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Define the counts and richness · Work Shannon and Hill measures · Compare Simpson and evenness · Handle sampling and detection · Practice and follow the evidence
Two communities each contain four recorded species and 100 individuals. One is even and one has a dominant species. Which biodiversity statistic answers which question? Work every calculation, then check what field evidence is still missing.
Chapter index
Define the counts and richness · Chapters 1–3
1. Richness does not describe the abundance pattern · 2. Define the survey boundary before using any index · 3. Species richness counts presence, not dominance
Work Shannon and Hill measures · Chapters 4–5
4. Shannon entropy separates evenness from a species count · 5. An effective number of species makes Shannon easier to read
Compare Simpson and evenness · Chapters 6–8
6. Simpson concentration gives greater weight to dominance · 7. Compare the three Hill orders on the same scale · 8. Evenness can be named, but not mistaken for ecosystem health
Handle sampling and detection · Chapters 9–10
9. Why the sampling rule changes a two-draw probability · 10. Sampling completeness and spatial structure still matter
Practice and follow the evidence · Chapters 11–12
11. Frequently asked questions and new datasets · 12. Use each statistic for what it actually establishes
Imagine two invented ecological surveys, each counting 100 individuals and four identified species. Both field summaries report species richness of four, which is correct. But their populations need not resemble one another. Site A has 25 individuals of each species; Site B has 70 of the first species and 10 of each of the remaining three.
The first site is perfectly even across the four recorded species. The second has one dominant species and three less abundant ones. Both contain four observed species and 100 counted individuals, yet the uncertainty about a randomly selected individual’s species differs greatly.
We will distinguish richness from evenness by calculating Shannon entropy, Simpson concentration, effective Hill numbers and finite sampling probabilities. Then we will ask what these calculations support about an ecological assemblage and what they cannot tell us about conservation value.
Every count is stipulated for a teaching case. These are not observations of a named forest, reef, wetland or protected area. There is no claim that high evenness is always good or that a naturally dominant habitat needs corrective management.
Use four abstract taxa A, B, C and D. Each of the 100 individuals is assigned to exactly one taxon. Site A records counts (25,25,25,25). Site B records (70,10,10,10). The counts refer to individuals, not detections, visits, biomass or survey hours.
Let N=100 and p_i=n_i/N. Site A therefore has relative abundances (0.25,0.25,0.25,0.25). Site B has (0.70,0.10,0.10,0.10). Each probability vector sums to one, so neither calculation leaves unallocated individuals.
The sample sizes match and each taxon occurs in both sites. That arrangement isolates the role of abundance distribution. If one sample had been collected across a larger area, or a different method detected cryptic organisms better, its comparison would need additional standardisation.
A real ecological study must also define the sampling unit: a quadrat, transect, net catch, camera-trap capture, tissue sample or individual organism may not represent the same thing. Counts should not be combined until their meanings are compatible.
| Recorded taxon | Site A | Site B |
|---|---|---|
| A | 25 | 70 |
| B | 25 | 10 |
| C | 25 | 10 |
| D | 25 | 10 |
| Total | 100 | 100 |
Observed species richness S is simply the number of species with positive count. Both sites have S=4. The richness index does not distinguish one individual of a taxon from seventy individuals of that taxon.
Richness is useful, particularly for documenting taxonomic presence and identifying whether additional species are detected over time. But a reader must not infer relative abundances, population viability or community structure solely from this count.
The phrase ‘observed richness’ matters. A survey may have missed rare organisms, seasonal migrants, nocturnal taxa or cryptic species. Incomplete detection can cause observed richness to differ from the site’s true richness, and that gap does not disappear just because the index is reported to several digits.
The Nature Index overview of biodiversity measurement discusses sampling effort, detectability and non-independent observations. Our worked example keeps counting complete by stipulation; field work requires separate evidence for completeness.
The Shannon index is H’ = -Σ p_i ln(p_i), with natural logarithms. It captures the uncertainty in species identity for a randomly drawn individual under the observed relative-abundance model.
For Site A, H’ = -4(0.25 ln0.25) = ln4 ≈ 1.386294. Equal representation maximises Shannon entropy for a fixed richness of four. Selecting one individual leaves four equally likely candidate species.
For Site B, H’ = -[0.70 ln0.70 + 3(0.10 ln0.10)] ≈ 0.940448. A randomly chosen individual is more likely to belong to the dominant taxon, reducing the uncertainty despite unchanged richness.
Shannon values are dimensionless, and a different logarithm base changes their numerical expression. When comparing studies, check whether natural logs or base-two logs were used. Never subtract or compare numbers without a common convention.
Hill diversity of order one is D1 = exp(H’). It can be interpreted as the number of equally abundant species that would yield the same Shannon entropy. This conversion places the index on an effective-number scale.
For Site A, exp(ln4)=4. For Site B, exp(0.940448)≈2.561. Both sites still literally contain four observed species. Site B’s effective number is not a newly discovered fractional organism but an equivalent-diversity comparison.
For a perfectly even community, observed richness and the Hill q=1 effective number agree. When abundances are uneven, q=1 tends to be lower than richness because a frequently encountered species contributes more heavily to the assemblage.
The approach traces to Hill’s foundational article on diversity and evenness. Utrecht University’s practical biodiversity tutorial also works through Hill numbers with relative abundances.
Define λ = Σ p_i². For Site A, λ = 4(0.25²) = 0.25. For Site B, λ = 0.70² + 3(0.10²) = 0.49 + 0.03 = 0.52. This is called Simpson concentration.
Under independent draws with replacement from the stipulated species proportions, λ is the probability that both draws belong to the same species. Site B has a much higher matching chance because its dominant species is drawn frequently.
The Gini–Simpson form 1−λ is 0.75 in A and 0.48 in B. The reciprocal Hill diversity of order two, D2=1/λ, is 4 in A and 1/0.52≈1.9231 in B.
Because people use the word ‘Simpson index’ for several related formulas, a careful article must specify concentration, complement or reciprocal explicitly. Quoting one without its mathematical definition can reverse a comparison accidentally.
Hill q=0 diversity is observed richness, D0=S, which equals four in both sites. Hill q=1 is the exponentiated Shannon index, and Hill q=2 is the inverse Simpson concentration.
Site A has D0=D1=D2=4 because all four species are equally abundant. Site B has D0=4, D1≈2.561 and D2≈1.9231. The effective diversity decreases as the order of the measure places greater emphasis on common species.
This is a diversity profile rather than a single absolute verdict about biodiversity. q=0 is particularly sensitive to the presence of rare species; q=2 pays relatively greater attention to dominant species. The researcher should explain why one profile component answers the particular question.
The iNEXT methodological paper describes richness and Hill-number diversity in a sampling-standardisation framework. In our case the counts are constructed exact inputs, not rarefaction estimates or a survey of unseen taxa.
| Measure | Site A | Site B |
|---|---|---|
| Observed richness | 4 | 4 |
| Shannon H′ | 1.386294 | 0.940448 |
| Hill q=1 exp(H′) | 4 | 2.561 |
| Simpson concentration λ | 0.25 | 0.52 |
| Gini–Simpson 1−λ | 0.75 | 0.48 |
| Hill q=2 1/λ | 4 | 1.9231 |
| Pielou J | 1 | 0.6784 |
Pielou’s Shannon evenness is J=H’/lnS for S greater than one. It compares observed Shannon entropy with its maximum given the same richness.
For A, J=1.386294/1.386294=1. For B, J≈0.940448/1.386294≈0.6784. A is completely even in its recorded abundance distribution; B is not.
Higher J should not be presented as a universal conservation goal. Natural assemblages sometimes have strongly dominant plants, microbes or other organisms without being degraded. A site containing a threatened taxon can be highly important even when that species is rare.
Community evenness also says nothing by itself about food webs, genetic variation, ecosystem resilience, pathogen dynamics, species’ conservation status, or the area sampled. The biological interpretation must include those other dimensions when relevant.
The Simpson concentration calculation imagines independent draws with replacement. What if two distinct recorded individuals are selected without replacement from the finite collection of one hundred?
The probability that both are of the same species becomes Σn_i(n_i−1)/(N(N−1)). For A, four groups contribute 25×24 each, giving 2400/9900≈0.2424. For B, 70×69+3(10×9)=5100, giving 5100/9900≈0.5152.
These differ slightly from 0.25 and 0.52 because the first selection changes what remains in the sample. Neither answer is wrong. They belong to two different selection procedures, and the denominator reveals which experiment was assumed.
In real ecological work, a repeated detection, a non-independent sample, or a clustered field design adds further complications. Asking how observations were selected should come before applying a familiar probability formula.
Counts from camera traps, nets, photographs, audio recordings or molecular surveys have different detection and classification processes. Rare species may be missed if surveys are short, weather varies, or organisms occupy places that were not sampled.
Rarefaction and extrapolation can help compare effort or coverage, but only under suitable models and documented sampling data. These methods do not turn a poorly identified count into a definitive census of a habitat.
Spatial arrangement is another missing dimension. Two sites with identical species totals and proportions might contain mixed patches or sharply segregated habitat zones. Same Histogram, Different Neighbourhood Patterns shows why frequency counts alone cannot recover where observations occur.
Before ranking real sites, examine taxonomic identification, survey coverage, abundance units, spatial replication, seasonal variation, and uncertainty intervals. A correct diversity index answers a defined descriptive question, not every ecological question at once.
How an ecological field comparison should be checked
Before treating the two invented assemblages as field evidence, write down the exact ecological question. If the aim is to count all recorded taxa, observed richness is relevant. If the aim is to understand how likely an encountered individual is to represent a common taxon, Shannon or Simpson measures may be more informative. If the aim is to locate a threatened species, neither global index tells the whole story.
Sampling area and effort should be standardised before inferring abundance differences. One site sampled in a ten-minute transect and another surveyed for several hours may produce different counts even when their true communities are similar. Where feasible, investigators document effort, replicate locations, observer methods and seasonal conditions, then consider sample coverage or appropriate rarefaction methods.
Detection probabilities may vary across species. A conspicuous bird, a cryptic amphibian and a microbe do not have equal probabilities of detection through every method. An index computed from detections can therefore reflect a mixture of true abundance and detectability. That is not a reason to discard diversity indices, but a reason to be precise about whether the input measures organisms, observations or occupancy.
A numerical sensitivity check makes the issue concrete. Change Site B from counts (70,10,10,10) to (60,20,10,10) while retaining N=100 and S=4. Simpson concentration falls from 0.52 to 0.60²+0.20²+0.10²+0.10²=0.42. Hill q=2 diversity rises from about 1.923 to 1/0.42≈2.381. Moving ten counted individuals from the dominant species to a less common species changes evenness without changing richness.
The revised Shannon entropy is −[0.60ln0.60+0.20ln0.20+2(0.10ln0.10)]≈1.0889, giving Hill q=1 effective diversity of approximately 2.971. Both abundance-sensitive measures rise, but not by identical amounts because they weight rare and dominant taxa differently. A researcher should not expect every diversity metric to respond equally to the same changed count vector.
Spatial context must also be preserved. A site-level vector that pools two very different habitat patches might conceal a dominant species in one patch and a different dominant species in another. Pooling can answer a site-wide question while simultaneously hiding the location of sensitive habitat or a species of concern. When the question is spatial, retain mapped observations rather than only pooled totals.
Finally, biodiversity decisions must be tied to biological purpose. A lower effective species number does not automatically prove poor habitat condition, and a higher number does not automatically prove resilience. Species identities, ecological interactions, conservation status and pressures can matter more than small differences between two index values. Report the indices as evidence, not as an unqualified grade for nature.
Can richness be equal while diversity differs? Yes. That is exactly what our two invented sites demonstrate. Both have four recorded taxa, but their Shannon-effective and Simpson-effective diversities differ substantially.
Does Site A automatically deserve more protection? No. A protection decision also concerns threats, rare taxa, habitat function, connectivity, irreplaceability and local conservation priorities. An evenness score cannot settle those issues.
Why does Simpson order two give a smaller effective number than Shannon order one in B? Order two responds more strongly to the dominant species. Both indices are consistent with the same abundances and simply weight their distribution differently.
Practice: change Site B counts to (40,20,20,20). The new Simpson concentration is 0.40²+3×0.20²=0.28, so D2=1/0.28≈3.5714. Richness remains four, but diversity rises as abundances become more even.
For a second exercise, add a fifth species with one recorded individual while holding N=100 by moving one individual from an existing taxon. Richness becomes five, but the exact changes to Shannon and Simpson depend on which taxon lost an individual. Write the whole new count vector before calculating.
In our original datasets, both sites have 100 individuals and four species. Yet D1 equals four versus about 2.561 and D2 equals four versus about 1.923. The difference lies in abundance evenness, not species presence.
Read The Gold Standard of Ecology for the subject framework, then The Gold Standard of Conservation Biology for broader biodiversity decisions. For the role of position and neighbourhood, see the connected Geostatistics example.
For methodology, consult Hill’s diversity framework, Utrecht’s Hill-number tutorial, and iNEXT’s sampling coverage discussion. Their work supports the method; our field sites and counts are fictional.
The practical lesson is to retain the observation table and sampling design behind each index. A headline count may be useful, but interpreting ecology requires knowing what was observed and how its distribution was measured. Return through The Gold Standard hub.