## Simpson’s Diversity Index

Simpson’s Diversity Index is a measure both of species richness (i.e. the number of different species present) and species evenness (i.e. how evenly distributed each species is).

$$D = \frac{(N\;\times\;(N\;-\;1))}{(Σn\;\times\;(n\;-\;1))}$$
• $$D$$ = Simpson’s Diversity Index
• $$n$$ = the number of individuals of each species
• $$N$$ = the total number of individuals

## Mann Whitney U test

Mann Whitney U is a statistical test that is used either to test whether there is a significant difference between the medians of two sets of data.

The Mann Whitney U test can only be used if there are at least 6 pairs of data. It does not require a normal distribution.

There are 3 steps to take when using the Mann Whitney U test

### Step 1. State the null hypothesis

There is no significant difference between _______ and _______

### Step 2. Calculate the Mann Whitney U statistic

[llatex]U_1= n_1 \times n_2 + 0.5 n_2 (n_2 + 1)\;- ∑ R_2[/latex]

[llatex]U_2 = n_1 \times n_2 + 0.5 n_1 (n_1 + 1)\;- ∑ R_1[/latex]

• $$n_1$$ is the number of values of $$x_1$$
• $$n_2$$ is the number of values of $$x_2$$
• $$R_1$$ is the ranks given to $$x_1$$
• $$R_2$$ is the ranks given to $$x_2[latex] ### Step 3. Test the significance of the result Compare the value of U against the critical value for U at a confidence level of 95% / significance value of P = 0.05. If U is equal to or smaller than the critical value (p=0.05) the REJECT the null hypothesis. There is a SIGNIFICANT difference between the 2 data sets. If U is greater than the critical value, then ACCEPT the null hypothesis. There is NOT a significant difference between the 2 data sets. ### Worked example A biologist is investigating whether there is a difference in woodland flora between two contrasting areas of woodland, Site A and Site B. Eight randomly placed frame quadrat samples in each of two contrasting areas of woodland produced the following % cover of dog’s mercury, a common woodland plant. Here are the results. ### Step 1. State the null hypothesis There is no significant difference in species richness between Site A and Site B. ### Step 2. Calculate Mann Whitney U statistic (a) Give each result a rank. Calculate the sum of the ranks for the two columns. Now arrange the data values in order, and give each value a rank in order from smallest to highest (b) Calculate [latex]∑R_1$$ and $$∑R_2$$

∑R_1` is the sum of the ranks in the first column (Site A) = 49

$$∑R_2$$ is the sum of the ranks in the first column (Site B) = 87

$$n_1$$ = 8 and $$n_2 = 8$$

(c) Calculate $$U_1$$ and $$U_2$$

$$U_1 = 8\times 8 + 0.5 \times 8 \times (8 + 1) – 87 = 13$$ $$U_2 = 8 \times8 + 0.5 \times 8 \times (8 + 1) – 49 = 51$$

### Step 3. Test the significance of the result

In this example, $$U_1$$ = 13 and $$U_2$$ = 51

Select the smaller or the values. In this case $$U_1$$ is the smaller of the two values, so U=13

The critical value at p=0.05 significance level for $$n_1=8$$ and $$n_2=8$$ is 13. Since our calculated value of 13 = 13, the null hypothesis can be rejected.

In conclusion, there is a significant difference in species richness between Site A and Site B.

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