In a [[Linear Regression|linear regression]] model, coefficient of determination $\rho^2$/$r^2$ (square of the [[Pearson Correlation Coefficient|correlation coefficient]]) is the proportion of the [[Variance|variance]] in $Y$ that can be accounted by a linear relationship with $X$: $ \begin{align*} \text{Coefficient} &= \frac{\text{Explained Variation}}{\text{Total Variation}} \\ &= \frac{s_x^2 b_1^2}{s_y^2} \\ &= \frac{s_x^2 b_1^2}{s_y^2}\frac{n - 1}{n - 1} \\ &= \frac{S_{xx}b_1^2}{S_{yy}} \\ &= \frac{S_{xx} S_{xy}^2}{S_{yy}S_{xx}^2}\\ &= \frac{S_{xy}^2}{S_{yy}S_{xx}} \\ &= \paren{\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}}^2 \\ &= r^2 \end{align*} $ The rest of the variation occur from [[Random Variable|randomness]] or the impact of other variables.