Quiz: Simpson's diversity index calculator

The formula you need

D = Σ n(n − 1) ÷ N(N − 1) Simpson's index of diversity = 1 − D

n is the number of individuals of each species and N is the total of all individuals. D is the chance that two individuals chosen at random (without putting the first back) are the same species, so 1 − D is the chance that they are different.

A method that works every time

1. Add up all the counts to get N.
2. For each species, multiply n by one less than itself: n × (n − 1). A species with one individual gives 1 × 0 = 0.
3. Add those products to get Σn(n − 1).
4. Work out N × (N − 1).
5. Divide step 3 by step 4 to get D, then subtract from 1.

Setting the numbers out in a table with columns n and n(n − 1) makes step 2 hard to get wrong and is exactly what examiners want to see.

A worked question

A quadrat contains 6 daisies, 4 clover plants and 2 plantains. Then N = 12 and

Σn(n − 1) = 6×5 + 4×3 + 2×1 = 30 + 12 + 2 = 44 N(N − 1) = 12 × 11 = 132 D = 44 ÷ 132 = 0.3333 1 − D = 0.6667, which is 0.67 to two decimal places

If you had used proportions instead, Σp² = (0.5² + 0.333² + 0.167²) = 0.3889, and 1 − Σp² = 0.61. That is a different, slightly smaller answer, which is why the quiz tells you when you have used the wrong form.

What the answer means

1 − D is always between 0 and 1. A value near 0 means one species dominates; a value near 1 means many species in similar numbers. With few species the maximum is lower: three equally common species of 4 individuals each give 1 − D = 1 − 36 ÷ 132 = 0.73, not 1. Keep that in mind when you sanity-check an answer: if you get more than 1 or less than 0, something has gone wrong, most often N² in place of N(N − 1) or a missing bracket on the calculator.

Using the calculator for the arithmetic

Work out Σn(n − 1) and N(N − 1) separately and write both down before dividing; typing the whole expression in one go invites bracket errors. On most calculators, 1 − 44 ÷ 132 gives the right answer because division is done first, but 1 − 44 ÷ 12 × 11 does not.

Common slips the quiz catches

• Stopping at D. The question asks for 1 − D, the index of diversity.
• Squaring n instead of multiplying by n − 1.
• Using Σp², the formula for proportions, when the course uses n(n − 1).
• Rounding D early. Keep four decimal places until the last step.