vazelinahS

2021-08-19

To find the measure of diversity H for the following proportions of each:
hemlock, 0.521. beech, 0.324. birch, 0.081. maple, 0.074.

Tasneem Almond

Modeling diversity of species:
One measure of the diversity of the species in an ecological community is modeled by the formula,
$H=\left[{P}_{1}{\mathrm{log}}_{2}{P}_{1}+{P}_{2}{\mathrm{log}}_{2}{P}_{2}+\cdots +{P}_{n}{\mathrm{log}}_{2}{P}_{n}\right]\dots \dots \dots ..\left(1\right)$
Where ${P}_{1},{P}_{2},\cdots ,{P}_{n}$ are the proportions of sample that belong to the each of n species found in the sample.
Given that ${P}_{1}=0.521,{P}_{2}=0.324,{P}_{3}=0.081$ and ${P}_{4}=0.074$
Substitute these values in equation (1) to get the following,
$H=-\left[0.521{\mathrm{log}}_{2}0.521+0.324{\mathrm{log}}_{2}0.324+0.081{\mathrm{log}}_{2}0.081+0.074{\mathrm{log}}_{2}0.074\right]\dots \dots \dots \dots ..\left(2\right)$
Change-of -Base theorem:
For any positive real numbers x, a and b, where $a\ne q1$ and $b\ne q1$, the following holds.
${\mathrm{log}}_{a}x=\frac{{\mathrm{log}}_{b}x}{{\mathrm{log}}_{b}a}$
By using the change of base theorem the terms in the equation (2) becomes the following.
${\mathrm{log}}_{2}0.521=\frac{\mathrm{log}0.521}{{\mathrm{log}}_{2}}$
$=\frac{-0.2832}{0.3010}$
$=-0.9406$
${\mathrm{log}}_{2}0.324=\frac{\mathrm{log}0.324}{{\mathrm{log}}_{2}}$
$=\frac{-0.4895}{0.3010}$
$=-1.6259$
${\mathrm{log}}_{2}0.081=\frac{\mathrm{log}0.081}{\mathrm{log}2}$
$=\frac{-1.0915}{0.3010}$
$=-3.6259$
${\mathrm{log}}_{2}0.074=\frac{\mathrm{log}0.074}{\mathrm{log}2}$
$=\frac{-1.1308}{0.3010}$
$=-3.7563$
Substitute in equation (2) to get the following,
$H=-\left[0.521\left(-0.9406\right)+0.324\left(-1.6259\right)+0.081\left(-3.6259\right)+0.074\left(-3.7563\right)\right]$
$=-\left[-0.4900759-0.52680271-0.29370068-0.27796849\right]$
$=-\left[-1.5886\right]$
$=1.5886$
$\approx 1.589$
So, the measure of diversity H is 1.589.
Final statement:
The measure of diversity H is 1.589.

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