Reliable Uncertainties: Error Correlation, Rotated Error Bars, and Linear Regressions in Three-Isotope Plots and Beyond

Correlated errors of experimental data are a common but often neglected problem in physical sciences. Various tools are provided here for thorough propagation of uncertainties in cases of correlated errors. Discussed are techniques especially applicable to three-isotope plots common in geo- and cosmochemistry, where common denominators in isotope ratios, as well as instrumental effects, such as mass-dependent isotope fractionation, could lead to significant correlation of errors. Furthermore, various techniques for calculating linear regressions are compared to each other, showing that the method suggested by Mahon (1996) [1] gives the best results after correcting for some typographical errors therein and eliminating a minor mistake. We also provide a method for calculating linear regressions through a fixed point, avoiding previously made mistakes.