R² = \displaystyle \frac{\sum_{i=1}^n(\widehat{y_i}-\bar{y})^2}{\sum_{i=1}^n(y_i-\bar{y})^2}
R² = \displaystyle \frac{\sum_{i=1}^n(ax_i+b-\bar{y})^2}{\sum_{i=1}^n(y_i-\bar{y})^2}
or, \displaystyle b = \bar y - a \bar x donc :
R² = \displaystyle \frac{\sum_{i=1}^n(a(x_i-\bar{x}))^2}{\sum_{i=1}^n(y_i-\bar{y})^2}
R² = \displaystyle a^2\frac{\sum_{i=1}^n(x_i-\bar{x})^2}{\sum_{i=1}^n(y_i-\bar{y})^2}
or, \displaystyle a=\frac {\sum_{i=1} ^n (x_i-\bar x)(y_i-\bar y)}{\sum_{i=1}^n (x_i-\bar x)^2} donc
R² = \displaystyle [\frac {\sum_{i=1} ^n (x_i-\bar x)(y_i-\bar y)}{\sum_{i=1}^n (x_i-\bar x)^2}]^2\frac{\sum_{i=1}^n(x_i-\bar{x})^2}{\sum_{i=1}^n(y_i-\bar{y})^2}
R² = \displaystyle \frac{[\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})]^2}{\sum_{i=1}^n(x_i-\bar{x})^2\sum_{i=1}^n(y_i-\bar{y})^2}
R² =\displaystyle [\frac{\sum_{i=2}^n(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i=1}^n(x_i-\bar{x})^2\sum_{i=1}^n(y_i-\bar{y})^2}}]^2
R² = r² cqfd