Estimating the Lognormal Size Distribution of Spheres by Plane Sampling

In stereological material characterization log-normal distributions are often used. For instance, spherical distribution parameters are estimated by measuring cross-section circle diameters. These cross-sections are selected randomly but, without details of that randomization it is not possible to fit a single distribution. This work defines the randomization and creates an unambiguous distribution that avoids Bertrand's paradox. Two particular randomizations are demonstrated and simulation verified. Suggestions are given for selecting the appropriate randomization.