Determination of the straight line of best fit to observational data of two related variates when both sets of values are subject to error

Abstract

Abstract The general equation to a straight line of best fit to observational data of two related variates x and y is obtained by minimizing the general expression \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_{i = 1}^n {(y_i - ax_i - b)^2 } $\end{document} ϕ( a ) involving the departures of both variates from the time. It is shown that the only expression for ϕ (a) which produces consistent results with change of unit is ϕ (a) = constant. a k , where k depends on the relative errors ( e x , e y ) in measurements of the two variates. The line of best fit is shown to be where c is obtained from the equation ( k + 2) c 2 – 2 ( k + 1) rc + k = 0, in which r is the coefficient of correlation between x and y , and k is a given function of e x and e y . The usual regression lines and line with a slope equal to the geometric mean of the slopes of the two regression lines, arise as special cases of the general equation. Examples are given to show general application.