ANOVA

ANOVA works by partitioning total variance into a between-group component and a within-group component. The F ratio asks whether the variation between group means is large relative to the variation inside the groups. A significant F tells you that at least one group differs — it does not tell you which one.

That is what post-hoc tests are for: Tukey HSD when group sizes and variances are comparable, Games-Howell when variances are unequal. Report F, both degrees of freedom, the p-value and an effect size such as eta squared or partial eta squared, because significance alone says nothing about magnitude.

The assumptions are independent observations, approximately normal residuals and homogeneity of variance (checked with Levene's test). If homogeneity fails, Welch's ANOVA is the robust alternative; if the outcome is ordinal or badly non-normal, the Kruskal-Wallis test is the usual non-parametric substitute.