Estimating the Average Diameter of a Population of Spheres from Observed Diameters of Random Two-Dimensional Sections
Size distributions of chondrules, volcanic fire-fountain or impact glass spherules, or of immiscible globules in silicate melts (e.g., in basaltic mesostasis, agglutinitic glass, impact melt sheets) are imperfectly known because the spherical objects are usually so strongly embedded in the bulk samples that they are nearly impossible to separate. Hence, measurements are confined to two-dimensional sections, e.g. polished thin sections that are commonly examined under reflected light optical or backscattered electron microscopy. Three kinds of approaches exist in the geologic literature for estimating the mean real diameter of a population of 3D spheres from 2D observations: (1) a stereological approach with complicated calculations; (2) an empirical approach in which independent 3D size measurements of a population of spheres separated from their parent sample and their 2D cross sectional diameters in thin sections have produced an array of somewhat contested conversion equations; and (3) measuring pairs of 2D diameters of upper and lower surfaces of cross sections each sphere in thin sections using transmitted light microscopy. We describe an entirely probabilistic approach and propose a simple factor of 4/x (approximately equal to 1.27) to convert the 2D mean size to 3D mean size.