Error Estimates in the Sampling From Particle Size Distributions

It is often necessary to estimate the properties of particle size distributions from limited samples taken from large populations. When the distributions are broad, and higher order moments required, as in the case of volume based particle size distributions, the inferred parameters d_3,50 (volume median diameter) and GSD (geometric standard deviation) can have high intrinsic errors not immediately obvious to the measuring scientist. We show that there is a critical number of particles, N_crit, which must be counted or else the error may blow up catastrophically. N_crit is very sensitive to the width of the distribution, and is approximately proportional to GSD¹¹ We develop formulae to estimate the random sampling error inherent in measured values of the d_3,50 and GSD for the log-normal distribution; compare the predictions to a typical experimental particle size measurement; and then generalize to the median of any arbitrary moment, d_{r, 50}.