For quantitative data both taxonomic diversity (a) and average taxonomic distinctness (a*) are sample-size independent (Figures 2e and 2f). Whatever the hierarchy or subsample size, a has been shown to be exactly unbiased and a* is close to being so (except for very small subsamples). For nonquantitative data (a species list), when random subsamples of a fixed number of 'species' are drawn from the full list, the mean value of a+ is unchanged (Figure 2g), and its exact unbiasedness in all cases has again been demonstrated. This lack of dependence of a+ (in mean value) on the number of species in the sample has far-reaching consequences for its use in comparing historic data sets and other studies for which sampling effort is uncontrolled, unknown or unequal. VarTD also has this desirable sampling property (Figure 2h).
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