QM BenchQuality engineering tools

Gage R&R simulator: see what your measurement system really does

Build a gauge with the sliders: repeatability, operator differences, interaction, bias, linearity and resolution. The simulator runs a real crossed Gage R&R study (ANOVA method, the same calculation as the Gage R&R calculator) and shows what the measurement error does on the shop floor: good parts rejected, bad parts shipped and a Cpk that is lower than the process deserves. Start with the ten guided lessons.

%GRR of tolerance
13.5 %
theory 14.4 %
%GRR of study var.
14.5 %
theory 20.2 %
ndc
9
theory 6, need ≥ 5
Verdict (AIAG)
Conditional
< 10 % / 10–30 % / > 30 %
Variance components (ANOVA method, 10 parts × 3 operators × 3 trials = 90 readings)
Total Gage R&R
14.5 % study var.
13.5 % tolerance
Repeatability
14.1 % study var.
13.1 % tolerance
Reproducibility
3.2 % study var.
3.0 % tolerance
Part-to-part
98.9 % study var.
92.1 % tolerance

Interaction is not significant and pooled into repeatability. Operator × part: 0.0 % study var.

Measurements by part and operatorOperator AOperator BOperator C
9.8279.92010.01310.10610.199P1P2P3P4P5P6P7P8P9P10
100 % inspection: true size vs measured value (500 parts)
true size (mm) · red dashed = LSL / USLmeasured · blue = acceptance limits
correct acceptgood part rejectedbad part acceptedcorrect reject
What the measurement error costs (expected, all parts)
Bad parts accepted
4
ppm · customer risk
Good parts rejected
13
ppm · your cost
True processAs measured
σ0.07000.0715
Cp1.431.40
Cpk1.431.40
Out of tolerance1827 rejected

Acceptance limits 9.700 – 10.300 mm (tolerance 9.700 – 10.300 mm).

Formulas with the current numbers
  • σ gauge = √(EV² + AV² + INT²) = √(0.012² + 0.008² + 0.000²) = 0.0144
  • %GRR tolerance = 6·σ gauge / (USL − LSL) = 6·0.0144 / 0.60 = 14.4 %
  • %GRR study variation = σ gauge / √(σ gauge² + σ parts²) = 20.2 %
  • ndc = 1.41 · σ parts / σ gauge = 1.41 · 0.070 / 0.0144 = 6.8
  • σ observed = √(σ process² + σ gauge²) = 0.0715, Cp observed = 0.60 / (6·0.0715) = 1.40
  • Resolution = 0.2 % of tolerance (rule of ten: ≤ 10 %)

The study above estimates these values from 90 readings, so it scatters around the theory. That is why AIAG asks for enough parts, operators and trials.

Ten guided lessons

  1. What %GRR tells you. Read a Gage R&R result: %GRR, ndc and the verdict.
  2. Operators measure differently. Recognise reproducibility (appraiser variation) and its causes.
  3. The gauge itself is noisy. Recognise repeatability (equipment variation).
  4. Resolution and ndc. See the rule of ten and the number of distinct categories.
  5. Wrong part selection. See why the study parts must cover the process variation.
  6. Bias: invisible to Gage R&R. Understand that a Gage R&R does not detect a systematic offset.
  7. Linearity. See a bias that changes over the measuring range.
  8. Measurement error lowers Cpk. See that the observed variation = process + measurement.
  9. False accept, false reject and guard bands. Understand the two decision errors at the tolerance limits.
  10. Operator × part interaction. Recognise an interaction in the ANOVA.

How the simulator works

Every part has a true size drawn from the process (mean and process σ). A reading is the true size plus the errors of the measurement system:

reading = true size + bias + linearity·(size − nominal) + operator effect + operator×part effect + repeatability noise, rounded to the resolution

The study design follows AIAG MSA: by default 10 parts, 3 operators and 3 trials. The parts are spread over the selected range of the process, and the ANOVA separates repeatability, reproducibility, the interaction and the part-to-part variation. Because a study is a sample, the estimate scatters around the true value. Press Run a new study a few times to see how much.

Reading the result

IndicatorFormulaRule of thumb (AIAG)
%GRR of tolerance6·σGRR / (USL − LSL)< 10 % good, 10–30 % conditional, > 30 % not acceptable
%GRR of study variationσGRR / σtotalsame limits; depends on part selection
ndc1.41 · σparts / σGRR≥ 5
Resolutionstep / tolerance≤ 10 % (rule of ten)

Five things the simulator makes visible

  • Repeatability vs reproducibility point to different fixes. High repeatability: gauge, fixture, clamping, environment. High reproducibility: method, measuring point, training.
  • Part selection changes the verdict. Ten nearly identical parts make a good gauge look bad in %GRR of study variation and ndc, while %GRR of tolerance stays the same.
  • Bias is invisible to Gage R&R. Set a bias and %GRR does not move, but the decision panel shows good parts rejected on one side and bad parts accepted on the other. Check bias with a type 1 gauge study and linearity and bias study.
  • The gauge eats your Cpk. σ observed = √(σ process² + σ gauge²). With a %GRR of tolerance of 30 %, a process with true Cp 1.67 is reported at about 1.49; at 50 % only at about 1.28.
  • Every gauge makes decision errors near the limits. Guard bands reduce shipped defects but scrap more good parts. The lasting fix is a better measurement system.

Worked example

Tolerance 10.00 ± 0.30 mm, process σ 0.07 mm, repeatability 0.012 mm, reproducibility 0.008 mm (the default settings). σ gauge = √(0.012² + 0.008²) = 0.0144 mm. %GRR of tolerance = 6 · 0.0144 / 0.6 = 14.4 %: conditionally acceptable. ndc = 1.41 · 0.07 / 0.0144 = 6.8, so 6 distinct categories. True Cp = 0.6 / (6 · 0.07) = 1.43; observed Cp = 0.6 / (6 · √(0.07² + 0.0144²)) = 1.40. Try halving the repeatability and see which numbers move.

Frequently asked questions

What is a Gage R&R simulator?

It generates measurements from a gauge whose errors you control (repeatability, reproducibility, interaction, bias, linearity and resolution), runs a crossed Gage R&R study on them with the ANOVA method and shows the result next to the true values. You see cause and effect, which a real study never shows you.

What is an acceptable %GRR?

AIAG MSA: below 10 % the measurement system is generally acceptable, 10 to 30 % may be acceptable depending on the application, the cost of the gauge and the risk, above 30 % it is not acceptable. The number of distinct categories (ndc) should be 5 or more.

What is the difference between repeatability and reproducibility?

Repeatability (equipment variation, EV) is the variation when one operator measures the same part several times with the same gauge. Reproducibility (appraiser variation, AV) is the difference between operators. High EV points to the gauge, fixture or environment; high AV points to the method and training.

Why is %GRR of study variation different from %GRR of tolerance?

%GRR of tolerance compares 6σ of the gauge with the tolerance width, so it tells you whether the gauge can judge good and bad parts. %GRR of study variation compares the gauge with the variation of the parts in the study, so it depends on how you selected the parts. Narrow part selection makes it look worse.

Does a Gage R&R detect bias?

No. A Gage R&R measures variation only. A gauge that reads 0.05 mm too high on every part can still have an excellent %GRR. Use a type 1 gauge study (Cg, Cgk), a bias study against a reference master, a linearity study and regular calibration.

How does measurement error affect Cpk?

The observed variance is the process variance plus the gauge variance: σ observed² = σ process² + σ gauge². A poor gauge therefore lowers the observed Cp and Cpk even when the parts are good. That is why MSA comes before capability studies in PPAP.

What are guard bands?

Acceptance limits moved inside the tolerance by about the measurement uncertainty. They reduce the number of bad parts accepted (customer risk) at the cost of rejecting more good parts (producer risk). The simulator shows both numbers as you move the guard band.

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