The rich-Rao quadratic entropy

The Rao quadratic entropy ( Q ) is a fundamental measure of biodiversity, widely used in ecology, which quantifies the average dissimilarity among individuals in a community. However, Q has been sometimes criticized for not fulfilling certain properties desired in applications like conservation biology, notably: its maximum value does not necessarily correspond to maximum species richness, and its unit of measurement is not species number. For these applications, I introduce the Rich-Rao quadratic entropy ( RQ ), defined as the product of Q and observed species richness ( S ). I demonstrate that RQ satisfies key properties unsatisfied by Q , including weak species monotonicity (ensuring that the loss of any species results in a decrease in RQ ). If interspecies dissimilarities are bounded between 0 and 1, RQ can be interpreted the average number of distinctive species encountered by a randomly chosen species in the community. I show that RQ also retains key statistical properties of Q , such as concavity, and can be additively partitioned into α, β, and γ components. Unlike measures of Equivalent Species Richness, RQ maintains the fundamental property that diversity is zero in the absence of inter-individual dissimilarity. Applying RQ and Q to bat and bird functional diversity across six European countries reveals that Q and RQ can provide conflicting rankings, particularly for birds, where RQ variation is strongly driven by species richness. The Q approach is best suited for mechanistic studies, while RQ provides a synthetic, readily communicable measure that balances dissimilarity and species richness, making it valuable for informing conservation priorities. • Rich-Rao quadratic entropy ( RQ ) is defined as the product of Q and species richness ( S ). • RQ overcomes Q 's limitations, notably fulfilling weak species monotonicity. • RQ provides a synthetic measure for conservation, unlike mechanistic Q . • Bat and bird functional diversity rankings differ between RQ and Q . • RQ offers a balanced, communicable metric for setting conservation priorities.