The largest particle in a sample from an exponential, gamma or lognormal distribution

The size of the largest particle in a sample of hailstones or raindrops is of some interest. In the case of hail, it provides an indication of the potential for damage. In the case of rain, some scientists take it as an indication of the need to truncate any applicable drop-size distribution function. Such samples come from much larger populations of hydrometeors, and a simple gedanken experiment shows that the largest particle in a sample is most unlikely to be the largest particle in the population. This paper examines the sampling distribution of the maximum particle size, as a function of the sample size (number of particles observed).

This distribution is calculated analytically for exponential size distributions, and numerical results are provided for gamma and lognormal distributions. The results show that the maximum particle size in a sample is unlikely to approach the maximum size in the underlying population, even with a sample of many thousands of particles. For example, in a sample of 5,000 drops from an exponential distribution with mass-weighted mean diameter 2 mm (Do = 1.84 mm) there would be only a 20% chance of finding a drop as large as 5 mm even if the distribution extends to infinite size. Consequently, an observation of “maximum particle size” provides little more than a lower bound on the maximum size in the population, and establishes no basis for truncating any applicable size distribution function.