# What is Simpson’s diversity index?

Computes Simpson’s diversity index from the number of individuals of each species: Simpson’s index D, the index of diversity 1 − D and the reciprocal index 1 ÷ D, with the species count.

- Page: https://www.acalculator.org/biology/simpsons-diversity-index-calculator
- JSON spec: https://www.acalculator.org/biology/simpsons-diversity-index-calculator.json
- Version: 892aa2efd413

## Default answer

Example with the default inputs (Individuals of each species [25, 15, 40, 20], Formula Σ n(n − 1) ÷ N(N − 1)): Simpson’s index of diversity is 0.722222, with D = 0.277778, for 4 species.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| n | Individuals of each species | How many individuals of each species were counted, separated by commas, spaces or new lines. |
| f | Formula | Σ n(n − 1) ÷ N(N − 1) for counts from a sample (Simpson’s estimate), or Σ p² for proportions of a whole community. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| index | Simpson’s index of diversity (1 − D) | The chance that two individuals picked at random are different species. |
| d | Simpson’s index (D) | The chance that two individuals picked at random are the same species. |
| reciprocal | Simpson’s reciprocal index (1 ÷ D) | One divided by D: from 1 up to the number of species. |
| richness | Species (S) | The number of species with a count above 0. |
| total | Total (N) | The sum of all the counts. |

## Method

D = Σ n(n − 1) ÷ (N(N − 1)), or D = Σ (n ÷ N)² for proportions; Simpson’s index of diversity = 1 − D; reciprocal index = 1 ÷ D.

## Assumptions

- Each count is the number of individuals of one species; N is their sum and S the number of species with a count above 0.
- Σ n(n − 1) ÷ N(N − 1) is Simpson’s estimate from a sample (two individuals drawn without replacement); Σ p² treats the counts as proportions of the whole community and also accepts decimals.
- Higher 1 − D and 1 ÷ D mean more diversity; D itself is lower when diversity is higher.

## Worked examples

1. n = 0.25 or 0.15, f = infinite gives d = 0.285, index = 0.715, reciprocal = 3.508772, richness = 4. Source: NIST, Dataplot Reference Manual, SIMPSON DIVERSITY INDEX (D = Σ (f_i ÷ n)², or Σ p_i² for proportions; lower values mean more diversity; Simpson 1949; proportions 0.25, 0.15, 0.40, 0.20 give 0.2850000), https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/simpson.htm (retrieved 2026-10-18).
2. n = 25 or 15, f = finite gives d = 0.277778, index = 0.722222, reciprocal = 3.6, richness = 4, total = 100. Source: Simpson, E. H. (1949), Measurement of diversity, Nature 163, 688 (the index Σ π_i² and its estimate Σ n_i(n_i − 1) ÷ (N(N − 1))), https://doi.org/10.1038/163688a0; Tiffeau-Mayer, A. (2024), Unbiased estimation of sampling variance for Simpson’s diversity index, arXiv:2310.03439, Eqs. 1 and 2 (the probability that two individuals drawn at random are the same species, Σ p_i², and Simpson’s estimate Σ n_i(n_i − 1) ÷ (N(N − 1)) for pairs drawn without replacement), https://arxiv.org/abs/2310.03439 (retrieved 2026-10-18).
3. n = 10 or 0, f = finite gives d = 0.473684, index = 0.526316, reciprocal = 2.111111, richness = 2. Source: Simpson, E. H. (1949), Measurement of diversity, Nature 163, 688 (the index Σ π_i² and its estimate Σ n_i(n_i − 1) ÷ (N(N − 1))), https://doi.org/10.1038/163688a0.
4. n = 1 or 1, f = finite gives d = 0, index = 1, richness = 3. Source: Simpson, E. H. (1949), Measurement of diversity, Nature 163, 688 (the index Σ π_i² and its estimate Σ n_i(n_i − 1) ÷ (N(N − 1))), https://doi.org/10.1038/163688a0.

## FAQ

### How do I calculate Simpson’s diversity index?

For each species, multiply its count n by n − 1 and add the results. Divide by N(N − 1), where N is the total count: that is D. For counts of 25, 15, 40 and 20: (600 + 210 + 1,560 + 380) ÷ (100 × 99) = 2,750 ÷ 9,900 = 0.278. The index of diversity is 1 − D = 0.722.

### What is the difference between D, 1 − D and 1 ÷ D?

D is the chance that two individuals picked at random are the same species, so a lower D means more diversity. 1 − D, Simpson’s index of diversity, is the chance that they differ, so higher means more diverse. 1 ÷ D, the reciprocal index, runs from 1 up to the number of species.

### Should I use n(n − 1) or p²?

Use Σ n(n − 1) ÷ N(N − 1) for counts from a sample: Simpson showed it is an unbiased estimate of the chance that two individuals drawn without replacement match. Σ p² uses the proportions n ÷ N directly, as NIST’s Dataplot does, and suits proportions or percent cover. The two are close for large samples.

### What is a good Simpson’s diversity index?

There is no fixed scale; compare sites counted the same way. 1 − D is 0 when there is only one species and gets closer to 1 as there are more species with more even numbers. A community of 4 species in the proportions 25%, 15%, 40% and 20% has 1 − D = 0.715 (Σ p²).

### Why is there no reciprocal index for some counts?

When every species has a count of 1, no two individuals can be the same species, so Σ n(n − 1) is 0 and D is 0. Then 1 ÷ D has no value, and the page leaves it out.

### How is Simpson’s index different from the Shannon index?

Both grow with the number of species and with how evenly the individuals are spread among them. Simpson’s D adds up p², so the most common species make up most of it. The Shannon index adds up −p × ln p, which gives rare species more weight.

## Sources

- Simpson, E. H. (1949), Measurement of diversity, Nature 163, 688. The index Σ π_i², the chance that two individuals taken at random belong to the same group, and its estimate Σ n_i(n_i − 1) ÷ (N(N − 1)) from a sample. https://doi.org/10.1038/163688a0 (reprinted with permission at https://people.wku.edu/charles.smith/biogeog/SIMP1949.htm)
- NIST, Dataplot Reference Manual, SIMPSON DIVERSITY INDEX: D = Σ (f_i ÷ n)² for counts, or Σ p_i² for proportions; lower values indicate more diversity; example with proportions 0.25, 0.15, 0.40, 0.20 gives 0.2850000. https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/simpson.htm (retrieved 2026-10-18)
- Tiffeau-Mayer, A. (2024), Unbiased estimation of sampling variance for Simpson’s diversity index, arXiv:2310.03439: the index Σ p_i² (Eq. 1) and Simpson’s unbiased estimate Σ n_i(n_i − 1) ÷ (N(N − 1)) for pairs drawn without replacement (Eq. 2). https://arxiv.org/abs/2310.03439 (retrieved 2026-10-18)