A New Index for Measuring the Non-Uniformity of a Probability Distribution

This paper proposes a new index, the “distribution non-uniformity index (DNUI)”, for quantitatively measuring the non-uniformity or unevenness of a probability distribution relative to a baseline uniform distribution. The proposed DNUI is a normalized, distance-based metric ranging between 0 and 1, with 0 indicating perfect uniformity and 1 indicating extreme non-uniformity. It satisfies our axioms for an effective non-uniformity index and is applicable to both discrete and continuous probability distributions. Several examples are presented to demonstrate its application and to compare it with two distance measures, namely, the Hellinger distance (HD) and the total variation distance (TVD), and two classical evenness measures, namely, Simpson’s evenness and Buzas and Gibson’s evenness.