Simpson Index (Dominance and Diversity)

You determine the relative frequency p_i of each species, that is the species count divided by the total. Then you square each proportion and sum the squares to the dominance index D = Σ p_i². Example: three species with 10, 6 and 4 individuals give the proportions 0.5, 0.3 and 0.2 at $N = 20$, so $D = 0.25 + 0.09 + 0.04 = 0.38$. D is the probability of randomly drawing the same species twice. As a diversity measure one usually uses $1 - D = 0.62$ or the reciprocal index $\frac{1}{D} = 2.63$. For small samples one uses the more accurate finite form $D=\frac{\sum_i n_i(n_i-1)}{N(N-1)}$, which draws without replacement. For the same data it gives $D =\frac{ 132}{380} = 0.347$.

The pure Simpson index D = Σ p_i² measures dominance: the probability that two randomly drawn individuals belong to the same species. A large D means few species dominate the community, that is low diversity. This is unintuitive, because a high value stands for low diversity. Therefore one usually forms the Simpson diversity index 1 − D, also called the Gini-Simpson index. It is the probability of drawing two different species and increases with diversity: 0 for one species only, near 1 for many equally frequent species. Example: $D = 0.38$ means $1 - D = 0.62$. A common mistake is to confuse D and 1 − D and thus reverse the statement. You should always state which form is meant.

The reciprocal Simpson index is the reciprocal of the dominance, that is $\frac{1}{D}$. Its great advantage is the intuitive interpretation: it gives the effective number of species, that is the number of equally frequent species that would produce the same dominance. Example: at $D = 0.38$, $\frac{1}{D} = 2.63$, so the community behaves like about 2.6 equally frequent species. If S equally frequent species occurred, $\frac{1}{D}$ would equal exactly S, because then each p_i = 1/S and $D = S\cdot{}(\frac{1}{S})^{2} =\frac{ 1}{S}$. The reciprocal index therefore always lies between 1 and the actual number of species. It is especially useful for comparing diversity across sites of different richness, because its unit "effective species" is easy to understand and allows linear comparisons.

The finite or unbiased form $D=\frac{\sum_i n_i(n_i-1)}{N(N-1)}$ is used when you draw a sample from a population and the individual counts are small. It models drawing without replacement: once you have drawn an individual of a species, one fewer of that species remains, which the factor (n_i − 1) and the denominator N(N − 1) account for. For large individual counts this form approaches the simple proportion form D = Σ p_i², because the difference between n_i and n_i − 1 then becomes negligible. For school and rough comparisons the proportion form usually suffices. Scientifically, especially with small samples, the finite form is more correct. It is important to always use the same form within a comparison so that the values stay consistent.

The Simpson index squares the species proportions. As a result, common species with large p_i contribute disproportionately strongly, while rare species with small p_i almost vanish after squaring. A species with proportion 0.5 contributes 0.25, a rare species with proportion 0.02 only 0.0004, that is practically nothing. Therefore few dominant species determine the value, and the addition or loss of rare species barely changes D. The Shannon index, by contrast, weights rare species much more strongly via the logarithm and reacts more sensitively to them. Which measure fits better depends on the question: if the dominance structure and common species matter, Simpson is suitable; if rare species are ecologically important, for example in conservation, the Shannon index or a comparison of both measures is more sensible.