A simple formula for calculating the lower limit of Shannonʹs diversity index

The purpose of this note is to demonstrate that the exact value of the lower limit of Shannonʹs diversity index (called Hmin′) can be calculated with a very simple formula when data are in the form of counts: Hmin′ = ln(Q) − ((Q - S + 1)ln(Q - S + 1))/Q where S = species richness. Q = total number of individuals. S and Q are integers and Q ≥ S. Hmin′ is generally neglected in ecological studies although it can be very different from zero. Hmin′ is more and more important when the ratio S/Q approaches unity and, then, the interpretation of Shannonʹs diversity or of an eveness index should take it into account. For this reason, we recommend the use of Hurlbertʹs index as an eveness measure rather than Pielouʹs index when Hmin′ > 0. Calculations realized on three examples show that results can be appreciably different. Depending on the cases, eveness differences between two communities can be increased or decreased if we use Hurlbertʹs instead of Pielouʹs index. Moreover, one example allowed us to show that an eveness calculated with Pielouʹs index, even if it is large ( = 0.67), can be in fact the minimal diversity which is mathematically possible. In all of these cases, the ecological interpretation of data can be highly simplified when using Hurlbertʹs index, owing to the fact that Hmin′ is integrated into the calculation of this index.