Measurement System Analysis (MSA / Gage R&R)

Formulas and fundamentals

Type 1: gauge capability (Cg/Cgk)

Type 1 (gauge capability) evaluates repeated measurements of a calibrated standard against a share of the tolerance. From the mean x_barg and the sample standard deviation sg (with n−1) of the n repeated measurements follow the pure-scatter capability index and the index reduced by the magnitude of the systematic deviation:

The tolerance share p is typically 0.20, the reference convention L selects between ±3s (L = 6, DE/Bosch/VDA default, 99.73 %) and ±2s (L = 4). In addition the resolution (requirement ≤ 5 %) is checked and a two-sided t-test with n−1 degrees of freedom decides whether the systematic deviation is significant:

Types 2 and 3: Gage R&R (average-range method)

Type 2 (Gage R&R with appraiser) splits the total variation by the average-range method into repeatability and reproducibility. From the mean range R_bar follows the equipment variation, from the range of appraiser means X_DIFF the reproducibility, together the measurement-system variation:

The part variation follows from the range of part means Rp, the total variation is:

The K factors are reciprocals of the constants d2 and d2* and depend on the number of measurements r, appraisers k and parts n. In Type 3 the appraiser term drops out:

Types 2 and 3: Gage R&R by ANOVA

Alternatively, the two-way analysis of variance (ANOVA) decomposes the variation via sums of squares and additionally detects the part × appraiser interaction. With n parts, k appraisers and r repetitions the sums of squares and mean squares are:

The variance components follow from the mean squares; a non-significant interaction (F test, p ≥ 0.25) is pooled into repeatability per the AIAG convention:

This yields the same variation components as the average-range method, but statistically more efficient:

For Type 3 the model reduces to the one-way ANOVA across the parts: EV = √(MS_within) and s²_part = (MS_between − MS_within)/r. The rating via %GRR, %Contribution and ndc is identical for both methods; the method is switchable per study in the calculator.

Rating: %GRR, %Contribution and ndc

The rating uses the ratio of measurement variation to total variation or to the tolerance:

%Contribution works with variance shares and sums with the other sources to 100 %. The resolution gives the number of distinct categories (truncated, not rounded):

A measurement system is capable at %GRR ≤ 10 %, conditionally capable up to 30 %, above that not capable; additionally ndc ≥ 5 is required.

Worked example

All sample data sets of the calculator describe the same part: a bolt diameter Ø 10 ±0.1 mm, that is T = 0.20 mm with USL 10.1 mm and LSL 9.9 mm, measured with a micrometer of 0.001 mm display increment. The same bolt is assessed from four angles: against the setting standard (type 1), by three appraisers by hand (type 2), on an optical measuring cell without an operator (type 3) and as a complete uncertainty budget (VDA 5 budget). The figures below appear as soon as you click Load sample data in the calculator.

Type 1: The micrometer measures the setting standard x_m = 10.000 mm 25 times, tolerance T = 0.20 mm, resolution RE = 0.001 mm, L = 6, p = 0.20. From x_barg = 10.00016 mm and sg = 0.001106 mm follow bias = 0.00016 mm, Cg = (0.2·0.20)/(6·0.001106) = 6.03 and Cgk = (0.1·0.20 − 0.00016)/(3·0.001106) = 5.98. The resolution is sufficient at %RE = 0.5 %; the bias t-value 0.72 stays below the critical value 2.06 (n − 1 = 24 degrees of freedom), so the systematic deviation of 0.16 µm is not significant and needs no correction. The gauge is capable by a wide margin: the repeatability spread amounts to about one display increment, and the assigned spread 0.2·T = 0.04 mm is roughly six times 6·sg.

Type 2: Ten of these bolts are measured three times each by three appraisers with the same micrometer. This gives R_bar = 0.00223 mm, X_DIFF = 0.00180 mm and Rp = 0.09578 mm; with K1 = 0.5908, K2 = 0.5231 and K3 = 0.3146 the components are EV = 0.00132 mm, AV = 0.00091 mm, GRR = 0.00160 mm, PV = 0.03013 mm and TV = 0.03017 mm. That yields %GRR (study var) = 5.3 %, %GRR (tolerance) = 4.8 % and ndc = 26, so the measurement system is capable. At %EV = 4.4 % against %AV = 3.0 % the repeatability dominates slightly; the appraiser influence is present but small. Switched to ANOVA the result is 5.2 % (study var), 4.9 % (tolerance) and ndc = 26; the appraiser × part interaction is not significant at p = 0.289 and is pooled into the repeatability.

Type 3: The same ten bolts pass three times through an optical measuring cell, without an operator. Without an appraiser share GRR = EV = 0.00112 mm, together with PV = 0.02999 mm and TV = 0.03001 mm, hence %GRR (study var) = 3.7 %, %GRR (tolerance) = 3.4 % and ndc = 37; ANOVA gives 3.5 %, 3.3 % and ndc = 39. The cell separates the bolts far better than the manual station because the appraiser share is gone: a typical result for optical or fully automatic measuring cells.

VDA 5 budget: For the same bolt the uncertainty budget of the manual measuring station is drawn up, T = 0.20 mm and k = 2. The measurement system contributes the calibration certificate U_CAL = 0.002 mm (u = U_CAL/2 = 0.001 mm), the resolution RE = 0.001 mm (u = RE/√12 = 0.000289 mm), the repeatability at the standard u_EVR = 0.003 mm and an uncorrected bias of 0.001 mm (u = |bias|/√3 = 0.000577 mm). Summed in quadrature these give u_MS = 0.00323 mm, U_MS = 0.00645 mm and Q_MS = 2·U_MS/T = 6.5 % against the 15 % limit. The measurement process adds the object influence u_OBJ = 0.002 mm and the temperature u_T = 0.001 mm: u_MP = 0.00393 mm, U_MP = 0.00785 mm and Q_MP = 7.9 % against the 30 % limit. Both figures sit well inside the green range; the largest single contribution is u_EVR, the repeatability, not the calibration certificate. The budget is set with a reserve: u_EVR = 0.003 mm and the bias of 0.001 mm lie above the values measured in type 1, sg = 0.001106 mm and 0.00016 mm. The Import type-1 study button replaces the assumption with the measured values.

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