18.2.6—Simpson's index of diversity (D)

Spearman’s rank correlation tests for association between two variables using their ranks. It is useful when the data are not quantitative, are not normally distributed, or show a non-linear but monotonic relationship.

1. Pair the two observations from each sample, then state the null hypothesis that there is no correlation.
2. Rank each variable separately, using the course convention for tied values when ties occur.
3. For every pair, calculate the difference in rank, d, then calculate d²; add the d² values to obtain Σd².
4. Substitute Σd² and the number of samples n into the supplied Spearman equation to calculate rₛ.
5. Determine the direction from the sign and the strength from how close rₛ is to +1 or -1; values near 0 indicate little rank association.
6. Compare the calculated value with the critical value for n and the stated probability level, then reject or retain the null hypothesis with a conclusion in context.

The test evaluates a monotonic rank relationship: as one variable increases in rank, the other tends to increase or decrease in rank. It does not require a straight-line relationship, but a non-monotonic pattern is not captured reliably by one coefficient.

A significant rank correlation is evidence of association in the sampled data, not proof that one variable causes the other. A third factor, sampling pattern or coincidence may explain the relationship. This is distinct from Pearson’s linear correlation and Simpson’s index.