Compare observed richness at a common number of individuals

Define the ecological target

Observed species richness usually increases as more individuals are examined, so unequal effort can confound comparisons. Individual-based rarefaction asks how many species would be expected in a random smaller subsample from the observed pool. It standardizes one aspect of effort rather than estimating every species in the ecosystem.

Specify the observation model

For a pool of N individuals with ni in species i, the probability a draw of m without replacement misses that species is choose(N−ni, m)/choose(N, m). One minus that probability is its chance of appearing. Summing these appearance probabilities gives expected richness, because expected contributions add even though species appearances are not independent.

Work through the sampling logic

The two original pools each contain 10 individuals and three observed species. One has counts 8, 1, 1; the other 4, 3, 3. They share observed pooled richness but differ in evenness. At small standardized draws, the more even pool is more likely to contribute several species. At a full draw of all 10, both necessarily return all three observed species.

Challenge the inference

The expectation may be fractional because it averages across possible subsamples; a realized sample has integer richness. The model does not extrapolate beyond the observed pool, infer undiscovered species, correct identification errors or equalize habitat area and sampling gear. Equal rarefied richness does not imply equal function, genetic diversity or conservation value.

Design a defensible survey

For a defensible comparison, retain the original counts, taxonomy, sampling unit and effort. Decide whether individual-based or another standardization matches the study. A smaller common draw discards some precision; it does not erase nonrepresentative field collection. Use the virtual counts to understand the probability rather than collect organisms.

Worked original packet

At m=2, the 8, 1, 1 pool has expected richness 44/45+0.2+0.2=62/45≈1.378. The 4, 3, 3 pool has expectation 2/3+8/15+8/15=26/15≈1.733. Both observed pools contain three species, but their expected small-sample richness differs.

Investigation

Compare both pools at draws 1, 2, 5 and 10. Identify the endpoints forced by the design and explain why the intermediate expectation depends on relative counts.

Changed-case transfer

For an observed pool with two species represented 5 and 5 times, what is expected richness at m=1 and m=2?

Feedback

At m 1, exactly one species. At m 2, each species appears with probability 1−choose(5, 2)/choose(10, 2)=7/9; expectation 14/9≈1.556. This is a subsample expectation, not unseen richness.

Sources and evidence boundary

Original instructional prose, values and exercises. Context checked 30 October 2026: primary agency source. All observations are synthetic. No organism collection, trapping, tagging, release or unsafe-site access is requested; a real study requires qualified design, permission and supervision. Public feedback is practice, not independently verified mastery.