ANOVA Conceptual understanding I am trying to understand ANOVA.
When we look at the null hypothesis we are trying to make a statement about the means. But what we indeed calculate is the variances, and make statements about the null hypothesis based on the F-test. 
My question is how does calculating variances help in understanding about the mean, or rather make a statement about the means? Kindly clarify. 
Thanks,
G Ravi Kiran.
 A: The denominator of the F statistics, $(IJ-I)^{-1}\sum_{i,j}(y_{ij} - y_{i.})^2$ in the simple 1-way ANOVA case, is indeed an estimate of the residual variance; but the numerator is not really a variance: you have something like $\sum_i(y_{i.}-y_{..})^2$ scaled (in the simple 1-way ANOVA), which is the sum of squared deviations of the sample group means to the global mean. What ANOVA does is to check whether this sum of squares is "large" as compared to the residuals variance, using the distribution in the null case (equal mean groups).
A: I've found visualizations like this to be particularly helpful:
https://demonstrations.wolfram.com/VisualANOVA/
While we are technically comparing variances, you can imagine the "mean-squared between" measure of variance as a measure of differences in means among more than 2 groups. The ANOVA reduces to a t-test when there are only 2 groups. The mean-squared between (the numerator of the F ratio) reduces to a simple difference in means and the mean-squared within (the denominator of the F ratio) reduces to the pooled variance, just like you'll see in a t-test.
