Analysis of variance
Analysis of variance (ANOVA) uses an F-test to compare the means of three or more groups by partitioning total variation into between-group and within-group components. A significant result means at least one group mean differs.
FrameworkHypothesis testing
See it move
A retailer tests three store layouts by measuring weekly sales at several branches. ANOVA partitions the total sum of squares in sales into a between-group component of 500, reflecting how much the layouts differ, and a within-group component of 300, reflecting ordinary branch-to-branch noise. The ratio of these, F = MSB ÷ MSW, tests whether the layouts' mean sales genuinely differ.
The formula
Variables
- Mean square between groups = SSB ÷ (k − 1)
- Mean square within groups = SSW ÷ (n − k)
- Number of groups
- Total number of observations across all groups
Reject H₀ (equal group means) when F exceeds the critical F-value at significance level α with (k − 1) and (n − k) degrees of freedom.
Variables
- Sum of squares between groups: variation explained by group membership
- Sum of squares within groups: unexplained residual variation
ANOVA partitions total variation into explained (between-group) and unexplained (within-group) components.
Check yourself
A pharmaceutical company tests three vitamin supplement formulations. Each is given to 20 participants for eight weeks and blood-level outcomes are recorded. To determine whether mean outcomes differ across the three formulations, the analyst proposes ANOVA rather than three separate pairwise t-tests. What is the key statistical reason for this choice?