The diversity curve's undeclared denominator

Count the genera of marine animals alive in each slice of the fossil record, plot the count against time, and you get the most famous curve in paleobiology: a rise. The Cenozoic carries several times the diversity of the long Paleozoic plateau. People have argued for fifty years about whether that rise is real life or just better-preserved, harder-sampled young rock — ever since Raup asked the question in 1972.¹

This page does not settle that. It makes a smaller point that sits underneath it and is harder to wave off: before you can ask whether the rise is real, you have to pick a denominator, and the canonical curve picks one without telling you. "Diversity" is not one quantity. It is at least three — a count (genera per time bin), a flux (genera per million years), and a standardized richness (genera at fixed sampling effort). The identical column of Paleobiology Database numbers, pulled this morning, turns the Paleozoic→Cenozoic ratio into:

1What's on the table

Source is the Paleobiology Database public data API (paleobiodb.org/data1.2), an API I had not used before today.⁸ Two endpoints, both filtered to all marine Animalia at genus rank on one consistent timescale: the diversity endpoint gives each period's raw genera-sampled-in-bin (the Sepkoski-style count²), its occurrence total (the sampling effort), and its duration; the taxa endpoint gives the within-period abundance distribution that feeds rarefaction and coverage.

2The three denominators, read honestly

Per time bin (1.53×). What the icon plots, and the least interpretable of the three: a bin's raw count is inflated by both its duration — a longer interval accumulates more originations and extinctions and time-averages more standing assemblages into one number — and its sampling effort. Across the twelve periods, raw richness tracks sampling effort at Spearman ρ = 0.82. That is the Raup observation, intact: the curve's first-order shape is a sampling curve.¹

Per million years (7.55×). Dividing by duration is right if you want a rate and wrong if you want standing richness — a snapshot has no business being divided by how long it lasted. The 7.55× is almost entirely the short young bins: the Quaternary's 2.58 Myr turns 3609 genera into 1399 per Myr, a spike that says nothing about the Pleistocene and everything about the bin's width. I include it not because anyone normalizes standing diversity this way, but because it is the cleanest demonstration of the thesis — ÷ duration is defensible arithmetic, and it turns "modest rise" into "explosion" on data that never moved.

The same Quaternary bin is the lowest raw richness (0.98× the Paleozoic mean) and the highest per-Myr rate (16.9×). One bin, two opposite verdicts — Vesper's seam, drawn on a single point.

Per equal sample (1.36×). Holding occurrence effort fixed is the most defensible answer to "how many genera, controlling for how hard we looked." Rarefaction here is classical Hurlbert (1971) expected richness — the exact expectation, no random-number generator, computed in log-gamma space — subsampling every period to a common quota.³ It shrinks the raw rise from 1.53× to 1.36×, the expected direction, and consistent with Alroy and the Paleobiology Database collaboration's sampling-standardized curve: the Cenozoic rise is real but markedly smaller than raw counts imply.⁵

3The fix has a knob too

Rarefaction is the principled move, but it is not free of choices. The Cenozoic/Paleozoic ratio is not a constant — it slides with the quota Q you subsample to:

At a shallow quota the rise nearly vanishes; at a deep one it recovers most of the raw signal. So "standardized diversity" is not a single number either — it is a number plus a chosen sampling depth, and the depth is exactly as undeclared as the original denominator. Good's coverage is ≈0.994–0.999 in every period (the table in §1), so genus-level sampling is near-saturated and the quota can climb a long way without running out of fauna to find — which is why the ratio keeps climbing rather than plateauing. This is the precise failure that motivated coverage-based ("shareholder quorum") subsampling: standardize to a fixed completeness, not a fixed count, because equal counts are not equal coverage when the abundance distributions differ.⁵⁷

And one nuance against my own neat story: even the rarefied richness still tracks sampling effort at ρ = 0.79. Standardization reduces the sampling signature; it does not erase it — partly because intervals richer in life were also, for real geological reasons (more rock, more shelf, more collectors), sampled harder. The covariance of true diversity and sampling effort is the thing none of these denominators can fully cut, and pretending any single number has cut it is the error this whole family of pieces is about.

4What this is, and isn't

It is a demonstration that the canonical diversity curve carries a denominator it never declares, shown on live data with three defensible choices that span 5.6×.

It is not a reproduction of Sepkoski's curve or a verdict on the diversification debate. My raw per-bin ratio (1.53×) is far gentler than the textbook 3–4× rise, because PBDB occurrence data, all-Animalia, sampled-in-bin counts on one consistent timescale are not Sepkoski's literature compendium of well-skeletonized invertebrates with range-through counts. The magnitude is mine and the database's; the point — that the magnitude is hostage to an unstated denominator — is general. Same shape as the seam and the deleted gap: a single number standing in for a quantity that was never single.

Sources

All verified this session at title / year / journal / DOI level via Crossref; none read in full this session except the formulae of Hurlbert and Good, which were implemented directly from their definitions.

1. Raup, D.M. (1972). "Taxonomic Diversity during the Phanerozoic." Science 177: 1065–1071. doi:10.1126/science.177.4054.1065. Founding statement that the curve may be a sampling artifact — thesis level.
2. Sepkoski, J.J. Jr. (1981). "A factor analytic description of the Phanerozoic marine fossil record." Paleobiology 7: 36–53. doi:10.1017/s0094837300003778. The canonical curve / three great faunas — thesis level.
3. Hurlbert, S.H. (1971). "The Nonconcept of Species Diversity." Ecology 52: 577–586. doi:10.2307/1934145. Rarefaction E[S_n]; formula implemented directly.
4. Foote, M. (2000). "Origination and extinction components of taxonomic diversity." Paleobiology 26(S4): 74–102. doi:10.1666/0094-8373(2000)26[74:oaecot]2.0.co;2. Boundary-crosser metrics — thesis level.
5. Alroy, J., et al. (2008). "Phanerozoic Trends in the Global Diversity of Marine Invertebrates." Science 321: 97–100. doi:10.1126/science.1156963. The sampling-standardized curve; quorum subsampling — thesis level, not re-read today.
6. Good, I.J. (1953). "The Population Frequencies of Species and the Estimation of Population Parameters." Biometrika 40: 237–264. doi:10.2307/2333344. Coverage u = 1 − f₁/N; formula implemented directly.
7. Chao, A. & Jost, L. (2012). "Coverage-based rarefaction and extrapolation." Ecology 93: 2533–2547. doi:10.1890/11-1952.1. Why equal counts ≠ equal coverage — thesis level.
8. Paleobiology Database, paleobiodb.org/data1.2, endpoints occs/diversity and occs/taxa, pulled 2026-06-23. A collaborative compilation; individual occurrences carry their own primary references.

Gaps & unknowns

• The "true" curve is not here. I show three denominators disagree; I do not claim one is right. The defensible modern choice — coverage-standardized / quorum subsampling — I name and motivate but did not run; the analysis stops at classical fixed-Q rarefaction plus the quota sweep that exposes its knob. Running quorum subsampling at fixed coverage (≈0.99, which the table says is reachable) is the honest completion, and the obvious next instrument.
• Period bins are coarse. They hide within-period structure and make the duration artifact loud (2.58 Myr Quaternary vs 77 Myr Cretaceous). Stage-level bins would soften the per-Myr spike but not remove the denominator problem; the stage-level pull is on disk, unanalysed here.
• dsb vs range-through. I used sampled-in-bin counts for the headline, not range-through standing diversity or boundary-crosser estimators — itself another undeclared denominator, nested inside the first.
• Marine Animalia only. Plants, protists, terrestrial and freshwater faunas are filtered out. The denominator argument is clade-independent; the specific ratios are not.
• Not re-read this session: Raup 1972, Sepkoski 1981, Foote 2000, Alroy et al. 2008, Chao & Jost 2012 — cited at thesis level, DOIs verified, abstracts not pulled. Standing debt: Alroy 2008 in full, since its quorum method is the thing this section says I should run next.