The measures are based not just on the species abundances (denoted by x, the number of individuals of species in the sample) but also the taxonomic distances (!j), through a classification tree, between every pair of individuals (the first from species and the second from species j). For a standard Linnean classification these are discrete distances, the simple tree in Figure 3 illustrating path lengths of zero steps (individuals from the same species), one step (same genus but different species,) and two steps (different genera). Figure 3 uses a simple linear scaling whereby the largest number of steps in the tree (two species at greatest taxonomic distance apart) is set to ! = 100. Thus, for a sample consisting only of the five species shown, the path between individuals in species 3 and 4 is !34 = 100, between species 1 and 2 is !12 = 50, between two individuals of species 5 is !55 = 0, etc. Average taxonomic diversity of a sample is then defined as

where the double summation is over all pairs of species i and j (i, j = 1, 2, ..., S; i < j), and N / x , the total number of individuals in the sample. a, then, is simply the average taxonomic distance apart of every pair of individuals in the sample or, to put it another way, the expected path length between any two individuals chosen at random. a is therefore a natural extension of Simpson's index (see Simpson Index), from the case where the path length between individuals is either 0 (same species) or 100 (different species) to a more refined scale of intervening relatedness values (0 = same species; 20 = different species in the same genera; 40 = different genera but same family, etc.). To remove the dominating effect of the species abundance distribution {x/}, leaving a measure which is more nearly a pure reflection of the taxonomic hierarchy, a is divided by the Simpson index to give average taxonomic distinctness

Another way of thinking of this is as the expected taxo-nomic distance between any two individuals chosen at random from the sample, provided those two individuals are not from the same species.

A further form of the index takes the special case where quantitative data are not available and the sample consists simply of a species list (presence/absence data). Both a and a* reduce to the same coefficient

where S is the observed number of species in the sample and the double summation ranges over all pairs i and j of these species (i < j). Put simply, the average taxonomic distinctness (AvTD) A+ of a species list is the average taxonomic distance between all its pairs of species. This is a very intuitive definition of biodiversity, as average taxonomic breadth of a sample.

150 100 50

No. of species, S

No. of species, S

150 100 50

100 1000 10000 100000

Richness (Margalef's d)

Richness (Margalef's d)

10 100 1000 10000 100000

100 1000 10000 100000

10 100 1000 10000 100000

Shannon, H'

50 40 30 20 10

Shannon, H'

10 100 1000 10000 100000 Taxonomic diversity, A

0.75

0.55

0.35

Evenness (Pielou's J')

10 100 1000 10000 100000 Taxonomic diversity, A

Evenness (Pielou's J')

0.75

0.55

0.35

100 1000 10000 100000 Taxonomic distinctness, A*

100 1000 10000 100000 Taxonomic distinctness, A*

70 60 50 40 30 20

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