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Crossed How-To¶

This walkthrough runs a complete crossed gage study: five parts, three operators, two measurements each. Every operator measures every part, which is what makes the study crossed.

The finished report says the measurement system is usable but not comfortable, so there is something to read rather than a clean pass.

The data is on this page rather than in a file to download. Press Copy for Excel, then paste it into a blank worksheet.

The data¶

Five machined shafts, measured to two decimal places in millimetres. Three operators each measured every shaft twice, in random order. The specification is 9.00 to 11.00 mm.

The layout is one row per measurement, with the part and the operator named in their own columns. This is the layout the MSA tools read.

Part	Operator	Diameter
1	A	9.97
1	A	9.93
1	B	9.93
1	B	9.98
1	C	10.05
1	C	10.09
2	A	10.45
2	A	10.43
2	B	10.42
2	B	10.35
2	C	10.46
2	C	10.45
3	A	9.60
3	A	9.49
3	B	9.56
3	B	9.60
3	C	9.70
3	C	9.63
4	A	10.19
4	A	10.14
4	B	10.23
4	B	10.13
4	C	10.26
4	C	10.19
5	A	9.70
5	A	9.80
5	B	9.74
5	B	9.86
5	C	9.82
5	C	9.92

Thirty rows in all: five parts, times three operators, times two measurements.

Look down the Diameter column before running anything. The parts are genuinely different from one another, spanning about 9.5 to 10.5, while the two measurements of any one part by any one operator agree to within about 0.1. That gap between the part spread and the measurement spread is what a gage study measures, and seeing it in the raw data first makes the report easier to read.

Steps¶

Put the data in Excel

Press Copy for Excel above the table. In Excel, open a blank worksheet, click cell A1, and press Ctrl+V. You should have headers in row 1 and data in rows 2 through 31.
Select the data and start the analysis

Select A1:C31, including the header row. From the Excel ribbon: QXL Stat Tools > MSA / Gage R&R > Crossed.
Assign the columns on the Data tab
- Tick Diameter under Measurement Columns.
- Tick Part under Part.
- Tick Operator under Operator (Optional).
Leave Reference (Optional) empty. This study has no known true values, so it reports variation but not bias.
Enter the specification limits

In the Specification Limits grid, on the Diameter row, type 9 under LSL and 11 under USL.
Leave the Options tab alone

The defaults are what this walkthrough uses: Automatic (balanced: EMS; unbalanced: REML) for the estimation method, Misclassification sweep charts on, and Part Variation is Not Representative of the Population clear.

The data are balanced, every part and operator combination holding exactly two measurements, so Automatic chooses expected mean squares.
Choose Finish

A new worksheet named MSA Crossed appears.

What the report says¶

Two analysis of variance tables, not one¶

The first thing to notice is that there are two tables, headed ANOVA (All Terms) and ANOVA (Terms used for the Gage R&R calculation).

That is interaction removal, and it runs by default on a crossed study even though the dialog offered no control over it. The Part × Operator interaction has a p-value of about 0.79, far above the 0.25 threshold, so the data give no evidence that the operators disagree more about some parts than about others. The term is dropped and its variation pooled into repeatability.

The first table marks it: the row reads Part × Operator (removed).

ANOVA (All Terms), the model as declared:

Source	DF	Adj SS	Adj MS	F	P
Part	4	2.50945	0.62736	411.16	<0.001
Operator	2	0.04533	0.02266	14.85	0.002
Part × Operator (removed)	8	0.01221	0.00153	0.56	0.793
Repeatability	15	0.04075	0.00272

ANOVA (Terms used for the Gage R&R calculation), after the interaction was pooled away:

Source	DF	Adj SS	Adj MS	F	P
Part	4	2.50945	0.62736	272.47	<0.001
Operator	2	0.04533	0.02266	9.84	<0.001
Repeatability	23	0.05296	0.00230

Repeatability picked up the interaction's 8 degrees of freedom and its sum of squares: 15 plus 8 is 23, and 0.04075 plus 0.01221 is 0.05296. Every component below is computed from the second table.

The Gage R&R table¶

Source	Variance	Std Dev	Study Var (6 x SD)	% Contribution	% Tolerance
Total Variation (TV)	0.108515	0.32942	1.97650	100.00%	98.83%
Total Gage R&R (GRR)	0.004339	0.06587	0.39521	4.00%	19.76%
Repeatability (EV)	0.002302	0.04798	0.28790	2.12%	14.40%
Reproducibility (AV)	0.002036	0.04512	0.27074	1.88%	13.54%
Operator	0.002036	0.04512	0.27074	1.88%	13.54%
Part to part (PV)	0.104177	0.32276	1.93659	96.00%	96.83%
Part	0.104177	0.32276	1.93659	96.00%	96.83%

Seven rows, not five. Under Reproducibility (AV) sits its one member term, Operator, and under Part to part (PV) sits its member term, Part. With only one term in each group the member row repeats its total exactly; in a study with more terms in a group, the members are what the total is the sum of.

and beneath it:


%GRR	20.00%
%EV	14.57%
%AV	13.70%
%PV	97.98%
P/Tol Ratio	0.1976
Number of Distinct Categories (ndc)	6 (CI: 1.507 to 20.29)

The distinct-categories cell carries its confidence interval alongside the count, because this study reports intervals. The interval is on the underlying continuous ratio, which is why its ends are not whole numbers.

The two percentages that disagree, and why¶

This report is a clear illustration of a trap. Read the Total Gage R&R row:

% Contribution is 4.00%
%GRR, which is percent study variation, is 20.00%

Both are right. % Contribution compares variances and %GRR compares standard deviations, and one is the square root of the other: the square root of 0.04 is 0.20. A measurement system accounting for 4 percent of the variance accounts for 20 percent of the standard deviation.

The two answer different questions. Note also that %GRR (20.00) and the Total Gage R&R % Tolerance (19.76) are close here but are not the same quantity: one divides by the total standard deviation, the other by the tolerance width. Their being nearly equal on this data is a coincidence of these numbers, not a rule.

Reproducibility is real, and small¶

The Operator p-value is 0.001 in the second table, so the operators do differ from one another. The difference is small, though: reproducibility is 1.88 percent of the variance, slightly less than repeatability's 2.12 percent.

So the two halves of the measurement error are about the same size here. Neither the equipment nor the operators dominates.

What the Stats Advisor says¶

Two verdicts are worth reading together on this report.

Total GR&R (% Total Std Dev) is 20.00, which falls in the 10 to 30 band, so the Advisor prints its marginal verdict in blue rather than its acceptable verdict in black.
Number of Distinct Categories (ndc) is 6, and the Advisor treats 5 or more as acceptable, so that verdict is black.

A study can sit on both sides at once, which is why the Advisor prints a separate verdict per statistic rather than one pass or fail for the whole study.

Things to try next¶

Clear the specification limits and run it again. The % Tolerance column, the P/Tol Ratio, the whole Probabilities of Misclassification block, the gage performance curve and the sweep charts all disappear, because every one of them is measured against the tolerance. Everything else is unchanged.
Choose Expected Mean Squares (EMS) explicitly instead of Automatic. The report is identical, because the data are balanced and that is what Automatic already chose. The Estimation method: row near the top of the sheet confirms which one ran.
Choose XbarR. The components are estimated from ranges instead, and the analysis of variance table, every confidence interval and the whole misclassification block disappear from the report.