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Gage Evaluation¶

This page defines every column and summary figure the Gage R&R table reports, once the variance components exist. It applies to every study and every estimation method: the components differ in how they were estimated, but what is done with them here does not.

The components themselves, and the rule that groups them, are on Variance Components.

Notation¶

Term	Description
\(\sigma^2_t, \sigma_t\)	the variance component of term \(t\) and its standard deviation
\(\sigma_{\text{gage}}, \sigma_{\text{part}}, \sigma_{\text{total}}\)	the total gage, part to part and total standard deviations
\(k\)	the study variation multiplier, the dialog's Study variation multiplier (k):, default \(6\). On an MSA page \(k\) is never a control limit sigma multiplier
\(SV_t\)	the study variation of term \(t\)
\(LSL, USL\)	the lower and upper specification limits
\(\bar{x}\)	the grand mean
\(\sigma_{\text{hist}}\)	a supplied historical process standard deviation
\(ndc\)	the number of distinct categories

Standard deviation¶

Each reported component is also reported as a standard deviation, which is simply its square root:

\[ \sigma_t = \sqrt{\sigma^2_t} \]

Because the floor at zero is applied to the variance before the square root is taken, this is always defined.

Study variation¶

The study variation of a component is its standard deviation multiplied by \(k\), the study variation multiplier:

\[ SV_t = k\,\sigma_t \]

\(k\) is set by Study variation multiplier (k): on the Extended options group and defaults to 6, which spans a range of plus and minus three standard deviations. A value of zero or below selects the default rather than being used.

Percent contribution¶

Percent contribution compares variances:

\[ \%\text{Contribution}_t = 100 \times \frac{\sigma^2_t}{\sigma^2_{\text{total}}} \]

Because the parts are variances and variances add, these percentages sum to 100 across the components.

Percent study variation¶

Percent study variation compares standard deviations:

\[ \%\text{Study Variation}_t = 100 \times \frac{\sigma_t}{\sigma_{\text{total}}} \]

The multiplier \(k\) cancels, so this is the same number whether it is formed from standard deviations or from study variations.

On the report this quantity is not a column. It appears as four named rows beneath the results table, one for each aggregate, under the abbreviations the gage literature uses:

Row	Which component's percent study variation
`%GRR`	Total Gage R&R
`%EV`	Repeatability, the equipment variation
`%AV`	Reproducibility, the appraiser variation
`%PV`	Part to part, the part variation

A row whose component the study did not produce reports NA. The member components of reproducibility and of part to part have no row of their own here, so their percent study variation is not reported at all even though the aggregate ones are.

These two percentage columns answer different questions and do not agree, and neither is wrong. Percent contribution is a ratio of variances; percent study variation is a ratio of standard deviations, which is the square root of the same ratio. A component contributing 9 percent of the variance contributes 30 percent of the standard deviation. Percent study variation does not sum to 100 across the components, because standard deviations do not add.

Tolerance width¶

The tolerance width depends on how many specification limits were supplied:

\[ \text{Tolerance} = \begin{cases} USL - LSL & \text{both limits supplied} \\ 2\,\lvert \bar{x} - L \rvert & \text{exactly one limit } L \text{ supplied} \\ \text{not reported} & \text{no limit supplied} \end{cases} \]

With one limit the width is twice the distance from the grand mean \(\bar{x}\) to that limit, which treats the supplied limit as one side of a symmetric tolerance about the study mean. With no limit the tolerance width and everything derived from it are not reported.

Percent of tolerance¶

Percent of tolerance compares a component's study variation against the tolerance width:

\[ \%\text{Tolerance}_t = 100 \times \frac{SV_t}{\text{Tolerance}} = 100 \times \frac{k\,\sigma_t}{\text{Tolerance}} \]

Unlike the two percentage columns above, this one does move with \(k\). It is not reported when no specification limit was supplied.

Percent of historical process¶

When a historical standard deviation is supplied, each component is also reported against it:

\[ \%\text{Historical Process}_t = 100 \times \frac{\sigma_t}{\sigma_{\text{hist}}} \]

This column is not reported when no historical value was supplied. Note that it uses the historical value directly as the denominator, with no subtraction; that is different from how the historical value enters the misclassification calculation.

Precision to tolerance ratio¶

The precision to tolerance ratio compares the whole measurement system against the tolerance:

\[ P/T = \frac{k\,\sigma_{\text{gage}}}{\text{Tolerance}} \]

It is not reported when no specification limit was supplied.

Number of distinct categories¶

The number of distinct categories estimates how many separate groups of parts the measurement system can tell apart:

\[ ndc = \left\lfloor 1.41 \times \frac{\sigma_{\text{part}}}{\sigma_{\text{gage}}} \right\rfloor, \qquad ndc \geq 1 \]

The ratio is truncated toward zero and then floored at 1, so the reported count is never zero and never negative. A ratio large enough to exceed the integer range is clamped to the largest reportable integer.

Two properties of this number are worth stating because they are easy to assume otherwise:

There is exactly one of it. It is always computed from the estimated part to part and total gage standard deviations, even when a historical standard deviation was supplied. No historical variant of the count exists.
It is a truncation, so a reported count of 4 covers a range of underlying ratios. Where a confidence interval on the count is reported, that interval is on the underlying continuous ratio, not on the truncated integer.

When it is not reported¶

The count is not reported at all when the total gage standard deviation is exactly zero, because the ratio it is a truncation of would divide by zero. The report prints a sentence in place of the number rather than a blank cell or a zero, since a zero would read as a measured count of zero categories.

The test is on total gage R&R, not on repeatability alone. A study in which every repeated measurement repeats its own value exactly has no repeatability variation, but if its operators still disagree it has a real gage variation, and its count is reported.