The Shannon equation and resulting index is the mathematical foundation of the theory of information, which is linked to entropy (order in a system) and proposed to use as measure of diversity in Ecology H= -∑(pi logpi). When H=0, all objects (species) are basically the same (sampling unit contains single species) with no diversity because there is a single state. However with a minimum of two species equally distributed, with two states, the entropy associated would be H=1, and the more species the more H increases. Therefore H is low under strong dominance of one single species and higher as the number of species increases in an assemblage, especially if they are equally distributed. In order to calculate the entropy of a probability distribution thus, it is necessary to calculate the relative frequency of a given state, when the probability of observing each state when selecting an object (i.e. species) at random. I have here assumed that objects are distributed evenly, however when they are not, as it is usually in nature, entropy (H) should decrease. Thus when the relative frequencies of species are known, the Shannon formula for entropy can be used to measure dispersion of the qualitative variable and is widely used to measure species assemblage, computed for each sampling site separately. Thus H encapsulates in one index, number of species and relative abundances and can be extended to systematic categories (family, genus).

While this index was first developed to directly interpret species diversity as resulting from the spatio-temporal, biological and physico-chemical environment surrounding these species, diversity can also be interpreted in terms of number of species and evenness, which basically compares the diversity to the corresponding maximum (when all species are equally represented) value of diversity (Hmax). Evenness is therefore expressed as (H/Hmax), which is a ratio and can be defined as similarity in species relative abundance in a community. From preliminary visualization of your data, it seems that diversity and richness are very similar in the three sites, however evenness is different because S2 and S3 are more similar (same region?, connected habitats) than S1 (different region?, isolated habitat/patch).

Hope this helps. Good luck!

Thank you very much sir, very informative,