How to calculate gage R&R with the average and range method
The method is the one in the AIAG Measurement Systems Analysis (MSA) manual, 4th edition. With n parts, r trials and operators A, B and C:
Repeatability: EV = R̄̄ × K1, where R̄̄ is the average of every operator’s range on every part
Reproducibility: AV = √((X̄diff × K2)² − EV² ÷ (n × r)), where X̄diff is the highest minus the lowest operator average. If the value under the root is negative, AV = 0
Gage R&R: GRR = √(EV² + AV²)
Part variation: PV = Rp × K3, where Rp is the highest minus the lowest part average
Total variation: TV = √(GRR² + PV²)
% of total variation: %GRR = 100 × GRR ÷ TV (same for EV, AV and PV)
% of tolerance: %Tol = 100 × 6 × GRR ÷ (USL − LSL)
Distinct categories: ndc = 1.41 × PV ÷ GRR, rounded down
The factors depend on the size of the study:
| Factor | Depends on | Values |
|---|---|---|
| K1 | trials | 2: 0.8862 · 3: 0.5908 |
| K2 | operators | 2: 0.7071 · 3: 0.5231 |
| K3 | parts | 2: 0.7071 · 3: 0.5231 · 4: 0.4467 · 5: 0.4030 · 6: 0.3742 · 7: 0.3534 · 8: 0.3375 · 9: 0.3249 · 10: 0.3146 |
First, check the range chart: every range (operator, part) should sit below UCL = D4 × R̄̄, with D4 = 3.267 for 2 trials and 2.574 for 3. A range above the limit is a reading that went wrong: find the cause and measure again.
Worked example
The data already in the calculator: a micrometer measuring a shaft diameter of 25.00 ± 0.10 mm (tolerance 0.2 mm), 10 parts, 3 operators (A, B and C) and 3 trials.
- Average range per operator: R̄A = 0.0101, R̄B = 0.0152 and R̄C = 0.0118 mm. Their average: R̄̄ = 0.01237 mm
- Range chart: UCL = 2.574 × 0.01237 = 0.0318 mm. The largest range is 0.029 mm: none out
- EV = 0.01237 × 0.5908 = 0.00731 mm
- Operator averages: A = 25.0051, B = 25.0099 and C = 24.9999 mm, so X̄diff = 0.0100 mm
- AV = √((0.0100 × 0.5231)² − 0.00731² ÷ (10 × 3)) = √(0.00002736 − 0.00000178) = 0.00506 mm
- GRR = √(0.00731² + 0.00506²) = 0.00889 mm
- Part averages: the highest is part 9 (25.0832) and the lowest part 5 (24.9354), so Rp = 0.1478 mm and PV = 0.1478 × 0.3146 = 0.04649 mm
- TV = √(0.00889² + 0.04649²) = 0.04733 mm
| Source | Std. deviation (mm) | % TV | % tolerance |
|---|---|---|---|
| Repeatability (EV) | 0.007306 | 15.4% | 21.9% |
| Reproducibility (AV) | 0.005058 | 10.7% | 15.2% |
| Gage R&R (GRR) | 0.008886 | 18.8% | 26.7% |
| Part variation (PV) | 0.04649 | 98.2% | 139.5% |
And ndc = 1.41 × 0.04649 ÷ 0.00889 = 7.4, which gives 7.
Result: at 18.8% of total variation and 26.7% of tolerance, the micrometer may be acceptable, depending on the application. An ndc of 7 shows it tells the parts apart well. Most of the R&R is repeatability, meaning the gage itself and the way it is used, more than differences between operators.
How to read the result
- %GRR under 10%: the measurement system is acceptable.
- 10% to 30%: may be acceptable, depending on how important the characteristic is, the cost of the gage and the cost of a wrong call.
- Over 30%: not acceptable. The gage or the method needs to improve.
- ndc of 5 or more: the system can tell parts apart within the process variation.
- Repeatability (EV) larger than reproducibility (AV): the problem is the gage, the fixturing or the method. Maintenance, a better gage or a holding fixture usually help.
- Reproducibility (AV) larger: operators measure differently. Training and a clearer measuring instruction usually help.
A high R&R is not only about the gage, but a calibrated gage is where it starts. To check a certificate, use the calibration acceptance check, and to build the uncertainty of a measurement, the measurement uncertainty calculator. Once the measurement system passes, the next step is process capability with the Cpk calculator.
Common mistakes
- Picking parts that are too alike. The 10 parts have to span the real process variation. With near identical parts, PV shrinks, %GRR goes up and ndc drops, even with a good gage.
- Measuring in the same order every time, or letting operators see earlier readings. The order has to be random and blind.
- Ignoring the range chart. One range above the limit inflates EV. Repeat that reading before drawing conclusions.
- Comparing % of total variation with % of tolerance as if they were the same thing. They answer different questions.
- Mixing 5.15σ and 6σ. %Tol with 5.15σ comes out about 14% lower. Always state which convention you used.
Sources
- AIAG, Measurement Systems Analysis (MSA), 4th edition, 2010: average and range method, K1, K2 and K3 factors and acceptance guidelines.
- AIAG, Statistical Process Control (SPC), 2nd edition: the D4 constant of the range chart.