# Expected Mean Squares for a One-factor between-subject ANOVA (1 of 2)

As stated in another section, both the numerator and the denominator of the F ratio estimate the population variance when the null hypothesis is true. Since they are both unbiased estimates, the expected value of both MSB and MSE is σ². Symbolically, E[MSB] = E[MSE] = σ². This section covers the expected value of MSB and MSE when the null hypothesis is false.

The MSB is based upon the sample means: the greater the variance of the sample means, the greater the MSB. Therefore, when the null hypothesis is false and the population means are not equal, the expected value of MSB is greater than when the null hypothesis is true. It stands to reason that the more the population means differ from each other, the greater the expected value of MSB. Mathematical statisticians have derived the following formula for the expected value of MSB:

where σ² is the population variance, µi is the ith population mean, is the mean of the population means, n is the number of subjects in each group, and "a" is the number of population means.