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I've been digging into the formula for one way ANOVA test and I understand the we obtain the results from the ANOVA analysis with:

Group Sum of Squares/Total Sum of Squares

To see how much variance is explained. However, I also read that:

Total Sum of Squares = Error Sum of Squares + Group Sum of Squares

In that case, how could an F statistic be greater than 1? Shouldn't it always be a fraction of 1?

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The F is not GroupSS/TotalSS.

If it were you'd be right that it shouldn't exceed 1 ... but it isn't. You should re-read your sources (and preferably find several different sources).

In ordinary one-way ANOVA the overall F is the group mean square over the error mean square. Each mean square is the relevant sum of squares divided by the corresponding degrees of freedom.

If there's no group effects (they're all one population) then the F should be typically close to 1 (the numerator and denominator mean squares should both be unbiased estimates of $\sigma^2_\text{error}$). When there are group effects the expected group mean square includes a term related to the variation in group-means.

As a result (since there's usually at least some group effect), the F is typically greater than 1.

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    $\begingroup$ +1 A while back I went through the manual process of getting F here. It could possibly be of help provided R is not a barrier. $\endgroup$ – Antoni Parellada Oct 9 '16 at 23:10

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