### Visman’s sampling experiment

Visman’s work is based on the additive property of variances. His sampling experiment showed that the variance of the primary sample selection stage (the sampling variance) is the sum of the composition variance and the segregation variance. When I met Visman for the first time in Canada in the 1970s, we talked about his sampling theory. He agreed that the adjective *segregation* seems to suggest that some quantity of coal may have been more homogeneous at an earlier stage, and that term *distribution variance* more succinctly describes this component of the sampling variance. Thus, the composition variance is a measure for variability between particles in primary increments, and the distribution variance is a measure for variability between primary increments in the complete set that constitutes the sampling unit. I work with the distribution variance because it is an intuitive measure for intrinsic variability in sample spaces such as *in-situ* coal seams and ore blocks.

I was aware that ASTM Committee D-5 on Coal and Coke wanted to include Vismanâ€™s sampling experiment in *ASTM D2234 Standard Practice for Collection of a Gross Sample of Coal*. In fact, this ASTM Standard was the first internationally recognized document that specified the precision for ash content. Vismanâ€™s sampling experiment with small and large increments may still be found in Annex A1. *Test Method for Determining the Variance Components of a Coal*. It is based on taking pairs of small and large increments side-by-side from a stopped conveyor belt such that pairs are evenly spaced in the sampling unit. Each large increment was selected with a sampling frame, and its paired small increment was taken next to that frame. Each increment was weighed, air-dried, prepared and tested for ash on dry basis. In those early days, the variance of the set of small increments was called *random variance*, and the variance of the set of large increments was called *segregation variance*.

What ASTM D2234 did not determine but Visman defined in his 1947 PhD thesis is the variance of sample preparation and analysis. This variance is small when compared with the variance of the primary sample selection stage. In fact, Vismanâ€™s *C* turned out to be about 5%. It is possible to optimize sampling protocols by applying analysis of variance to the variances of the primary sample selection stage, the sample preparation stage, and the analytical stage.

But thereâ€™s so much more to Vismanâ€™s seminal sampling experiment than meets the eye. For example, Annex A1 in ASTM D2234 reports highly variable weights of both small and large increments. So much so that the mass-weighted average dry ash contents and its variance make more sense than the arithmetic means and its variance. Visman himself knew how to derive the weighted average and its variance but ASTM D-5 kept it simple. Whatâ€™s more, degrees of freedom for sets of measured values with variable weights are positive irrationals rather than positive integers. This fundamental concept in applied statistics and sampling practice is most annoying to those who want to do more with less and think degrees of freedom are for the birds. But I digress!

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