Sums of squares

The table below summarises the interpretation of the total, within-treatment and between-treatment sums of squares.

Sum of squares Interpretation
Overall variability of Y, taking no account of the factor.
Describes variability around the treatment means and is therefore variability that cannot be explained by the model.
Describes how far the treatment means are from the overall mean — i.e. the variability of the treatment. It can also be interpreted as the sum of squares explained by the model.

Coefficient of determination

Since the between-treatment (explained) and within-treatment (residual) sums of squares add to the total sum of squares, a useful summary statistics is the proportion of the total sum of squares that is explained by the model. This proportion is called the coefficient of determination and is denoted by R2.

Note the following properties of R2.

0  ≤  R2  ≤  1


Examples

The diagram below shows how R2 is calculated and interpreted for a few data sets.