Calculate the Shannon Diversity Index and Equitability Index for a population using individual species counts.

## Understanding the Shannon Diversity and Equitability Index

The Shannon Diversity Index is a measure of diversity that accounts for both the abundance and evenness of species present in a sample. It is widely used in ecology and other fields to quantify biodiversity.

### Formula for Shannon Diversity Index

The Shannon Diversity Index \( H' \) is calculated as:

$$ H' = - \sum_{i=1}^S p_i \ln(p_i) $$

Where:

- \( S \): Total number of species
- \( p_i \): Proportion of individuals belonging to species \( i \)

### Equitability Index

The Equitability Index, or Shannon Equitability Index, is a measure of how evenly the individuals are distributed across species. It is calculated by dividing the Shannon Diversity Index \( H' \) by the natural logarithm of the number of species \( S \):

$$ E = \frac{H'}{\ln(S)} $$

An Equitability Index close to 1 indicates that all species are equally abundant, while a value closer to 0 indicates that the population is dominated by a few species.

### Example Calculation

Assume a sample with 3 species with counts of 10, 20, and 30, respectively:

- First, calculate the total count of individuals: \( 10 + 20 + 30 = 60 \).
- Calculate \( p_i \) for each species:
- For species 1: \( p_1 = \frac{10}{60} = 0.1667 \)
- For species 2: \( p_2 = \frac{20}{60} = 0.3333 \)
- For species 3: \( p_3 = \frac{30}{60} = 0.5 \)

- Calculate \( p_i \ln(p_i) \) for each species:
- \( p_1 \ln(p_1) \approx 0.1667 \times -1.7918 = -0.2986 \)
- \( p_2 \ln(p_2) \approx 0.3333 \times -1.0986 = -0.3662 \)
- \( p_3 \ln(p_3) \approx 0.5 \times -0.6931 = -0.3466 \)

- Sum these values and multiply by -1:
$$ H' = -(-0.2986 - 0.3662 - 0.3466) = 1.0114 $$

- Finally, calculate the Equitability Index \( E \):
$$ E = \frac{1.0114}{\ln(3)} \approx 0.921 $$

Thus, the Shannon Diversity Index for this example is approximately 1.011, and the Equitability Index is approximately 0.921.

Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing more fun and curiosity to the world of science and research online.