The Logic: Partitioning Variability

Sum of Squares Total (SST): Total variability around the grand mean.

$$\text{SST} = \sum_{i=1}^{n}(y_i - \bar{y})^2$$

Sum of Squares Between (SSB): Variability of group means around the grand mean.

$$\text{SSB} = \sum_{j=1}^{k} n_j(\bar{y}_j - \bar{y})^2$$

Sum of Squares Within (SSW): Variability within each group around its own mean.

$$\text{SSW} = \sum_{j=1}^{k}\sum_{i=1}^{n_j}(y_{ij} - \bar{y}_j)^2$$

These three quantities satisfy: SST = SSB + SSW.

The F-statistic compares between-group and within-group variability, normalized by degrees of freedom:

$$F = \frac{\text{SSB}/(k-1)}{\text{SSW}/(n-k)}$$

A large F means between-group variability dominates within-group variability — evidence against the null. If p ≤ α, reject the null: at least one group mean differs.

In Python: scipy.stats.f_oneway(group1, group2, group3, ...)