Diversity (or biodiversity) is typically measured by a species count (richness) and sometimes with an evenness index; it may also be measured by a proportional statistic that combines both measures (e.g., Shannon‐Weiner index or ). These diversity measures are hypothesized to be positively and strongly correlated, but this null hypothesis has not been tested empirically. We used the results of Caswell's neutral model to generate null relationships between richness (S), evenness ( ), and proportional diversity (). We tested predictions of the null model against empirical relationships describing data in a literature survey and in four individual studies conducted across various scales. Empirical relationships between or and differed from the null model when <10 species were tested and in plants, vertebrates, and fungi. The empirical relationships were similar to the null model when >10 and <100 species were tested and in invertebrates. If >100 species were used to estimate diversity, the relation between and was negative. The strongest predictive models included and . A path analysis indicated that and were always negatively related, that empirical observations could not be explained without including indirect effects, and that differences between the partials may indicate ecological effects, which suggests that S and act like diversity components or that diversity should be measured using a compound statistic.