# ANOVA Conceptual understanding [duplicate]

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.

• ANOVA compares the variances within the groups and the variance between the group means. If the group means are quite spread out relative to the data, that is evidence of different population means. However, the ANOVA F-test does not compare the variances of the original groups. I suggest the JBStatistics videos on ANOVA. He makes a point to describe what “within” and “between” mean and that ANOVA is most certainly a test of group means, not group variances.
– Dave
Commented Apr 17, 2020 at 6:48

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).