Investigating equality: The poverty and riches indices

As noise is omnipresent, real-world quantities measured by scientists and engineers are commonly obtained in the form of statistical distributions. In turn, perhaps the most compact representation of a given statistical distribution is via the mean-variance approach: the mean manifesting the distribution's 'typical' value, and the variance manifesting the magnitude of the distribution's fluctuations about its mean. The mean-variance approach is based on an underlying Euclidean-geometry perspective. So very often real-world quantities of interest are non-negative sizes, and their measurements yield statistical size distributions. In this paper, and in the context of size distributions, we present an alternative to the Euclidean-based mean-variance approach: a mean-equality approach that is based on an underlying socioeconomic perspective. We establish two equality indices that score, on a unit-interval scale, the intrinsic 'egalitarianism' of size distributions: (i) the poverty equality index which is particularly sensitive to the existence of very small “poor” sizes; (ii) the riches equality index which is particularly sensitive to the existence of very large “rich” sizes. These equality indices, their properties, their computation, their application, and their connections to the mean-variance approach - are explored and described comprehensively.