ANOVAs

ANOVA, short for Analysis of Variance, is a statistical method used to compare the means of two or more groups. It tests whether observed differences in sample means reflect real differences in population means or arise from random variation. The standard one-way ANOVA partitions the total variability of the data into between-group variability and within-group variability. The F statistic is the ratio of the mean square between groups to the mean square within groups; large F values indicate that group means differ more than would be expected by chance. The null hypothesis states that all group means are equal; the alternative asserts that at least one mean is different. For a one-way design with k groups and N observations, degrees of freedom are k−1 for between groups and N−k for within groups.

Several extensions exist: two-way ANOVA examines two independent factors and their interaction; repeated measures ANOVA analyzes related observations on the same subjects; mixed-design ANOVA combines both. Assumptions include normality of residuals, homogeneity of variances (equal variances across groups), and independence of observations; sphericity is an additional concern for repeated measures. When assumptions fail, nonparametric alternatives such as Kruskal-Wallis or Friedman tests may be used, or data can be transformed. After a significant ANOVA, post hoc tests (e.g., Tukey's HSD, Bonferroni) help identify which means differ. Effect sizes such as eta squared, partial eta squared, or omega squared quantify the proportion of variance explained by the factor(s).

Reporting typically includes F values with corresponding degrees of freedom and p-values, plus effect size. ANOVA is widely used across fields to assess whether group membership or experimental conditions influence a dependent variable.