Chao1 Calculator: Estimate True Species Richness from Abundance Data
Estimate true species richness from abundance counts or from singletons and doubletons: Chao1 (classic and bias-corrected, with the (n−1)/n small-sample factor), variance, log-linear 95% confidence interval, ACE, unseen species and sample coverage (Good and Chao–Jost).
How it works
Chao1 = S_obs + [(n − 1)/n] × f₁² ÷ (2f₂) (f₂ = 0: S_obs + [(n − 1)/n] × f₁(f₁ − 1) ÷ 2) · bias-corrected = S_obs + [(n − 1)/n] × f₁(f₁ − 1) ÷ [2(f₂ + 1)] · var = f₂[(A/2)(f₁/f₂)² + A²(f₁/f₂)³ + (A²/4)(f₁/f₂)⁴], A = (n − 1)/n · CI = S_obs + T ÷ K to S_obs + T × K, T = Chao1 − S_obs, K = exp(z√ln(1 + var/T²)) · ACE = S_abund + S_rare/C + (f₁/C)γ², C = 1 − f₁/N_rare · Good coverage = 1 − f₁/n
Species you saw only once (singletons, f₁) are the clue to species you missed entirely: if many species turned up just once, more were probably present but went unsampled, and if most rare species were seen twice (doubletons, f₂), the sample is close to complete. Chao1 turns the ratio of singletons to doubletons into an estimate of the number of undetected species and adds it to the observed richness, giving a lower bound on true richness. Current software (EstimateS 9.1 onward, iNEXT, SpadeR) includes the small-sample factor (n − 1)/n, which this calculator uses too, so results can be slightly lower than the textbook S_obs + f₁²/2f₂. The bias-corrected form avoids dividing by zero when there are no doubletons and is EstimateS's default. The confidence interval is the log-linear interval of Chao (1987), which assumes the number of unseen species is log-normally distributed, so the lower limit never falls below the observed count and the interval is skewed upward. ACE (abundance-based coverage estimator) uses all species with 10 or fewer individuals rather than only f₁ and f₂; when those counts are very uneven (CV above 0.5), Chao recommends reporting the larger of classic Chao1 and ACE. Good's coverage, 1 − f₁/n, estimates the share of the individuals in the community that belong to species already detected; the Chao–Jost version also uses f₂ and is slightly more accurate.
Worked example
A beetle survey records 17 species and 97 individuals (25, 18, 12, 9, 7, 5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1): 5 singletons and 3 doubletons. With (n − 1)/n = 96/97 = 0.9897, Chao1 = 17 + 0.9897 × 25 ÷ 6 = 17 + 4.12 = 21.1 species. The variance is 3 × [0.4948 × 1.667² + 0.9795 × 1.667³ + 0.2449 × 1.667⁴] = 23.40 (SE 4.84), and the log-linear 95% interval is 17.7 to 42.5 species. The bias-corrected Chao1 is 17 + 0.9897 × 20 ÷ 8 = 19.5. ACE, using the 14 species with 10 or fewer individuals (42 individuals, coverage 0.881, γ² = 0.514), gives 21.8. About 4 species remain unseen, and Good's coverage is 1 − 5/97 = 94.8%.
Sources: EstimateS 9 User's Guide (Colwell): Chao1, bias-corrected form, (n−1)/n factor, log-linear CI, ACE · EstimateS User's Guide PDF
Frequently asked questions
How do you calculate Chao1?
Count the species seen once (f₁) and twice (f₂) and add f₁² ÷ (2 × f₂) to the observed species count, multiplied by (n − 1)/n in current software. With 17 species, 5 singletons, 3 doubletons and 97 individuals: 17 + 0.9897 × 25 ÷ 6 = 21.1.
Is Chao1 an upper or lower bound?
A lower bound on true richness. It is conservative when many rare species remain undetected, so the real number may be higher, especially when coverage is low.
Should I use classic or bias-corrected Chao1?
EstimateS uses the bias-corrected form by default and always when there are no doubletons. Many papers report the classic form when f₂ > 0. If the abundance distribution is very uneven (CV above 0.5), Chao recommends the larger of classic Chao1 and ACE.
Why is the confidence interval not symmetric?
Chao's log-linear interval treats the number of unseen species as log-normal, so it cannot fall below the observed richness and stretches further upward than downward, which matches how uncertain the unseen tail really is.
What about incidence (presence/absence) data?
Use Chao2, which replaces singletons and doubletons with species found in exactly one and exactly two sampling units, and the number of individuals with the number of sampling units.
What sample coverage is good enough?
Above about 90–95% the sample represents the community well. Lower coverage means more sampling would still find new species and richness estimates are less reliable; compare sites at equal coverage with rarefaction.
Related calculators
This calculator is for educational purposes. Double-check important results before acting on them.