In mineral exploration and mining applications, it is sometimes necessary to reduce sampling error in order to improve representivity of individual sample assays. Historic efforts to reduce sampling error have relied on the belief that larger samples exhibit less sampling error. Justifications for this belief traditionally have coupled the binomial theorem with an ideal geological material consisting of ore and gangue particles of equal size and shape (an equant grain model) and empirical tests of geological materials, to illustrate an inverse relationship between sample size and sampling variance. To date, no proof exists of this sample size–sampling variance relationship in real geological materials exhibiting variations in grain size, shape, composition, and degree of liberation.
Using first principles calculus and the formula for the mean, sample mass and sampling variance are proven to be inversely proportional in all geological materials. Distribution assumptions and physically ideal geological models are not used in this proof. Furthermore, algebraic manipulations of expressions describing the variances of equant grain models consistent with binomial, hypergeometric, and Poisson distributions reveal that these models also exhibit an inverse sample size–sampling variance relationship. Thus, the sampling behaviors of these models are numerically consistent with the sampling behaviors of real geological materials, and these models can be used to estimate sampling errors in real geological materials using simple sampling parameters.