Least Squares Regression Lines

A least squares regression line is a regression line which is drawn in such a way that the sum of the squares of the vertical distances between each point and the line is as small as possible.

The equation for a least squares regression line (where x is the explanatory variable and y the response) is:


x is the mean of x, and y is the mean of y.
Sx and Sy are the standard deviations of x and y

b is the gradient of the line and a is the y-intercept. Calculators and software can find a and b given the values of x and y

Least square regression lines always pass through the point on the graph (x, y)

r² is the proportion of the total variation in the y values explained by a least squares regression line. It is a measure of how well the regression line explains the relationship between x and y.