Dr. Ray L. Winstead
IUP Professor of Biology

A Summary of the Logic of the F Test in ANOVA

1. MSwithin is an estimate of the population variance based upon the deviation of scores about their respective group means. It is not influenced by treatment effects.

 2. MSbetween (= MSamong) is also an estimate of the population variance if the null hypothesis is true. It is based upon the deviations of group means about the grand mean. Since it is influenced by any treatment effects that exist in the population, it is only an estimate of the same population variance if those treatment effects are assumed to be zero, that is, if the null hypothesis is true.

 3. Since the two variance estimates are independent and since the logic of hypothesis testing demands that the null hypothesis be tentatively assumed true, the ratio of these two variance estimates is distributed as F:

                                                    F = (Msbetween) / (Mswithin)

 4. Since under conditions of the null hypothesis the two mean squares are estimating the same population value, this ratio should approach a value of 1.0 in the long run. The observed (calculated) value of F is compared to the sampling distribution of such ratios to determine the probability that such an F value could be obtained merely by sampling "error".

 5. If the observed F ratio is very large such that the probability is quite small that an F of this size should be obtained merely by chance, then perhaps the assumption of the null hypothesis of no treatment effects was not appropriate. If there were treatment effects, MSbetween would be sensitive to them but MSwithin would not. Therefore, an improbably large F value probably means that treatment effects, in fact, do exist in the population and the null hypothesis should be rejected.

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Dr. Ray L. Winstead
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