Diagnosing excess sampling variance: empirical validation and practical frameworks beyond Gy's Fundamental Sampling Error

Abstract Gy's Fundamental Sampling Error (FSE) formula establishes a theoretical lower bound on the variance of sampling particulate materials, conditional on independent particle selection and a spatially representative sampling protocol; a companion paper (Minnitt, 2026) develops the probabilistic foundations of this interpretation. The present paper addresses the practical question that follows: when observed sampling variance exceeds the Gy prediction, how should the excess be diagnosed and managed? The observed-to-predicted variance ratio R = s²obs/s²FSE is developed as a routine diagnostic, and two empirical tests ground its use. First, a gold-ore calibration dataset comprising 32 replicate assays at each of 14 nominal fragment sizes shows the heterogeneity invariant scaling as d1.16 rather than Gy's nominal d³, demonstrating that R is meaningful only when computed with a calibrated size exponent; the classical exponent produces spurious diagnostic signals spanning several orders of magnitude. Second, controlled tracer experiments seeding density-graded marker grains into three matrices show that a one-point scoop generates significant positive inclusion correlation (over-dispersion) while a riffle splitter reproduces the independence baseline — a direct, calibration-free measurement of the second-order term responsible for grouping and segregation effects: the sampling device, not the material, creates the inclusion dependence. A survey of extended frameworks — hypergeometric, negative binomial, geostatistical, Bayesian, Monte Carlo, and discrete element methods — is then organised, together with the empirical results, into a five-step diagnostic workflow for industrial practice.