One-Factor Within Groups and Between Groups Variation

In a one-factor (one-way) analysis of variance, we collect samples from $k$ groups and want to measure how variation in the data splits into variation between the group means and variation within each group. We use the following notation:

• $n_i$ = sample size of group $i$
• $\bar{x}_i$ = sample mean of group $i$
• $x_{ij}$ = the $j$-th observation in group $i$
• $N = n_1 + n_2 + \cdots + n_k$ = total sample size
• $\bar{x}$ = grand mean, the mean of all $N$ observations:

$$\bar{x} = \dfrac{1}{N}\sum_{i=1}^k \sum_{j=1}^{n_i} x_{ij} = \dfrac{1}{N}\sum_{i=1}^k n_i\, \bar{x}_i.$$

To measure how much the group means differ from one another, we compute the between-groups sum of squares:

$$SS_B = \sum_{i=1}^k n_i (\bar{x}_i - \bar{x})^2.$$

Each group contributes a squared deviation of its mean from the grand mean, weighted by its sample size. A large $SS_B$ indicates that the group means are spread far apart.

For instance, suppose three groups of size $n_1 = n_2 = n_3 = 4$ have means $\bar{x}_1 = 2,\ \bar{x}_2 = 5,\ \bar{x}_3 = 8$. The grand mean is

$$\bar{x} = \dfrac{4 \cdot 2 + 4 \cdot 5 + 4 \cdot 8}{12} = \dfrac{60}{12} = 5,$$

so

$$SS_B = 4(2-5)^2 + 4(5-5)^2 + 4(8-5)^2 = 36 + 0 + 36 = 72.$$