
| Item | Formula | Constants and notes |
|---|---|---|
| Repeatability EV | EV = R̿ × K1 | K1: 0.8862 for 2 trials per part; 0.5908 for 3 trials |
| Reproducibility AV | AV = √[(X̄DIFF × K2)² − EV² ÷ (n × r)] | K2: 0.7071 for 2 appraisers; 0.5231 for 3. n = number of parts, r = trials per part; if the value under the root is below 0, AV is 0 |
| GR&R | GRR = √(EV² + AV²) | — |
| Part variation PV | PV = RP × K3 | K3: 0.3146 for 10 parts |
| Total variation TV | TV = √(GRR² + PV²) | — |
| Percentages | %GRR = 100 × GRR ÷ TV | %EV, %AV and %PV calculated the same way |
| Number of distinct categories | ndc = 1.41 × (PV ÷ GRR) | Rounded down to a whole number |
The K constants in the table are the 4th edition values, and the resulting EV, AV and PV are in units of standard deviation. When comparing with old reports, first check which edition’s constants they used, so the values are comparable.
Worked example (hypothetical data)
A dimension with a tolerance of ±0.25 mm (full width 0.50 mm), 10 parts × 3 appraisers × 3 trials. After compiling: R̿ = 0.020 mm, X̄DIFF = 0.010 mm, RP = 0.400 mm. First confirm every range is below UCLR = 0.020 × 2.58 = 0.0516 mm, then continue:
| Item | Calculation | Result | Share of total variation |
|---|---|---|---|
| EV | 0.020 × 0.5908 | 0.0118 mm | 9.3% |
| AV | √[(0.010 × 0.5231)² − 0.0118² ÷ 30] | 0.0048 mm | 3.8% |
| GRR | √(0.0118² + 0.0048²) | 0.0127 mm | 10.1% |
| PV | 0.400 × 0.3146 | 0.1258 mm | 99.5% |
| TV | √(0.0127² + 0.1258²) | 0.1265 mm | — |
| ndc | 1.41 × 0.1258 ÷ 0.0127 | 13 | — |
| %GRR (to tolerance) | 100 × 0.0127 ÷ (0.50 ÷ 6) | 15.3% | — |
Interpretation: %GRR is 10.1% of total variation and 15.3% of tolerance, both in the 10%–30% band; ndc is 13, more than 5. EV (9.3%) is clearly larger than AV (3.8%), so the variation comes mainly from the gauge itself; improvement should start with clamping and locating and gauge resolution, not with more operator training.
4. Method 2: Analysis of variance (ANOVA)
ANOVA applies a two-way analysis of variance to the same data (a crossed design of parts × appraisers × trials), splitting the variation into four parts: parts, appraisers, the appraiser × part interaction, and repeatability:
| Source of variation | Variance estimate | Assigned to |
|---|---|---|
| Repeatability | MSE | EV |
| Appraisers | (MSA − MSAP) ÷ (n × r) | AV (reproducibility) |
| Appraiser × part interaction | (MSAP − MSE) ÷ r | AV (reproducibility) |
| Parts | (MSP − MSAP) ÷ (k × r) | PV |
MS is the mean square for each source, and k is the number of appraisers; estimates below 0 are taken as 0. Take the square root of each variance to get its standard deviation; GRR, TV, the percentages and ndc are then calculated the same way as in the average and range method.
ANOVA shows one thing more than the average and range method: the interaction. A large interaction means certain people measure certain parts particularly inconsistently — for example, for a part of a certain shape, some people habitually measure from the side and others from the top. The average and range method mixes this difference into the other terms, so it cannot be seen.
| Comparison | Average and range | ANOVA |
|---|---|---|
| Calculation | By hand or in Excel | Usually with statistical software |
| Interaction | Cannot be seen | Can be calculated separately |
| Missing values or unequal trials | Not suitable | More flexible |
| Use | Routine gauges, quick floor assessment | Finding causes, formal reports for customers |
Calculated from the same data, the two methods usually give close but not identical results; the report should state clearly which one was used.
5. How to interpret the results
| %GRR | General guideline in the AIAG MSA manual | Common action |
|---|---|---|
| Below 10% | Generally acceptable | Use as normal and reassess periodically |
| 10%–30% | May be acceptable depending on the application: judge by the importance of the measurement, gauge cost and rework or repair cost, and get customer agreement | Assess whether to improve the gauge or measurement method |
| Above 30% | Unacceptable | Find the cause, improve, then repeat the study |
ndc (number of distinct categories) indicates how many “measurably different” levels this measurement system can divide the parts into; the manual recommends at least 5. An ndc below 5 means the gauge can barely see the differences between parts, and when used for statistical process control (SPC), control charts will be distorted.
Should the denominator be total variation or tolerance?
When the gauge is used for process control and looking at process variation, use total variation TV as the denominator for %GRR; when the gauge is used to judge whether products pass or fail, look at GRR as a share of tolerance: %GRR (to tolerance) = 100 × GRR ÷ (tolerance ÷ 6). The two figures answer different questions; ideally list both on the report and state which is which.
Whether EV or AV is larger changes the direction of improvement
| Result | What it means | Check first |
|---|---|---|
| EV larger than AV | The gauge itself is not repeatable enough | Resolution, clamping and locating fixtures, gauge wear, vibration and temperature |
| AV larger than EV | Different people measure inconsistently | Work standards, how readings are taken, measurement location, force applied, training |
| Large interaction (ANOVA) | Certain people measure certain parts particularly inconsistently | The shape of specific parts, the definition of the measurement point |
Whether a result is acceptable in the end depends on your customer’s requirements and the makers’ specifications for the gauge and components; this page summarizes the manual’s general guidelines and calculations, and actual acceptance criteria should follow your company’s quality system procedures and customer requirements.
6. Common mistakes
- Parts chosen too narrowly: part variation PV is underestimated, so %GRR (to total variation) is inflated. Choose parts from different lots and times, covering the actual process range.
- Appraisers know which part they are measuring: they unconsciously “line up” with their previous reading, making repeatability look better than it is. Measure blind and shuffle the order each round.
- Calculating anyway with insufficient resolution: readings keep repeating and the range is 0, so repeatability looks “very good” when the gauge simply cannot tell the difference. Confirm 10 to 1 first, then do the GR&R.
- Treating GR&R as calibration: GR&R looks at spread, not bias; a gauge that reads off across the board can still pass GR&R. Calibration, bias and linearity must be done separately.
- Looking at one indicator alone: looking at %GRR to tolerance without ndc, or at total variation without tolerance, easily leads to misjudgment. The indicators answer different questions and should be read together.
- Applying a crossed design directly to destructive measurement: for tests such as pull, peel and cross-section, the part is destroyed after one measurement and cannot be measured repeatedly by everyone; use a nested design instead, or select parts from the same lot with very small differences to stand in as “the same part.”
- Spelling it “Guage”: the correct spelling is Gauge (also Gage in American usage); watch for this when searching and writing reports, as the misspelling will not find the material.
Standards and references
The calculations and interpretation guidelines in this article are compiled from the sources below; refer to the current version published by each issuing body.
- AIAG, Measurement Systems Analysis (MSA) Reference Manual, 4th Edition. Issuer: Automotive Industry Action Group (AIAG)
- NIST/SEMATECH e-Handbook of Statistical Methods: 2.4 Gauge R & R studies. Issuer: National Institute of Standards and Technology (NIST)
Related reading
These related articles on measurement error and evaluating measurement systems give a fuller picture together:
- “How to Measure ESD Values So They Count: Surface Resistance Measurement Steps and Common Mistakes (LDPE Bag Test)”
- “Where Thermocouple Wire Measurement Error Comes From | IEC 60584-1 Tolerance Classes and the Whole Measurement Chain”
- “Should You Bring In New Equipment? JIAHON Completes the Evaluation Report for You”
- “How JIAHON Solved Adhesive Height Inspection When Standard Equipment Couldn’t: A Full Record of a Real Development Project”
Related products
JIAHON supplies the measurement and calibration equipment mentioned in this article:
- ACL-800 weighted-probe surface resistance tester
- SIMCO FMX-004 static field meter
- Wickon SMT reflow profiler Q12 / Q24
- FLUKE 712B RTD calibrator
Trademarks: ACL, SIMCO, Wickon, FLUKE and the model names on this page are trademarks or model names of their owners, used here to identify the products. JIAHON is not the authorized distributor of ACL, SIMCO, Wickon or FLUKE in Taiwan.
From: JIAHON Engineering