Goodness-of-fit and R-square

Although we can see from the residual plot that a linear model is a good fit for our data, it would be desirable to quantify just how good it is. Also called the coefficient of determination, R² varies between zero and one and indicates the explanatory power of the linear regression model. It calculates the proportion of variation in the dependent variable explained, or accounted for, by the independent variable.

Generally, the closer R² is to 1, the better the regression line fits the points and the more the variation in Y is explained by X. R² can be calculated using the following formula:

Here, var(ε) is the variance of the residuals and var(Y) is the variance in Y. To understand what this means, let's suppose you're trying to guess someone's weight. If you don't know anything else about them, your best strategy would be to guess the mean of the weights within the population in general. This way, the mean squared error of your guess compared to their true weight...