LEAST SQUARES LINES AS A CALCULUS I TOPIC

Abstract

ABSTRACT Fitting lines to data points is a common procedure in applied mathematics and the sciences. A frequently used technique is to fit a least squares line or regression line through a set of data points. The regression line is found as a local minimum of the function that measures the sum of the squares of the vertical distances (or horizontal distances or orthogonal distances) from the data points to the least squares line. In the literature finding the least squares line is always treated as an optimization problem of multivariable calculus. We show in this article that in all three cases (vertical, horizontal and orthogonal distances) the problem can easily be reduced to a one variable optimization problem. The key to this approach is the fact that the center of gravity of a set of data points lies on either of the three regression lines. This is of course well known and is usually established by showing that the coordinates of the centroid satisfy the equation of the regression line. In this article we first establish that the centroid has to be on the regression line, and then we use this fact to reduce the problem of finding the regression line to a one variable problem.