Proof of Coefficient of Determination

Given: $R^2=r_{XY}^2$

Correlation Coefficient

The correlation coefficient $r_{XY}$ is defined as: $r_{XY}=\frac{Cov(X,Y)}{{\sigma }_X{\sigma }_Y}$

Covariance and Standard Deviations

From the linear relationship $Y=a+bX$: $Cov(X,Y)=b{\sigma }_X^2$ ${\sigma }_Y=|b|{\sigma }_X$

Correlation Squared

$r_{XY}=\frac{b{\sigma }_X^2}{{\sigma }_X{\sigma }_Y}$ ${\sigma }_Y=|b|{\sigma }_X$ $r_{XY}=\frac{b{\sigma }_X^2}{{\sigma }_X(|b|{\sigma }_X)}$ $r_{XY}=\frac{b}{|b|}$ $r_{XY}=\pm 1$

When squared, $r_{XY}^2=1$.

Coefficient of Determination

$R^2=r_{XY}^2$ $R^2=(\pm 1{)}^2$ $R^2=1$

Thus, $R^2$ represents the proportion of the variance in $Y$ that is predictable from $X$, and for a perfect linear relationship, $R^2=1$.