10.1.3: Coefficient of Determination or R-squared

The coefficient of determination \(R^{2}\) (or \(r^{2}\)) is the fraction (or percent) of the variation in the values of \(y\) that is explained by the least-squares regression of \(y\) on \(x\). \(R^{2}\) is a measure of how well the values of \(y\) are explained by \(x\).

For example, there is some variability in the dependent variable values, such as grade. Some of the variation in student’s grades is due to hours studied and some is due to other factors. How much of the variation in a student’s grade is due to hours studied?

When considering this question, you want to look at how much of the variation in a student’s grade is explained by the number of hours they studied and how much is explained by other variables. Realize that some of the changes in grades have to do with other factors. You can have two students who study the same number of hours, but one student may have a higher grade. Some variability is explained by the model and some variability is not explained. Together, both of these give the total variability.

\[\begin{aligned} \text{(Total Variation)} &= \text{(Explained Variation)} & + \quad \text{(Unexplained Variation)} \\ \quad \sum \left(y - \bar{y}\right)^{2} &= \quad \sum \left(\hat{y} - \bar{y}\right)^{2} & + \sum \left(y - \bar{y}\right)^{2} \end{aligned}\]

Coefficient of Determination

The proportion of the variation that is explained by the model is \[R^{2} = \frac{\text{Explained Variation}}{\text{Total Variation}} = \frac{SSR}{SST} \nonumber\]

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