Understand the relationship
The reasoning behind the result
Richness and abundance balance answer different questions
S = number of taxa with nᵢ>0; pᵢ = nᵢ/N
Observed richness S counts the positive taxon records. It does not distinguish four equally common taxa from four taxa with nearly all abundance concentrated in one. Proportions p describe that balance after dividing each quantity by the community's own total N.
A zero row remains visible but does not add an observed taxon. Different names for the same taxon should be reconciled before entry; this workbench does not identify synonyms or combine different taxonomic levels. A missing observation is not an observed zero.
Shannon entropy becomes an effective number after exponentiation
H_b=−Σpᵢ log_b(pᵢ); effective taxa=b^H_b
Each taxon contributes according to both its proportion and its logarithm. The natural-log result is measured in nats; base-two results are in bits. Changing the base changes the numerical entropy, so values from different bases should not be compared without conversion.
Exponentiating in the same base gives the number of equally abundant taxa that would have the same entropy. Two equal taxa therefore have entropy ln(2) in nats and effective diversity two. This effective number is a description of the recorded composition, not an estimate of taxa missed by sampling.
Simpson's name covers several explicitly different quantities
D=Σpᵢ²; Gini–Simpson=1−D; inverse Simpson=1/D
D is the probability that two independent draws with replacement from the recorded composition have the same taxon. Its complement is the probability of different taxa. The reciprocal is another effective number of equally common taxa. The workbench labels each expression rather than calling all three simply a Simpson index.
For genuine counts, drawing two distinct individuals without replacement instead gives same-taxon probability Σnᵢ(nᵢ−1)/[N(N−1)]. That finite-sample expression requires at least two recorded individuals and is not defined for biomass or relative-abundance weights. It differs from Σpᵢ² even when both are valid summaries of the same counts.
Evenness has a real boundary at one observed taxon
J=H_b/log_b(S), for S>1
Pielou evenness compares entropy with the maximum for the observed richness. Equal proportions give J=1 when at least two taxa are present. Using the same logarithm base in the numerator and denominator makes J independent of that base.
At S=1 both H and log(S) are zero, so this ratio is undefined. An empty observation also has no abundance distribution. The workbench preserves richness zero for an all-zero record but does not turn the undefined diversity or evenness into a reassuring ecological score.
Community comparisons retain their own denominators
Each community is normalized by its own recorded total, and the rank curves sort taxa independently. The same rank in two curves need not refer to the same taxon. Use the labelled contribution tables or the separate similarity workflow to compare taxon identity.
Equal richness can coexist with very different concentration, while equal entropy can coexist with different taxon identities. No index difference alone establishes a treatment effect, ecological improvement or adequate sampling. Such interpretations need compatible methods and a study design beyond this descriptive calculation.
Follow the numbers
Equal counts give two effective taxa
- Two taxa with 50 individuals each give N=100 and proportions p1=p2=0.5.
- H=−2×0.5×ln(0.5)=ln(2)≈0.693147 nats. exp(H)=2 effective taxa and J=H/ln(2)=1.
- D=0.5²+0.5²=0.5 and 1−D=0.5. For two distinct drawn individuals, same-taxon probability is [50×49+50×49]/[100×99]=49/99≈0.494949.
The replacement convention changes the two-draw probability, while richness and evenness describe other parts of the composition.
Quick guide
How to use this calculator
- Choose one or two communities and distinguish individual counts from abundance weights.
- Enter taxon labels and complete quantities, preserving meaningful zeros and consistent taxonomic resolution.
- Choose the main index and Shannon log base; the companion results show how the indices differ.
- Inspect the taxon contributions and rank profile. Empty or single-taxon boundaries are labelled instead of assigned invented index values.
Calculation method
Calculation and interpretation
Separate how many taxa were observed from how evenly their abundance is distributed.
pᵢ=nᵢ/Σn; H=−Σpᵢ log_b(pᵢ); D=Σpᵢ²; Gini–Simpson=1−D; effective taxa=b^H and1/D; J=H/log_b(S)
Worked example
Equal counts give two effective taxa
The replacement convention changes the two-draw probability, while richness and evenness describe other parts of the composition.
pᵢ=nᵢ/Σn; H=−Σpᵢ log_b(pᵢ); D=Σpᵢ²; Gini–Simpson=1−D; effective taxa=b^H and1/D; J=H/log_b(S)
Supported inputs
Precision and limits
Observed composition only
The workbench does not estimate unseen taxa, correct imperfect detection or judge ecosystem health. Comparisons depend on compatible methods, effort, coverage and taxonomic resolution.
Counts and weights are different data
Count-based without-replacement probabilities require genuine individual counts. Biomass, cover, proportions and other weights are not automatically valid inputs to individual-based rarefaction.
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