Using Least Squares for Error Propagation

Abstract

The method of least-squares (LS) has a built-in procedure for estimating the standard errors (SEs) of the adjustable parameters in the fit model: They are the square roots of the diagonal elements of the covariance matrix. This means that one can use least-squares to obtain numerical values of propagated errors by defining the target quantities as adjustable parameters in an appropriate LS fit model. Often this will be an exact, weighted, nonlinear fit, requiring special precautions to circumvent program idiosyncrasies and extract the desired a priori SEs. These procedures are reviewed for several commercial programs and illustrated specifically for the KaleidaGraph program. Examples include the estimation of ΔH°, ΔS°, ΔG°, and K°(T) and their SEs from K° (equilibrium constant) values at two temperatures, with and without uncertainty in T, which is included using the effective variance method, a general-purpose LS procedure for including uncertainty in independent variables. In some cases, the target quantities can be obtained from the original data analysis, by redefining the fit model to include the quantity of interest as an adjustable parameter, automatically handling correlation problems. Examples include the uncertainty in the fit function itself, line areas from spectral line profile data, and the analysis of spectrophotometric data for complex formation.