Deconvolution of the Error Associated with Random Sampling

Abstract

In this work empirical models describing sampling error ( Δ ) are reported based upon analytical findings elicited from 3 common probability density functions ( PDF ): the Gaussian, representing any real-valued, randomly changing variable x of mean μ and standard deviation σ ; the Poisson, representing counting data: i.e. , any integral-valued entity’s count of x (cells, clumps of cells or colony forming units, molecules, mutations, etc.) per tested volume, area, length of time, etc. with population mean of μ and ; binomial data representing the number of successful occurrences of something ( x⁺ ) out of n observations or sub-samplings. These data were generated in such a way as to simulate what should be observed in practice but avoid other forms of experimental error. Based upon analyses of 10 4 Δ measurements, we show that the average Δ ( ) is proportional to ( σ_x•μ^-1 ; Gaussian) or (Poisson & binomial). The average proportionality constants associated with these disparate populations were also nearly identical ( ; ± s ). However, since for any Poisson process, . In a similar vein, we have empirically demonstrated that binomial-associated were also proportional to σ_x•μ^-1. Furthermore, we established that, when all were plotted against either or σ_x•μ^-1, there was only one relationship with a slope = A (0.767 ± 0.0990) and a near-zero intercept. This latter finding also argues that all , regardless of parent PDF , are proportional to σ_x•μ^-1 which is the coefficient of variation for a population of sample means ( ). Lastly, we establish that the proportionality constant A is equivalent to the coefficient of variation associated with Δ ( ) measurement and, therefore, . These results are noteworthy inasmuch as they provide a straightforward empirical link between stochastic sampling error and the aforementioned C_vs . Finally, we demonstrate that all attendant empirical measures of Δ are reasonably small (e.g., ) when an environmental microbiome was well-sampled: n = 16 - 18 observations with μ∼3 isolates per observation. These colony counting results were supported by the fact that the two major isolates’ relative abundance was reproducible in the four most probable composition observations from one common population.