We show how the spatial structure of species diversity can be analyzed using the correlation between the log abundances of the species in the communities, assuming that two communities at different localities can be described by a bivariate lognormal species abundance distribution. A useful property of this approach is that the log abundances of the species at two localities can be considered as samples from a bivariate normal distribution defined by only five parameters. The variances and the correlation can be estimated by maximum likelihood methods even if there is no information about the sampling intensity and the number of unobserved species. This method also enables estimation of over-dispersion in the sampling relative to a Poisson distribution that allows sampling adjustment of the estimate of β-diversity. Furthermore, we also obtain a partitioning of species diversity into additive components of α-, β- and γ-diversity. For instance, if the correlation between the log abundances of the species is close to one, the same species will be common and rare in the two communities and the β-diversity will be low. We illustrate this approach by analysing similarities of communities of rare and endangered species of oak-living beetles in south-eastern Norway. The number of recorded species was estimated to be only 48.1% of the total number of species actually present in these communities. The correlations among communities dropped rather quickly with distance with a scaling of order 200 km. This illustrates large spatial heterogeneity in species composition, which should be accounted for in the design of schemes of such devices for assessing species diversity in these habitat-types.