Variable second-order inclusion probabilities during the sampling of industrial mixtures of particles

During analysis and inspection of materials, the mass concentration of a compound in a sample is often used as an estimate for the mass concentration in the population. Depending upon the material and the analysis technique, sample masses vary from several grams to up to several kilograms. An equation for the relation between the sample mass and the variance of the sample concentration allows calculation of the minimum sample mass, defined as the sample mass for which the relative standard deviation of the sample estimator is equal to a maximum allowable value. Application of Gy's theory may result in inaccurate estimates for the minimum sample mass when the selections of the particles are dependent events. The minimum sample mass may be underestimated, resulting in a relatively high variance of the sample concentration, or overestimated, leading to higher sample processing costs. In this contribution, a variable second-order inclusion probability is proposed. Two estimators for the variance are derived, including an estimator based on the Horvitz–Thompson estimator. The estimators are compared using samples of aggregates. Copyright © 2006 John Wiley & Sons, Ltd.