One-Way ANOVA: Compare Three or More Group Means
What is Analysis of Variance?
ANOVA tests whether the means of three or more groups are equal by decomposing total variation into between-group variation (SSA, differences among factor levels) and within-group variation (SSE, random error) and forming the F statistic MSA/MSE. When the between-group variation is large relative to error, F is significant and at least one group mean differs. It controls the overall Type I error far better than performing pairwise t-tests.
When to Use It
Use it to compare several batches, suppliers, settings or treatments on a continuous response, and for factor comparisons in experiments. The classic assumptions are independent observations, approximately normal data within groups, and equal variances; the tool includes Levene's variance-homogeneity and normality checks. When variances are badly unequal, switch to Welch ANOVA or the Kruskal-Wallis nonparametric test, and randomize data collection so systematic factors do not confound the groups.
How to Use It (Step by Step)
Paste each group as a column, optionally run the homogeneity and normality checks, and the tool completes the sum-of-squares decomposition and outputs the ANOVA table (source, degrees of freedom, mean square, F, p-value) together with box or mean-interval plots. If the result is significant, proceed to multiple comparisons such as Tukey or Bonferroni to identify which groups differ, and evaluate the effect size (η², Cohen's f) to judge practical importance.
Key Formulas / Example
F = MSA/MSE = [SSA/(k−1)] / [SSE/(N−k)] with SST = SSA + SSE. Example: three groups with MSA = 40 and MSE = 10 give F = 4.0; compared with F_{2,27,0.05} ≈ 3.35, the result is significant at α = 0.05. For two groups, ANOVA's F equals the square of the two-sample t statistic (F = t²) and gives the same conclusion.