AVERAGES, ENTROPY AND OBSERVATIONS

This study discusses the probability Cv(n) that the average of n independent random variables is closer to the distribution expectation than some of the variables. The article reveals that the probability Cv(n) can be presented by a simple function of the standardized Normal distribution N(0,1) entropy and the number of variables n. Simulations based on randomly generated numbers, derived from three Normal Distributions – N(0, 1), N(0, 0.25), and N(0, 9), and three Uniform Distributions – U(-1, 1), U(-2, 2), and U(-10, 10) shows that the probability Cv(n) is independent of the distribution standard deviation σ and Cv(n) → 0 when n → ∞. Simulations show, that the greatest probability to be an average closer to the distribution expectation µ than some observation in the sample, is when n = 3. However, even for the Uniform distribution, known to be the distribution with the greatest entropy, the probability Cv(n=3) < 0.50. Thus, automatically taking the mean of n observations as the most certain value of a measured quantity is risky and, in most cases, a wrong decision. More complicated approaches must be used to select the most trustful value among observations.