ANOVA for Experimental Research
One-way analysis of variance tests whether $k$ group means can be treated as equal by comparing between-group variation with within-group variation.
Let $Y_{ij}$ be observation $j$ in group $i$, $\bar Y_i$ the mean of group $i$, $\bar Y$ the grand mean, $n_i$ the group size, and $N=\sum_i n_i$.
Sums of squares
Total variation is
\[SST=\sum_i\sum_j(Y_{ij}-\bar Y)^2.\]Between-group variation is
\[SSB=\sum_i n_i(\bar Y_i-\bar Y)^2.\]Within-group error variation is
\[SSE=\sum_i\sum_j(Y_{ij}-\bar Y_i)^2.\]The decomposition is
\[SST=SSB+SSE.\]This identity separates variation explained by group membership from residual variation within groups.
Mean squares and test statistic
The mean squares divide each component by its degrees of freedom:
\[MSB=\frac{SSB}{k-1}, \qquad MSE=\frac{SSE}{N-k}.\]The omnibus test statistic is
\[F=\frac{MSB}{MSE}.\]Under the classical null hypothesis
\[H_0:\mu_1=\mu_2=\cdots=\mu_k,\]and the standard independent, normal, equal-variance model, $F\sim F_{k-1,N-k}$. A large value indicates that group means differ more than within-group noise would normally explain.
What rejection means
Rejecting the omnibus null shows that at least one mean differs. It does not identify which means differ. Planned contrasts or multiplicity-adjusted post-hoc comparisons answer that follow-up question.
For two groups, the equal-variance one-way ANOVA test is equivalent to the pooled two-sample test:
\[F=t^2.\]Connection to regression
ANOVA can be written as a regression on indicator variables. This makes Indicator Variables and Interactions the bridge from group comparisons to factorial designs and covariate adjustment.