Variation, standard deviation and diversity data

What you’ll learn

• Calculate and interpret a sample standard deviation.
• Use the stated diversity-index formula and distinguish richness from evenness.

Understand the idea

Biological observations vary between individuals, so a mean alone does not fully describe a sample. The range uses only the largest and smallest observations. Sample standard deviation uses the squared distances of all observations from the mean and has the same units as the original measurements. State whether an error bar represents range, standard deviation or another quantity; its meaning cannot be inferred safely from its appearance.

Species richness counts the species present. Diversity also reflects how evenly individuals are distributed among those species. In the index used here, $D=N(N-1)/\sum n_i(n_i-1)$, where $N$ is the total number of individuals and $n_i$ the count for species $i$. For comparable sampling, larger values of this form of the index indicate greater diversity. Other named indices use different formulae, so identify the supplied version rather than relying on the name alone.

A numerical comparison is meaningful only when the samples represent the populations being compared. Match sampling effort and identification criteria, and avoid selecting only places where the desired organism is already visible. An unusual observation should be checked against records and measurement quality; being inconvenient is not a valid reason to delete it. Conclusions should reflect sample size, biological variation and limits on the area or population sampled.

Key formulae

$$\overline{x}=\frac{\sum x_i}{n}$$

$$s=\sqrt{\frac{\sum(x_i-\overline{x})^2}{n-1}}$$

$$D=\frac{N(N-1)}{\sum n_i(n_i-1)}$$

Worked example

A comparable sample contains 5 individuals of species A, 3 of B and 2 of C. Calculate the diversity index using the displayed formula.

1. The total is $N=5+3+2=10$, so the numerator is $10(9)=90$.
2. The denominator is $5(4)+3(2)+2(1)=20+6+2=28$.
3. The index is $D=90/28=3.214\ldots$. It is unitless and should be compared only with values using the same formula.

Answer: $D\approx3.21$.