On average, what value is expected for the F-ratio if the null hypothesis is true?

From Google:

If the null hypothesis is true, then the F ratio should be approximately one since MSB and MSE should be about the same. If the ratio is much larger than one, then it is likely that MSB is estimating a larger quantity than is MSE and that the null hypothesis is false.

Well, if the null hypothesis is true, the expected value for the F-ratio is around "F-unny." Just like a clown at a party, it likes to play around and entertain us. But remember, it's always important to take hypothesis testing seriously and not rely solely on humor.

The F-ratio, also known as the F-statistic, measures the ratio of two variances or mean square values. When the null hypothesis is true, meaning that the population means are equal, the expected value for the F-ratio is 1. In other words, if the null hypothesis is true, we expect the F-ratio to be close to 1 on average.

To determine the expected value of the F-ratio when the null hypothesis is true, we need to consider the distribution of the F statistic.

The F statistic follows an F-distribution with two degrees of freedom: one for the numerator (numerator degrees of freedom, DFnum) and one for the denominator (denominator degrees of freedom, DFden).

In the case of an F-test, the null hypothesis states that there is no significant difference between the group means or no significant relationship between the variables being compared. Thus, when the null hypothesis is true, the F-ratio should be close to 1.

Mathematically, the expected value of the F-ratio, denoted as E(F), can be calculated as:

E(F) = (DFden / (DFden - 2))

Since the null hypothesis assumes no significant difference, the numerator degrees of freedom (DFnum) is typically 1.

Therefore, if the null hypothesis is true, the expected value of the F-ratio would be 1.