Analysing the Dependence between Variance and Mean Estimated Values: A Theoretical and Experimental Approach

Abstract

The starting point of this paper is a known theoretical result - for normal populations, mean and variance estimators are statistically independent. By using an original statistical independence test method, we enriched the study by experimentally investigating the mean-variance data pairs in case of non-gaussian laws. We experimentally noticed that, for non-gaussian symmetrical distributions, mean and variance estimators are still uncorrelated, but not independent random variables. We further theoretically proved that, for symmetrical distributions, the two estimators, mean and variance, are uncorrelated. This new result provides the basis for an original skewness test method, i.e. if mean-variance data pairs are correlated, then the probability density function has an asymmetric shape.