Bias and precision of percentiles of bulk grain size

Percentiles such as D₅₀ and D₈₄, calculated from weights retained on different sieves, are widely used to characterize grain size distributions (GSDs) of bulk samples of sedimentary deposits or sediment fluxes. The sampling variability of such percentiles is not well known, and few sampling guidelines exist for reliable characterization of GSDs. We report results from computer sampling experiments on the variability of sample percentiles in different-sized samples from populations with a log-normal GSD by weight and different sorting coefficients. Sample sizes are scaled by the volume of a median-sized grain so that results can be applied to any log-normal GSD. Sampling is random for the GSD by number that is equivalent to a specified GSD by weight. Results show important differences from standard sampling theory applicable to pebble-count GSDs. In small bulk samples all percentiles, including the median, are underestimated (more so for smaller samples, coarser percentiles and poorer sorting), and precision does not improve with the square root of sample size until fairly large sample sizes are exceeded. Non-dimensional equations fined by eye to the results give good approximations to expected bias and precision in any percentile from 50 to 95 for any given sample size and population sorting coefficient. They are inverted to estimate the sample size required to avoid significant bias, or achieve specified precision, in any percentile of interest given estimates of the population D₅₀ and sorting coefficient. Target sample sizes are sometimes considerably smaller, but in other circumstances larger, than suggested by previous guidelines relating to estimation of the entire grain size distribution. Bias is likely in small samples of river bedload and good precision requires very large samples of poorly sorted gravel deposits.