Analysis of variance: a test of whether three or more group means are all equal. It compares the variation between the group means with the variation within the groups; a large ratio, the F statistic, says at least one mean differs, but not which.
Card 1 of 276. Answer: Analysis of variance: a test of whether three or more group means are all equal. It compares the variation between the group means with the variation within the groups; a large ratio, the F statistic, says at least one mean differs, but not which.
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One-way ANOVA splits the total variation into a between-groups part and a within-groups part. Each is divided by its degrees of freedom to give a mean square, and F = MS_between / MS_within. If all the means are equal, F should be near 1; it follows an F distribution under that null hypothesis. Ronald Fisher developed the method in the 1920s for agricultural experiments.
Example: three groups of five scores with means 5, 7 and 6, and the same spread inside each group. Between groups, SS = 5 × (1² + 1² + 0²) = 10 on 2 degrees of freedom, a mean square of 5. Within groups, SS = 6 on 12 degrees of freedom, a mean square of 0.5. F = 5 / 0.5 = 10, p ≈ 0.003.
Why not three t-tests? At α = 0.05 each, the chance of at least one false positive across three independent tests is 1 − 0.95³ ≈ 14%. After a significant ANOVA, a post-hoc test such as Tukey's HSD finds which groups differ. ANOVA assumes independent observations, roughly normal residuals and equal variances; Welch's ANOVA drops the last assumption, and the Kruskal-Wallis test is the rank-based alternative. With two groups, F is the square of the pooled t statistic.
