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Volume 2017 (2017), Article ID 8715605, 10 pages
Research Article

Advancing Shannon Entropy for Measuring Diversity in Systems

1Department of Mathematical Sciences, Kent State University, Kent, OH, USA
2Department of Sociology, Kent State University, 3300 Lake Rd. West, Ashtabula, OH, USA
3School of Social and Health Sciences, Abertay University, Dundee DD1 1HG, UK

Correspondence should be addressed to R. Rajaram; ude.tnek@marajarr

Received 31 January 2017; Revised 5 April 2017; Accepted 23 April 2017; Published 24 May 2017

Academic Editor: Enzo Pasquale Scilingo

Copyright © 2017 R. Rajaram et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


From economic inequality and species diversity to power laws and the analysis of multiple trends and trajectories, diversity within systems is a major issue for science. Part of the challenge is measuring it. Shannon entropy has been used to rethink diversity within probability distributions, based on the notion of information. However, there are two major limitations to Shannon’s approach. First, it cannot be used to compare diversity distributions that have different levels of scale. Second, it cannot be used to compare parts of diversity distributions to the whole. To address these limitations, we introduce a renormalization of probability distributions based on the notion of case-based entropy as a function of the cumulative probability . Given a probability density , measures the diversity of the distribution up to a cumulative probability of , by computing the length or support of an equivalent uniform distribution that has the same Shannon information as the conditional distribution of up to cumulative probability . We illustrate the utility of our approach by renormalizing and comparing three well-known energy distributions in physics, namely, the Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac distributions for energy of subatomic particles. The comparison shows that is a vast improvement over as it provides a scale-free comparison of these diversity distributions and also allows for a comparison between parts of these diversity distributions.