Estimation of mean particle size from single sections

The usual size descriptors for three-dimensional (3D) particles are volume (V), surface area (S) and, to a lesser extent, mean height (H), the mean being over isotropic directions in space. With these parameters we can construct 12 weighted means for the particle population, namely {E-Y(X), X = H, S, V and Y = N, H, S, V}, where E-Y(X) represents the population mean of the parameter X weighted by the parameter Y. The ordinary means are the number-weighted ones, namely {E-N(X), X = H, S, V}, whose estimation is only possible using a 3D probe (notably the disector) to sample the particles. In this paper we describe a general strategy to estimate E-Y(X) and consider in detail the estimation of E-S(V) and E-V(S); unbiased estimators have been proposed for both weighted means from single or independent sections, but to our knowledge the corresponding sampling and estimation procedures have not yet been illustrated with real data. We do this here for the tungsten particles in a cemented carbide.