Analysis of variance

Analysis of variance tests the null hypothesis that all group means are equal by comparing the variance between groups to the variance within groups. If the between-group variance is substantially larger than the within-group variance, the null hypothesis is rejected. The test statistic follows an F-distribution under the null hypothesis, with degrees of freedom determined by the number of groups and total sample size.¹ ANOVA assumes independence of observations, normality of residuals, and homogeneity of variances, though it is robust to moderate violations when sample sizes are balanced.

One-way ANOVA compares a single factor across multiple groups, while two-way ANOVA examines two factors and their interaction. More complex designs include repeated measures ANOVA, mixed models, and analysis of covariance (ANCOVA), which incorporates continuous covariates. Nonparametric alternatives, such as the Kruskal–Wallis test, relax normality assumptions. Multivariate analysis of variance (MANOVA) extends ANOVA to multiple dependent variables simultaneously.²

ANOVA is a cornerstone of designed experiments, from agricultural field trials to clinical trials and industrial quality control. It is used to compare treatment effects, assess interactions, and estimate variance components. However, ANOVA only indicates that at least one group differs; post-hoc tests such as Tukey's HSD or Bonferroni correction are needed to identify which groups. Violations of assumptions can lead to inflated Type I error rates, and unbalanced designs require careful handling.³

ANOVA's origins trace to agricultural experiments at Rothamsted Experimental Station, where Fisher developed the method to analyze crop yields. The term "variance" itself was coined by Fisher in 1918. A lesser-known extension is the Welch's ANOVA, which adjusts for unequal variances. ANOVA is also foundational to the analysis of variance components, used in quantitative genetics to estimate heritability. In computer science, ANOVA is applied to benchmark algorithm performance, and in ecology, it helps analyze species diversity across habitats.⁴

F-distribution

A continuous probability distribution used as the null distribution for ANOVA test statistics.

Post-hoc test

A follow-up test performed after ANOVA to identify which specific group means differ.

Covariate

A continuous variable that is controlled for in ANCOVA to reduce error variance.