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The Shannon-Weaver index of diversityIn assessing the vegetation (or other food sources) we have sampled from an environment, the first thing we are interested in is the proportion of the soil that has any vegetation at all on it, the percentage cover. Next, however, we need to measure the variability of the cover. How can we do this? One possible measure is the species richness - the total number of species (or other taxa) we have observed: clearly an environment where we find 10 species in a 1-sq-metre quadrant is more diverse than one where we find 5 in a square metre. However, this does not seem adequate. If in one environment that has 10 species per sq m., 90% of the cover is contributed by a single species, that is clearly less diverse than an environment with the same species richness, but in which all 10 species are found roughly equally. One way of assessing diversity is the Shannon-Weaver entropy statistic, discussed by Shannon and Weaver (1949). If pi is the proportion of cover contributed by the ith species, entropy is defined as the sum over all species of öpilog pi. The minus sign is used so as to get a positive result, since probabilities are always less than one, and the logs of numbers less than one are always negative. In communications theory, which is where the entropy statistic was first used (and from where it was introduced into cognitive psychology), the logs are sometimes taken to the base 2, in which case entropy is measured in the units of information called bits, familiar from descriptions of computers (one byte is 8 bits, one Kb is 1024 bytes, etc). However, in ecological work we always use "natural logarithms", otherwise known as logs to the base e, and usually abbreviated ln in computer languages and on calculator keypads. If we use natural logarithms, entropy is regarded as a units-free index number. The maximum value you can get is ln(S) where S is the number of species; in practice values above 5 bits are rarely observed. For more information see Krebs (1989), p. 361ff. Entropy is sometimes referred to as "information". Note too that you will sometimes see references to it as the Shannon-Wiener index; this acknowledges the contribution of the well known mathematician Norbert Wiener, the effective founder of cybernetics, with which information theory is closely linked. The Shannon-Weaver index is not of course specific to ecology - it has many other uses, and can, for example, be used to describe the variability of a stimulus set, or of an animal's or a person's behaviour. It is commonly used in the analysis of animal communication - for a recent discussion, see McCowan, Hanser & Doyle (1999). Calculating entropy is easy enough with a calculator provided it has an ln button. However, you need to note the following carefully: The actual value of entropy you get will depend how finely you have classified the plants (etc): if you have classified them to species level, you will get a higher numerical value than if you have just classified them into broad groups. So if you want to compare the diversities of two environments, make sure you classify with equal fineness in both. Entropy is not the only possible measure of diversity, and it is used more because of its mathematical convenience than because it could be shown to the perfect measure. Species richness, percentage cover and diversity do not exhaust what we can say about the vegetation in an environment: there is also the question of its spatial distribution: with a given level of species richness and diversity, the plants of a given species can either be clumped together, scattered at random, or distributed systematically. Additional information on the webThe following sources are all at a fairly similar level to the notes above, but they are all slightly different and may help you understand if you are still puzzled:
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