This paper presents a method of model population estimates that serves to distinguish elemental heterogeneity from error of analysis within a set of approximately 40 replicate microprobe analyses of a single mineral and within a single thin section. A set of muscovite analyses from a granite gneiss metamorphosed to the amphibolite facies is used as an example.
The method of model population estimates is a search for plausible parent populations to which the analyzed sample could belong at a 95% confidence level. Underlying assumptions governing construction of the models are (1) careful analysis of the real sample sufficed to minimize systematic errors; (2) nonsystematic analytical error is in part removed by charge-sum normalization of each analysis and is in part random; and (3) compositional heterogeneity, if present, follows a single equation of charge-coupled substitution.
The models are compared to the real sample at the 95% confidence level by an F test for standard deviations and a t test for correlations. Both a central model, which best fits the real analyzed sample, and a range of models that pass the tests for standard deviations and correlations may be ascertained. The result is an estimate of the allowed extent of compositional heterogeneity and analytical error.
Assumption of a single equation of charge-coupled substitution to cover all elemental heterogeneity of a given mineral is the simplest assumption beyond that of no heterogeneity at all. Should this assumption fail to produce a satisfactory model for a set of well-made mineral analyses, the presence of independent and competing substitutions should be suspected. Conversely, success of this modeling assumption suggests that any independent substitutions are obscured by analytical error but does not necessarily suggest their absence.
The method of model population estimates represents a logical advance in the use of microprobe analyses and may become useful to many types of analytical studies. For example, a large covariation in the extent of tschermakitic and alkali substitution is ascertained within the muscovite analyzed for this study. Application of this covariation to muscovite-plagioclase geothermometry yields a range of 57 °C in the apparent temperature of metamorphism for the sample. In contrast, use of a mean muscovite analysis for the studied sample yields a somewhat lower apparent range of only 10 °C.