Least-squares criterion

Out of every line you could draw through a scatterplot, the least-squares criterion keeps the one that minimizes $\sum (y - \hat{y})^2$, the total squared vertical miss, where $\hat{y}$ is read y-hat. Squaring makes large misses count heavily and stops positive and negative residuals from cancelling, since the residuals of this line always add to 0. For example, if the least-squares line for three points leaves residuals 3, -1, and -2, it scores $3^2 + (-1)^2 + (-2)^2 = 9 + 1 + 4 = 14$, and no other line scores lower on those three points. One consequence worth remembering: the least-squares line always passes through $(\bar{x}, \bar{y})$, the point of averages, read x-bar and y-bar.