Finding Linear Functions

Abstract

Drawing a straight line by eye through a scattered set of experimental points, presumed to represent a linear function, is a subjective business at best. Moreover, no valid statistical parameters comparable to the standard deviation of the mean or well-defined confidence limits can be calculated for a line so drawn because your line will not necessarily come out exactly the same as my line. In this chapter, we shall use the principle of least squares to generate the equation of one and only one straight line for any given set of xy data pairs. The line so obtained is the line that best fits the points subject to (1) the assumption of linearity and (2) the assumptions of the least squares method. We shall use the simplest case of a linear function passing through the origin to introduce the method and set up the ground rules. The more complicated case of a linear function not passing through the origin will be solved by a method that is general and will be extended to nonlinear functions in the next chapter and functions of many variables in the chapter after that.