Coefficient of Determination vs Residual

The coefficient of determination $r^2$ reports the share of the response's total variation that the least-squares line accounts for, so higher means a better linear fit. It is the square of the correlation coefficient $r$, and multiplying by 100 turns it into a percentage. For example, $r^2 = 0.64$ means 64% of the variation in the response is explained by the linear relationship with the explanatory variable, leaving 36% unexplained. Because it is a square, $r^2$ is never negative and never above 1.

A residual measures how far off the line is for a single point, so a positive residual sits above the line and a negative one below. It equals $\text{residual} = y - \hat{y}$, the observed response minus the predicted response $\hat{y}$ (read y-hat). For example, if the line predicts 72 but a student actually scored 80, the residual is $80 - 72 = 8$. Least-squares regression chooses the line that makes the sum of the squared residuals as small as possible.