Extraction of Physical Quantities from Isotope Mixing Diagrams

I derive a formula for the square of theMahalanobis distance that allowsa straight line to be fitted to data where uncertainties in xand yare correlated. I show how errors in the resulting gradient and intercept can be calculatedfrom the Cramér-Rao bound1. I provide VBA Excel® macros that perform these calculations. Using them I demonstrate that this approach reproduces the results of the standard York fit as implemented in the Excel® add in“Isoplot”. I show that the choice of normalising isotope can have a major influence on the quantities derived from fitting and the likelihood calculated for the model. This occurs because normalisation to an isotope determined with low precision produces error distributions that are poorly approximated by normal distributions, and so a fundamental condition required for the validity of the fitting process is violated. For this reason, calculations of relevant quantities should be carried out using normalisation to the most precisely measured isotope, usually one of high relative abundance. Graphical representations can be constructed in other, more familiar, normalisations based on these calculations if desired, but it should be noted that the error bars may not accurately reflect underlying distributions