Consider two random variables, X and Y. How can we determine which of them is greater, in some sense, than the other one?
One approach to this question is to define a map F from the set of random variables to the set of real numbers and interpret its value as the measure of "largeness" of a random variable (so that X = Y if and only if F(X) = F(Y)). Then F should atisfy some natural axioms that allow to use it as a measure of largeness. One such axiomatics was proposed by Cherny and Madan (2009) in connectio with analysis of performance of investment strategies. In this talk I will present a general family of measures F that are constructed as minimal hulls of the ratio of the mean of a random variable and its deviation from the mean. The main results of the work show that they satisfy the axiomatics of Cherny and Madan and provide an easy representation in terms of distribution functions of underlying random variables.