Decomposing Sums of Squares

Total Sum of Squares

Assume a factor A with p groups and the index i for each group. For each group i, we assign n observations. Let ymi be the value of an observation for student m in group i. With this terminology, the group mean (${\overline{A}}_i$) and the grand mean ($\overline{G}$) can be calculated with:

$${\overline{A}}_i=\frac{{\displaystyle \sum _{m=1}^n{\mathrm{y}}_{mi}}}{n};\overline{G}=\frac{{\displaystyle \sum _{m=1}^n{\displaystyle \sum _{i=1}^p{\mathrm{y}}_{mi}}}}{p\cdot n}$$

Applying the general definition of SS given in the previous section, the total sum of squared deviations of values ymi from their grand mean ($\overline{G}$) is:

$$\mathrm{TSS}={\displaystyle \sum _{i-1}^p{\displaystyle \sum _{m=1}^n{\left({\mathrm{y}}_{mi}-\overline{G}\right)}^2}}$$

This TSS can now be decomposed into two components: a component that captures variation between the groups (also called treatment or systematic sum of squares) and a component that captures variation within the groups (also called sum of squares of errors or unsystematic sum of squares) with the rationale described in the following subsections.