Issue 10, p. 31 (2020)

  Article

Material intrinsic heterogeneity: statistical derivation

  • Geoff Lyman  
 Corresponding Author
Principal, Materials Sampling & Consulting, materials-sampling-and-consulting.com

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The value of a fully statistical sampling theory is that it is possible to quantify a measure of material intrinsic heterogeneity and, on this basis, provide the entire distribution of the analyte content of potential samples to be extracted from the lot. The analyte content of a sample of a given mass is a random quantity as samples of nominally equal masses taken from a lot in a given state of comminution will not have exactly the sample analyte content. The analyte content of a sample is correctly described as a random variable and to characterise a random variable completely it is necessary to know either the probability density function or distribution function for the random variable, or all of the moments of the random variable (mean, variance and all the higher moments). The following discussion derives the fundamental sampling variance from a purely mathematical statistics basis, relying on the assumption that the number of particles of any one type (size and analyte content) that fall into a sample taken in a mechanically correct manner (following the principle of equiprobable sampling) follows a Poisson distribution. In addition, the Poisson distributions of particle numbers are statistically independent. A more fully argued substantiation of this fundamental assumption, partial experimental evidence and standard statistical introduction to the definition and properties of the Poisson distribution, and reasons for its use, can be found at the end of this article.
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