Cómo se calcula un Gauge R y R

Los datos del estudio

This is a crossed study: every operador measures every part, y each combination is repeated. Three operadores, ten piezas, three trials each gives 90 mediciones. "Crossed" is what makes it possible to separate operador effects from part effects - if each operador measured a different set of piezas, the two influences could never be untangled.

The characteristic is an outside diameter with a specification of 24.980 to 25.020 mm, so the tolerance is 0.040 mm. All values are in millimetres.

Paso 1: sumas de cuadrados

ANOVA begins by asking how much of the total spread in those 90 readings can be attributed to each possible cause. It does this by measuring how far each group average sits from the overall average, squaring those distances, y weighting each by how many readings the group contains.

Four sources are separated: differences between operadores, differences between piezas, the interacción between them, y the leftover scatter within each operador-part cell.

The within-cell term is the important one conceptually: it is the only source that involves no averaging. It is the raw scatter you get when the same operador measures the same part repeatedly, which is exactly the definition of repeatability.

Paso 2: la tabla ANOVA

Each sum of squares is divided by its grados de libertad to give a cuadrado medio - an average squared deviation per independent piece of information. Degrees of freedom are a-1 for operadores, b-1 for piezas, their product for the interacción, y a*b*(n-1) for the within-cell term.

Read the p-values carefully. Piezas differ enormously, which is exactly what you want - it means the sample spans a real range of the process. Operadors also differ significativamente (p = 0.0011), which points to a reproducibility problem worth investigating.

The interacción is not significativo (p = 0.5837). That is good news: the operadores disagree by a roughly constant amount across all piezas rather than disagreeing unpredictably on particular piezas. A constant offset usually means technique or training; a significativo interacción usually means certain piezas are physically awkward to measure.

Paso 3: componentes de varianza

Mean squares are not themselves the variances you want. Each cuadrado medio contains a mixture of the effect you are after plus the noise beneath it, so the components must be unpicked by subtraction. These are the expected-mean-square relationships.

Reproducibilidad combines the operador component y the interacción component, because both describe variation introduced by the people rather than by the gauge. Taking square roots converts each variance into a desviación estándar in millimetres.

GRR is the square root of EV squared plus AV squared, y TV is the square root of GRR squared plus PV squared. Confirm it yourself: 0.01014 plus 0.00533 is 0.01547, but GRR is 0.01145. The components combine as a right triangle, not by addition.

Step 4: the 5.15 multiplier

Styard deviations are converted into "study variation" by multiplying by 5.15, which spans 99 percent of a normal distribución. GaugeConnection uses 5.15, consistent with the AIAG MSA manual.

Some software uses 6 sigma instead, spanning 99.73 percent. The choice does not change which component dominates, y because the multiplier appears in both the numerator y the denominator it cancels out of %GRR when the basis is study variation. It does not cancel when the basis is tolerance.

Step 5: percentages y ndc

Each study variation figure is divided by a reference. GaugeConnection uses the tolerance when one is entered, y total variation otherwise. Both are shown here from the same study so you can see how much the choice matters.

At 28.63 percent of tolerance this sistema de medición falls in the conditionally acceptable by - usable, but only with justification, y not for a safety or regulatory characteristic. Note that the percentages down a column do not sum to 100: they are ratios of desviación estándars, y desviación estándars do not add.

That disagreement is the most useful thing on this page. A study can look survivable on %GRR y still be unable to distinguish piezas finely enough to support process control. Always read %GRR y ndc together; if they disagree, the more conservative one governs.

When a variance component comes out negative

The component formulas involve subtracting one cuadrado medio from another, y nothing prevents the result from being negative. A negative variance is impossible in reality, so it is truncated to zero. GaugeConnection does this, y so does every styard implementation.

A negative estimate is a signal, not an error. It normally means the true component is very close to zero y sampling noise pushed the estimate below it - most often seen for the interacción term when operadores genuinely behave consistently.

It becomes a concern when it appears on the part component, which implies the piezas in the study were too similar to distinguish. That is a study design problem, not a gauge problem, y the fix is to re-select piezas spanning the real process range rather than to reinterpret the output.