Is there an extension to the F-test? I'm new to statistics but I come from a mathematical background. So while learning about the t-test, which compares sample means, and the f-test which compares variances is was wondering if there is an extension to the f-test (correct me if I'm wrong), similar to the ANOVA for the t-test?
ANOVA is "used to analyze the differences among means (from Wikipedia)". So, as far as I understand it, it is an extension to the t-test.
I did some research and found out about the Bartlett's test, which in my understanding would be an extension to the f-test, is that correct?
My confusion comes mainly from how the terms are used in some software. For instance g*power. Here the anova is found in the section of the f-tests, which doesnt make any sense to me...
 A: You seem to be asking two questions in one. First, how come ANOVA compares 3+ means using an F-test which is designed to compare 2 variances. Second, is there a test to compare 3+ variances in the same way ANOVA compares 3+ means. I'll answer both:

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*Although the main question answered by the ANOVA does generalize that of the t-test as you have well understood (do any of these (>2) distributions have different means?), the technical way to answer the question is through an F test. The reason being that the smaller the residual variance is compared to the original variance, the further apart the means must be (and vice versa).
However, the F-test only tells you whether or not there seems to be a difference between some means, but it doesn't tell you in any way which means are concerned (no matter how obvious you think it is by looking at the data. This is why ANOVA's F-test is usually completed by honest t-tests, with p-values corrected for the amount of tests:


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*you call these tests post-hoc tests if you're going to compare a lot if not all possible pairs of means


*or you have the option of the more crafty contrasts, in case you know exactly where to look from the start and you don't want to ruin your statistical power by adjusting p-values for a huge number of useless tests.


*Bartlett's test, as you mention, does exactly the job of testing the equality of variances in a number of populations. Levene's test does this job as well, and a discussion on which one to use under which conditions can be found here: Bartlett's test vs Levene's test
A: The F test for ANOVA uses the ratio of two quantities, which when the population means are equal (and the assumptions hold) will both be independent estimates of $\sigma^2$. The one on the numerator estimates it from variation in group means, the one on the denominator estimates it from within-group variation about the group means (which will be consistent for $\sigma^2$ even when the population means differ).
When $H_0$ is false the numerator is biased upward as an estimate of $\sigma^2$, since it's got the effect of the variation in population group-means (under the alternative, the statistic is distributed as noncentral F).
That is, it is still effectively using a test for equality of two variance estimates, but it's using that to test equality of population means.
