The theory of sampling has progressed slowly over the last century. The concepts of correct sampling were known to mining engineers such as D W Brunton in the 19th century. It was recognised early in the 20th century that the Poisson distribution of particles carrying the target analyte was the key to the uncertainties between samples of a given mass. However, no efforts were made until the 1980s to determine the entire sampling distribution as a function of sample mass. The variance of sampling was made clear by Gy and others, but none attempted to determine the entire sample assay density function until J H Venter in South Africa applied the method of characteristic functions to the problem.Venter applied the method to a specific case of very low analyte concentrations in samples of very fine particle size where his assumption of constant sample mass was justified. When his general methodology is applied to the more general problem, it can be shown that the assumption of constant sample mass is no longer justified and leads to some anomalous results for sampling variance. Venter's method can be expanded to allow for a statistically variable sample mass and to take into account the size distribution of particles that carry no analyte.This paper presents the basis of Venter's work and shows how the approach can be easily modified to yield results for the variance that are consistent, for example, with the work of Gy. It is also demonstrated that the entire sampling distribution can be determined without undue computational burden when sufficient information on the material heterogeneity has been collected.CITATION:Lyman, G J, 2014. Determination of the complete sampling distribution for a particulate material, in Proceedings Sampling 2014 , pp 17-24 (The Australasian Institute of Mining and Metallurgy: Melbourne).