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Least Squares

The name of the least squares line explains what it does. We start with a collection of points with coordinates given by (xi, yi). Any straight line will pass among these points and will either go above or below each of these. We can calculate the distances from these points to the line by choosing a value of x and then subtracting the observed y coordinate that corresponds to this x from the y coordinate of our line.

Different lines through the same set of points would give a different set of distances. We want these distances to be as small as we can make them. But there is a problem. Since our distances can be either positive or negative, the sum total of all these distances will cancel each other out. The sum of distances will always equal zero.

The solution to this problem is to eliminate all of the negative numbers by squaring the distances between the points and the line. This gives a collection of nonnegative numbers. The goal we had of finding a line of best fit is the same as making the sum of these squared distances as small as possible. Calculus comes to the rescue here. The process of differentiation in calculus makes it possible to minimize the sum of the squared distances from a given line. This explains the phrase “least squares” in our name for this line.