Extensions to the Theory of Sampling 1. The extended Gy's formula, the segregation paradox and the fundamental sampling uncertainty (FSU).

The Theory of Sampling as developed by Pierre Gy is a complete theory that describes sampling errors and how to obtain a representative sample. Unfortunately Gy's formula for prediction of the Fundamental Sampling Error (FSE) can be difficult to use in practice, as it is only valid for binary materials with same size distribution of analyte containing fragments and matrix fragments. An extended Gy's formula for estimation of FSE is derived from Gy's definition of constitutional heterogeneity. This formula is exact with no assumptions and allows prediction of FSE for any particulate material with any number of particle classes in contrast to Gy's formula. The difference is that the only assumption made is that the sampled material can be divided into classes with similar properties for the fragments within each class. The extended Gy's formula is validated by model experiments sampling mixtures of 3-7 components with a riffle splitter with 18 chutes. In most cases the observed sampling error was well predicted by the newly derived, extended Gy's formula. However, in some experiments the observed sampling errors were lower than FSE. This can be explained by the sampling paradox, and the effect is calculated by a new function, the Fundamental Sampling Uncertainty, FSU. The observed results are typically in excellent agreement with the predictions (the predicted uncertainties were on average 0.5% points lower than the observed values). The extended Gy's formula described here is ideal for use in teaching of sampling methods because the experiments can be set up using materials with accurately known properties. The proposed new formula allows accurate prediction of FSE and FSU for complex materials that contain more than two types of particles.