Distribution of F ratio
Continuing our analysis in the same way as was done for the multi-group normal model, the mean residual and explained sums of squares are the sums of squares divided by their degrees of freedom. We define an F ratio,
The calculations are again organised in an analysis of variance table.
Simulation illustrating distribution of F
The diagram allows samples to be selected from a normal linear model with β1 = 0 and σ = 10.
Click Take sample several times to build up the distribution of the F ratio. Observe that it has an extremely skew distribution.
The theoretical F distribution is also shown in grey. The tail of the F distribution is longest when the sample size is small — the difference is not clear in the diagram here, but is important.
Note that the underlying slope is zero in this simulation — the explanatory variable does not affect Y. If the slope was non-zero, the F ratio's distribution would have a higher mean.