Chao's own 1984 formula for the Chao1 lower-bound richness estimator is D + f₁²/(2f₂), with no (n−1)/n bias-correction prefactor

statistics-of-the-unseenchao1good-turingsingletonsspecies-richnessprimary-source-verificationsourcing-provenance

Anne Chao's founding 1984 paper, "Nonparametric Estimation of the Number of Classes in a Population" (Scandinavian Journal of Statistics 11: 265–270), derives a lower-bound estimator for the total number of classes as the observed class count d plus a correction term built from singleton and doubleton counts: θ̂ = d + f₁²/(2f₂), where f₁ is the number of singletons and f₂ the number of doubletons. The paper's own prose frames the result as a lower bound whose "performance as an estimator ... is encouraging," and the formula is confirmed by recomputing the paper's own worked numerical examples from the frequency counts it prints. Three of its four examples supply those counts, and all three reproduce exactly under the unscaled form: the reverse side of the ancient-coin hoard (d = 178, f₁ = 156, f₂ = 19 → 818, the paper's 818); the obverse side (d = 141, f₁ = 102, f₂ = 26 → 341, the paper's 341); and Edwards & Eberhardt's penned cottontail rabbits (d = 76, f₁ = 43, f₂ = 16 → 134, the paper's 134, against a known true value of 135). The Smyrna-hoard example (d = 660, f₁ = 658, f₂ = 2) gives 108,901 against the paper's rounded 108,900. A version multiplied by (n−1)/n would have returned 815, 340 and 133 instead. The fourth example, Carothers's Edinburgh taxicabs, is not independently checkable from this paper — its Table 1 reports estimates for 14 data subsets whose underlying capture frequencies live in Carothers (1973), not in Chao.

This directly contradicts the formula claim-singletons-are-the-diagnostic-of-the-unseen had been carrying — "Chao1 = (n−1)/n · f₁²/2f₂" — sourced there from Folgert Karsdorp's blog (Tier 2), not from Chao's own paper. The (n−1)/n bias-correction factor is absent from the 1984 primary; wherever it entered circulation, it did not come from this document. That existing note has been updated in place to record the discrepancy.