A Coefficient of Linear Correlation Based on the Method of Least Squares and the Line of Best Fit
Given N points in a plane, corresponding to N pairs of values for two variables, X and Y , we find the line of best fit and the line of wtorst fit, by the method of least squares*. Then we derive a cofficient of correlation based on the sum of the squares of the distances of the points from these two lines. The line of best fit is in the line such that the sum of the squares of the distances of the points from it is a minimum. The line of wtporst fit is the one from which the sum of the squares of the distances of the points is a maximum. We shall refer to them, respectively, as the ntinimum and maxinum lines. For convenience we take the origin at the centre of gravity of the points, letting x and y denote deviations of X and Y , respectively, from their arithmetic means. 1. The two lines pass thru the arithmetic means of the X's