Understand the relationship
The reasoning behind the result
The random object is a subset of the entered individuals
Number of possible subsets = C(N,m)
N is the total recorded count and m is the number of individuals selected without replacement. Every subset of m distinct individuals is equally likely under this model. Individuals remain associated with their recorded taxon identities.
This differs from drawing sampling sites, resampling with replacement or predicting a new survey. The workbench computes expectations analytically and does not choose one random subset or delete data from the original record.
A taxon appears unless every selected individual comes from other taxa
P(taxon i omitted)=qᵢ=C(N−nᵢ,m)/C(N,m)
There are nᵢ recorded individuals of taxon i and N−nᵢ individuals of other taxa. The combinatorial ratio counts subsets that contain none of taxon i. Its inclusion probability is 1−qᵢ. If too few other individuals exist to fill the subset, omission is impossible.
Expected richness adds the inclusion probabilities for all taxa. A singleton has inclusion probability m/N, while a common taxon can be very likely to appear even in a small subset. A listed taxon with count zero has inclusion probability zero and adds nothing to expected richness.
Conditional variation accounts for dependence between taxa
qᵢⱼ=C(N−nᵢ−nⱼ,m)/C(N,m)
Taxon inclusion events are not independent because the subset has a fixed size. The variance therefore combines each taxon's Bernoulli variance with the covariance terms qᵢⱼ−qᵢqⱼ. Ignoring those terms would describe a different sampling process.
The resulting standard deviation measures how the number of observed taxa varies across subsets of this particular recorded collection. It does not quantify uncertainty about unobserved species, field detection or the true richness of a larger community. No normal-approximation confidence interval is attached.
Boundary sizes give exact deterministic results
A subset of zero individuals contains no taxa. A subset of one individual contains exactly one taxon when the original collection is nonempty. Drawing every original individual gives the original observed richness. These cases have zero conditional variance.
Expected richness at other sample sizes can be fractional even though any actual subset has a whole number of taxa. For example, an expectation of 11/6 summarizes the relative frequency of one-taxon and two-taxon subsets; it is not a possible fractional taxon count in one draw.
Interpolation within a record does not estimate unseen richness
The curve stops at the original number of individuals. A larger requested subsample is rejected instead of silently returning observed richness or extending a curve without a model. Genuine counts are required; multiplying percentages or biomass by an arbitrary factor changes the sampling universe and the answer.
Removing rare taxa, changing identification resolution or pooling incompatible samples also changes the question. Individual-based rarefaction does not correct those choices and does not establish how much future field effort would be sufficient.
Follow the numbers
Five of six two-individual subsets contain two taxa
- With counts [2,1,1], N=4 and m=2, there are C(4,2)=6 equally likely subsets of individuals.
- One subset contains the two individuals of taxon A and therefore one taxon. The other five subsets contain two taxa. Expected richness=(1×1+5×2)/6=11/6≈1.833333.
- The second moment is (1×1²+5×2²)/6=21/6. Variance=21/6−(11/6)²=5/36, giving standard deviation sqrt(5)/6≈0.372678.
The fractional expectation summarizes all possible subsets while each actual subset still contains one or two whole taxa.
Quick guide
How to use this calculator
- Enter the original genuine individual counts, with one distinct taxon per row.
- Choose a whole subsample size between zero and the total number of recorded individuals.
- Inspect expected richness, each taxon's inclusion probability and the conditional variation among possible subsets.
- Read the curve only within the entered collection. Do not use it as an unseen-richness estimate or a promise about a new field sample.
Calculation method
Calculation and interpretation
Explain how the size of a subsample changes expected observed richness within one finite record.
qᵢ=C(N−nᵢ,m)/C(N,m); E[Sₘ]=Σ(1−qᵢ); Var(Sₘ)=Σqᵢ(1−qᵢ)+2Σᵢ<ⱼ(qᵢⱼ−qᵢqⱼ)
Worked example
Five of six two-individual subsets contain two taxa
The fractional expectation summarizes all possible subsets while each actual subset still contains one or two whole taxa.
qᵢ=C(N−nᵢ,m)/C(N,m); E[Sₘ]=Σ(1−qᵢ); Var(Sₘ)=Σqᵢ(1−qᵢ)+2Σᵢ<ⱼ(qᵢⱼ−qᵢqⱼ)
Supported inputs
Precision and limits
Uniform finite subsampling
All subsets of the entered size are assumed equally likely. Clustered, weighted, site-based or with-replacement sampling requires a different model.
No ecological extrapolation
The workbench does not estimate unseen taxa, true community richness, sufficient future effort or a detection-corrected confidence interval.
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