Partitioning of Variance (ANOVA)

The fundamental principle of ANOVA, conceived by Ronald A. Fisher, revolutionized experimental design. Before ANOVA, researchers often used multiple t-tests to compare several groups, a practice that inflates the Type I error rate (the probability of a false positive). ANOVA provides a single test to check for any difference among group means. The technique works by decomposing the total variation in a dataset, measured by the Total Sum of Squares ([latex]SS_{Total}[/latex]), into two parts. The first part is the Sum of Squares Between groups ([latex]SS_{Between}[/latex]), which measures the variation of each group’s mean from the overall grand mean. This represents the variation explained by the grouping factor. The second part is the Sum of Squares Within groups ([latex]SS_{Within}[/latex]), which measures the variation of each observation from its own group’s mean. This represents the unexplained or random variation, often called error. If the variation between groups is substantially larger than the variation within groups, it provides evidence that the grouping factor has a significant effect on the outcome variable. This comparison is formalized through the F-statistic.