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Math Details¶

Every number on a Type 1 Gage Study report, and the rule that decides when it is reported at all.

A Type 1 study has no variance components and no analysis of variance, so none of the MSA Math pages apply to it. Everything it computes is on this page.

Notation¶

Term	Description
\(n\)	the number of usable measurements
\(x_1 \dots x_n\)	the usable measurements, in worksheet order
\(\bar{x}\)	their mean
\(S\)	their standard deviation
\(k\)	Number of Standard Deviations:, the study variation multiplier, default \(6\)
\(K\)	Percentage of Tolerance:, a fraction, default \(0.2\)
\(R\)	Reference Value, the part's known true value
\(\text{Res}\)	Resolution, the gage's resolution
\(\alpha\)	Alpha for the bias test:, default \(0.05\)
\(LSL, USL\)	the lower and upper specification limits
\(T\)	the tolerance width

What counts as a usable measurement¶

A value in the selected range that reads as a finite number is usable. Anything else is dropped and counted, and the count is printed as Dropped Values when it is not zero. \(n\) is the count of what survived, not the height of the range.

Summary statistics¶

\[ \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i , \qquad S = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}} \]

The study variation is the standard deviation stretched by the multiplier:

\[ SV = k\,S \]

reported under a label built from \(k\) itself, so it reads 6*StDev at the default and follows whatever you type.

The tolerance width¶

\[ T = USL - LSL \]

and only when both limits were supplied. One limit alone does not make a tolerance and nothing is substituted for the missing one, so every quantity below that divides by \(T\) is reported as N/A.

Two limits that are supplied but do not order, so that \(USL \le LSL\), are also treated as no tolerance, and the sheet says which of the two cases it met.

Bias and its test¶

\[ b = \bar{x} - R \]

A positive bias means the gage reads high. The bias is tested against zero with a one sample t test on the measurements:

\[ t = \frac{\bar{x} - R}{S / \sqrt{n}} , \qquad \nu = n - 1 \]

The T printed on the sheet is signed, so it carries the direction of the bias. The p value is two-sided and is taken from the magnitude of \(t\):

\[ p = 2\left(1 - F_{t,\,\nu}\!\left(|t|\right)\right) \]

The bias is called significant when \(p < \alpha\), and that is the only thing the comparison does: it turns the P-Value cell red and bold. No pass or fail verdict is printed. Without a reference value, \(b\), \(t\) and \(p\) are all N/A.

Capability¶

Four figures, each of which needs the tolerance width.

\[ C_g = \frac{K\,T}{SV} \]

\(C_g\) asks whether the gage's spread fits inside the reference span, which is the fraction \(K\) of the tolerance. It ignores where the readings sit.

\[ C_{gk} = \frac{\dfrac{K}{2}T - |b|}{\dfrac{SV}{2}} \]

\(C_{gk}\) asks the same question of the half of the span the readings are actually nearer to, so a bias pushes it below \(C_g\). A bias larger than the half span makes it negative, and that negative value is reported as the formula gives it rather than being floored at zero.

\[ \%\text{Repeatability} = \frac{SV}{T} \]

\[ \%\text{Repeatability and Bias} = \frac{K}{C_{gk}} \]

The second of those is left as N/A when \(C_{gk}\) is exactly zero, rather than dividing by it.

Both are stored as fractions and displayed by Excel as percentages, so a % Repeatability of \(0.149\) shows as \(14.90\%\).

The rules of thumb on the sheet

Cell notes on these four numbers give the customary acceptance levels: \(C_g\) and \(C_{gk}\) above \(1.33\), and both percentages below \(15\%\). They are notes on the cells, not tests the product performs, and nothing in the report changes because a figure falls short of one.

Resolution¶

\[ \%\text{Resolution} = \frac{\text{Res}}{T} , \qquad \text{acceptable when } \%\text{Resolution} \le 0.10 \]

Resolution Acceptable reads Yes or No on that comparison, and N/A when there is no percent of resolution to judge, which happens when either the resolution or the tolerance is missing. The threshold is the AIAG rule of tens, and the sheet says so in a note.

The run chart¶

The centre of the chart is the reference value when there is one, and the mean otherwise:

\[ c = \begin{cases} R & \text{when a reference value was supplied} \\ \bar{x} & \text{otherwise} \end{cases} \]

The band around it is the reference span, centred:

\[ \text{Upper} = c + \frac{K}{2}T , \qquad \text{Lower} = c - \frac{K}{2}T \]

so the two lines are \(K\,T\) apart, which is \(20\%\) of the tolerance at the default. Neither line is drawn when there is no usable tolerance.

The Y axis is scaled to hold both the readings and the band:

\[ y_{\min} = \min\!\left(\min_i x_i,\ \text{Lower}\right) , \qquad y_{\max} = \max\!\left(\max_i x_i,\ \text{Upper}\right) \]

and the axis is then rounded outwards to convenient numbers, so it never cuts off a point that is outside the band. Without a usable tolerance the range is the data's own minimum and maximum.

What stops a study¶

The options are checked before the measurements are, in this order: the percentage of tolerance, the number of standard deviations, alpha, then the resolution. Each must be a finite number, the first two strictly positive, alpha strictly between \(0\) and \(1\), and the resolution strictly positive if it was supplied at all.

The measurements are then checked: at least two usable values, and some variation among them. A set of readings that differ but not by enough for the sum of squared deviations to be computed at double precision is refused rather than reported as zero variation.

Every sentence and its trigger is listed on Options.

References¶

The acceptance levels quoted in the cell notes, the rule of tens for resolution, and the form of the Type 1 study itself follow:

Automotive Industry Action Group (2010). Measurement Systems Analysis Reference Manual, 4th edition. AIAG.
Dietrich, E. and Schulze, A. (2011). Measurement Process Qualification: Gage Acceptance and Measurement Uncertainty According to Current Standards, 5th edition. Hanser.