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# F-ratio

Explanations > Social ResearchAnalysis > F-ratio

## Description

The F-ratio is a test statistic for multiple independent variables. It is used in ANOVA calculations and calculated as:

F-ratio = MSM / MSR

... where MS = SS / df
SS = Sum of the Squares
df = degrees of freedom

Subscripted M means 'Model' and indicates the expected systematic variance. This is often measured as between-measures variation, and the subscript B is consequently often used here.

Subscripted R means 'Residual' and indicates the random, unsystematic variance. This is measured as within-measures variance, and the subscript W is consequently often used.

F can also be calculated with the Pearson correlation coefficient, r:

F = r2 / (1 - r2)(n - 2)

## Discussion

A Test Statistic is generally calculated as the ratio of intended, systematic variance divided by the undesirable and unexpected unsystematic variance.

Test statistic = systematic variance / unsystematic variance

This is typically measured as the sum of squares, SS, then this is:

Test statistic = SSM / SSR

However, with multiple variables, there are multiple sums of squares. Therefore the F-ratio takes the mean of these.

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