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MSA

These pages hold the mathematics the Measurement System Analysis tools share. Each one is included inline on the tool pages that use it, so a tool's own Math Details page carries the equations you need without a detour, and these pages are where the full treatment lives.

The Crossed, Nested and Extended studies share nearly all of this. Where a page applies to only one of them, it says so at the top.

Page	What it covers
Variance Components	The model every study fits, the rule that sorts the components into repeatability, reproducibility and part to part, the five aggregates, the floor at zero, and the fixed term substitute
Gage Evaluation	Every reported column: standard deviation, study variation, the two percentage families, tolerance width, percent of tolerance, the precision to tolerance ratio, and the number of distinct categories
Expected Mean Squares	The expected mean square system, its coefficients balanced and unbalanced, the method of moments solve, the F tests and their synthesized denominators, and the interaction removal loop
Restricted Maximum Likelihood	What REML maximises, which terms enter the fit, the non-negativity constraint, the reported diagnostics, and the model it cannot fit
XbarR Method	The range based method in full, and the three things it cannot produce. Crossed studies only
Probabilities of Misclassification	The five reported probabilities, the historical standard deviation substitution, the 8 sigma rule, and the mean shift sweep
Bias and Linearity	Bias and its t test, the linearity fit of bias on reference value, linearity and percent linearity, and the acceptance verdict

How the three estimation methods relate

All three answer the same question and report the same set of quantities. They differ in how they get the variance components, and in what they cannot produce:

	XbarR	Expected Mean Squares	REML
where the estimates come from	ranges	the analysis of variance table	the whole data set at once
study types	Crossed only	any	any
analysis of variance table	no	yes	yes
can report a negative estimate floored to zero	reproducibility only	yes	never
confidence intervals	none	balanced data only	balanced and unbalanced
misclassification block	none	yes	yes

Once the components exist, everything downstream is the same whichever method produced them: the grouping rule, the aggregates, and every column on the report. That is why Variance Components and Gage Evaluation are written without reference to the method.