The Average & Range method is useful because its logic can be audited with a spreadsheet or even by hand: repeated ranges estimate equipment variation, operator averages estimate appraiser variation, and the range of part averages estimates the part signal.
This article deliberately owns the calculation trail. The separate worked acceptability resource uses a larger 90-reading crossed ANOVA study and focuses on the engineering decision.
1. Define the crossed study
Operators A and B each measure all five parts twice. Because every operator measures every part, the design is crossed. The parts span approximately 10.02 to 10.54 mm, while repeated readings remain close.
The compact design is appropriate for teaching the calculation. A production study must follow the applicable sampling, operator, randomization, customer, and internal requirements.
| Part | A · Trial 1 | A · Trial 2 | B · Trial 1 | B · Trial 2 | Part mean |
|---|---|---|---|---|---|
| 1 | 10.018A range 0.003 | 10.021 | 10.024B range 0.004 | 10.020 | 10.02075 |
| 2 | 10.146A range 0.004 | 10.150 | 10.153B range 0.004 | 10.149 | 10.14950 |
| 3 | 10.278A range 0.004 | 10.274 | 10.281B range 0.005 | 10.276 | 10.27725 |
| 4 | 10.407A range 0.005 | 10.412 | 10.415B range 0.005 | 10.410 | 10.41100 |
| 5 | 10.536A range 0.004 | 10.532 | 10.541B range 0.004 | 10.537 | 10.53650 |
Do not arrange the real measurement sequence in this convenient table order. Randomize or blind the parts where practical so operators cannot reproduce a remembered value. The table is for calculation and review after data collection.
The calculation also assumes that the same characteristic, instrument, fixture, datum, measurement method, and environmental condition apply to every reading. If Operator B uses a different fixture position or one part is measured after a temperature change, the spreadsheet will still return a neat answer—but the estimated components will no longer describe one consistent measurement system.
The five part means were intentionally selected to provide a visible part signal. That supports this teaching example, but part selection in a real study should reflect the range in which the measurement system must make decisions. The purpose is not to maximize ndc. The purpose is to test discrimination across a defensible operating range.
2. Identify the design constants
The Average & Range method uses constants linked to the number of trials, operators, and parts. These constants convert observed ranges into the selected study-variation convention.
| Constant | Design input | Value used | Role |
|---|---|---|---|
| K1 | 2 trials | 4.56 | Converts the average repeated range into Equipment Variation (EV). |
| K2 | 2 operators | 3.65 | Converts the range of operator averages into the preliminary appraiser component. |
| K3 | 5 parts | 2.21 | Converts the range of part averages into Part Variation (PV). |
K1 is tied to the number of repeated trials because the expected range changes with the number of observations. K2 is tied to the number of operator averages in the range, and K3 is tied to the number of part averages. In other words, the constants are not calibration factors for the instrument; they are statistical factors for the study design and selected spread convention.
Use the unrounded constants required by the approved worksheet when they are available. The values shown here are the rounded constants used by the prepared calculator. Small rounding differences can change the last displayed digits without changing the engineering interpretation. Large differences usually indicate a different method, design, or study-variation convention and should be reconciled explicitly.
Keep the constants with the method. A value copied from a worksheet for a different number of trials, operators, or parts will silently change the answer. The constants here follow the 5.15σ Average & Range convention used by this prepared preview example.
3. Calculate every variation component
Average the repeated ranges
For each operator–part cell, subtract the smaller repeated reading from the larger. The ten ranges are 0.003, 0.004, 0.004, 0.004, 0.004, 0.005, 0.005, 0.005, 0.004, and 0.004 mm.
Convert R̄ into Equipment Variation
Equipment Variation represents repeatability: the study spread attributed to repeated measurements by the same operator on the same part.
Calculate the operator-average range
Operator A’s overall average is 10.27740 mm. Operator B’s is 10.28060 mm. Their range is therefore 0.00320 mm.
Correct Appraiser Variation for repeatability
The raw operator-average component contains some repeatability. The correction subtracts the appropriate EV variance before taking the square root. Here, n is the number of parts and r is the number of trials.
Combine EV and AV into Total Gage R&R
Independent variation components combine through their variances. Adding EV and AV directly would overstate the result.
Estimate Part Variation
The largest part mean is 10.53650 mm and the smallest is 10.02075 mm, so the range of part averages is 0.51575 mm.
Combine measurement and part variation
Total Variation represents the combined study spread from the measurement system and the selected parts.
Convert the components to percentages
Each study-variation percentage uses TV as the denominator. Always retain the denominator name when reporting the result.
Calculate the number of distinct categories
ndc compares the part signal with the measurement noise. The conventional result is truncated to a whole number.
Why the AV correction can become zero
In some datasets, the quantity inside the AV square root is negative. That means the estimated operator-average spread is too small to separate from the repeatability already present in those averages. The usual worksheet treatment is to set the estimated AV component to zero rather than report an imaginary number. This does not prove that operators are identical; it means the study did not resolve a positive between-operator component with this method and sample.
Keep full precision until the report
Rounding EV, AV, or PV before the next calculation can accumulate error in GRR, TV, and ndc. Carry the available precision through the formulas and round only the displayed results. The same rule applies to part and operator averages: calculate them from the raw readings, not from visually rounded intermediate values copied from a report.
4. Review the same calculation as product output
The panel below is generated from the exact 20 readings above. It keeps the result table beside the part, operator, and component graphs so the arithmetic and the data pattern can be reviewed together. Hover over chart marks for values.
Gage R&R — Average & Range
2 operators × 5 parts × 2 trials · Study variation convention: 5.15σ
| Source | Study Variation | %Study Variation |
|---|---|---|
| Total Gage R&R | 0.021600 | 1.89% |
| Repeatability (EV) | 0.019152 | 1.68% |
| Reproducibility (AV) | 0.009987 | 0.88% |
| Part-to-Part (PV) | 1.139808 | 99.98% |
| Total Variation (TV) | 1.140012 | 100.00% |
Components of Variation
Operator Mean Comparison
Measurements by Part
The large part-to-part signal drives the high ndc and small %GRR. That is legitimate only if these five parts represent the range for which the measurement decision must work.
Read the component chart first
The Part-to-Part bar dominates, while EV, AV, and Total GRR remain close to the baseline. This is the visual form of the 1.9% result. It shows the balance between signal and noise, but it does not diagnose why EV is larger than AV. For that, return to the cell ranges and observe the measurement method.
Use the operator chart to locate the offset
Operator B’s overall mean is 0.0032 mm higher than Operator A’s. The difference is small relative to the selected part span, yet it is the raw evidence behind the AV calculation. In a real study, a consistent offset should still be checked against zeroing, force, datum interpretation, and fixture practice—especially when the characteristic tolerance is tight.
Use the part chart to verify separation
The four readings for each part cluster closely, and the clusters are widely separated. That explains the high ndc. If one part showed a much wider operator gap or repeated range than the others, the pooled summary could hide a localized technique or geometry problem. A larger ANOVA study and interaction chart are better suited to testing that pattern explicitly.
5. Interpret the result without over-claiming
What supports acceptability
Total GRR is 1.9% of the study variation, repeatability is small, the operator-average difference is small, and ndc is 74. Within this prepared sample, the measurement system clearly resolves the selected part differences.
What still needs verification
The compact example does not prove representative sampling, randomization, bias, linearity, stability, calibration, adequate resolution for the real decision, or compliance with a customer-specific study procedure.
The part range is the dominant influence. If the same gage were studied on five nearly identical parts, PV and ndc would fall while EV might remain almost unchanged. That would not necessarily mean the gage deteriorated; it would mean the study is asking it to discriminate within a narrower range.
This example reports percentage of study variation, not percentage of tolerance. If specification limits matter, calculate %Tolerance using the applicable spread convention and compare it with the engineering tolerance. A strong %Study Variation result cannot substitute for a missing specification comparison.
Do not compare 1.9% here with 6.93% in the larger ANOVA resource as if they were competing calculations. They use different datasets, study sizes, calculation methods, interaction treatment, and study-variation conventions. The method label belongs beside the number.
6. What to include in an auditable calculation
- Characteristic, unit, specification, equipment, resolution, fixture, and method.
- Operator, part, and trial counts plus the exact measurement order.
- All raw readings with part, operator, and trial identity.
- The constant table and its design basis.
- Cell ranges, operator averages, part averages, and intermediate totals.
- EV, AV, GRR, PV, TV, percentages, ndc, and rounding convention.
- Study limitations, investigation evidence, conclusion, and approval authority.
7. Common calculation mistakes
- Using the wrong K table: constants for three trials or ten parts cannot be reused for this 2 × 5 × 2 design.
- Adding EV and AV directly: variation components combine through the square root of summed variances.
- Skipping the AV correction: operator averages already contain some repeatability, so the raw operator range is not AV by itself.
- Using the raw-part range: PV here comes from the range of part averages, not the largest minus smallest individual measurement.
- Rounding intermediate results: premature rounding can change the reported percentages and ndc.
- Dropping the denominator label: 1.9% of study variation is not automatically 1.9% of tolerance.
- Comparing methods without naming them: Average & Range and crossed ANOVA can differ because they estimate components differently.
Continue from calculation to decision
Use the larger worked resource for crossed ANOVA, all 90 measurements, the Gage Evaluation table, six diagnostic charts, and the engineering decision. Use the fundamentals guide when you need the concepts behind repeatability, reproducibility, percentage columns, and ndc.
To change these 20 measurements and recalculate, open the free MSA preview. For study setup, crossed ANOVA, Average & Range, six diagnostic charts, and printable reporting in one workflow, continue to Mechatrovich MSA Studio.
References and method sources
- AIAG, Measurement Systems Analysis (MSA), 4th Edition — industry reference for variable measurement-system studies and interpretation.
- NIST/SEMATECH e-Handbook, Gauge R&R Studies — design and analysis considerations for production measurement systems.
- NIST, Analysis of Repeatability — graphical and numerical treatment of repeatability.