Analysis of Variance

Analysis of variance (ANOVA) tests the null hypothesis that the means of $k\ge 2$ groups are all equal, by comparing between-group variability (how much group means differ from the overall grand mean) to within-group variability (how much individual observations differ from their own group's mean), via the F-statistic $F=\frac{M{S}_{\text{between}}}{M{S}_{\text{within}}}$, where each mean square is a sum of squares divided by its degrees of freedom. Under the null hypothesis of equal group means, $F$ follows an F distribution, a large F-statistic indicates the between-group variability is larger than within-group noise alone would explain, evidence that at least one group mean genuinely differs. ANOVA generalizes directly to more complex designs, factorial (two-way and higher) ANOVA for multiple categorical factors, repeated measures ANOVA for correlated observations, and ANCOVA for models incorporating a continuous covariate alongside categorical group membership.

• ANOVA is exactly equivalent to a two-sample t-test in the special case of exactly two groups, F = t², this equivalence is a useful mental check: ANOVA isn't a fundamentally different tool from the t-test, it's the natural generalization of the same mean-comparison logic to more than two groups.
• A significant overall ANOVA F-test tells you that at least one group mean differs from the others, but not which one(s), this is exactly why a significant ANOVA is typically followed by post-hoc pairwise comparison tests (like Tukey's HSD or Scheffe's test), which identify specifically which group pairs differ while controlling the overall false-positive rate across all the pairwise comparisons.
• ANOVA's decomposition of total variability into between-group and within-group components traces back to Ronald Fisher's work at Rothamsted Experimental Station in the 1920s, developed specifically for analyzing agricultural field trials with multiple treatment groups, the same origin as much of modern experimental design theory.