Shannon Index (Biodiversity)

First you determine the relative frequency p_i of each species, that is the number of individuals of the species divided by the total number of all individuals. Then you compute the product p_i·ln(p_i) for each species, sum these values over all species and finally reverse the sign. The leading minus is needed because the logarithm of numbers below 1 is negative. Example: three species with 10, 6 and 4 individuals give the proportions 0.5, 0.3 and 0.2 at $N = 20$. Then H' = −(0.5·ln0.$5 + 0.3\cdot{}ln0$.$3 + 0.2\cdot{}ln0$.2) = −(−0.347 − 0.361 − 0.322) = 1.03. The larger H', the more diverse the community. It is important to always use the same logarithm, usually the natural ln.

A high Shannon index indicates great diversity. It is driven by two factors: species richness, that is how many species occur, and evenness, that is how uniformly the individuals are distributed among the species. The index is maximal exactly when many species occur and all are roughly equally frequent. If, by contrast, a single species dominates strongly, the index stays low even when many species are present. A low value therefore points to a species-poor community or one dominated by few species. In practice you compare the index between sites or over time to detect changes in species diversity, for example after a disturbance or restoration. Absolute values are only meaningfully comparable with the same logarithm base and similar sampling.

Evenness measures how uniformly the individuals are distributed among the species, independent of how many species there are. It follows from the Shannon index divided by its maximum: $E = H$'/ln S, where S is the number of species. The maximum $H_{\max}=\ln S$ is reached when all species are equally frequent. Evenness lies between 0 and 1: a value near 1 means all species occur similarly often, a value near 0 that one species strongly dominates. Example: three species with proportions 0.5, 0.3 and 0.2 have H' = 1.03 and Hmax = ln3 = 1.099, so $E = 0.94$, a fairly even distribution. Evenness thus separates the even-distribution aspect from pure species richness, which H' alone mixes together.

In ecology the natural logarithm ln is most common; the unit is then called "nats". Some sources use the base-2 logarithm log₂, giving units in "bits", or the base-10 logarithm log₁₀. The choice only changes the numerical value, not the statement: a conversion factor transforms the values into each other, since $\log{}_{2}(x) =\frac{ \ln{}(x)}{ln2}$. What matters is that you always use the same base within a comparison, otherwise the values are not comparable. The maximum also depends on the base: with ln, $H_{\max}=\ln S$, with log₂ it would be log₂ S. For evenness the base cancels, because you divide H' by Hmax with the same base, so evenness always gives the same value between 0 and 1. Therefore check before every comparison which base was used.

Both measure diversity but weight differently. The Shannon index H' = −Σ p_i·ln(p_i) comes from information theory and reacts comparatively sensitively to rare species, because their information content −ln(p_i) is large. The Simpson index D = Σ p_i² is the probability of drawing the same species twice and emphasises common, dominant species more strongly; rare species barely influence it. As a diversity measure one usually uses 1 − D or the reciprocal $\frac{1}{D}$ with the Simpson index, which increase with diversity, while D itself measures dominance. In practice both complement each other: the Shannon index suits cases where rare species matter, the Simpson index where the dominance structure is decisive. Often one computes both to get a complete picture of the diversity.