The pattern of species abundances is central to ecology. But direct measurements of species abundances at ecologically relevant scales are typically unfeasible. This limitation has motivated a long-standing interest in the relationship between the abundance distribution in a large, regional community and the distribution observed in a small sample from the community. Here, we develop a statistical sampling theory to describe how observed patterns of species abundances are influenced by the spatial distributions of populations. For a wide range of regional-scale abundance distributions we derive exact expressions for the sampled abundance distributions, as a function of sample size and the degree of conspecific spatial aggregation. We show that if populations are randomly distributed in space then the sampled and regional-scale species-abundance distribution typically have the same functional form: sampling can be expressed by a simple scaling relationship. In the case of aggregated spatial distributions, however, the shape of a sampled species-abundance distribution diverges from the regional-scale distribution. Conspecific aggregation results in sampled distributions that are skewed towards both rare and common species. We discuss our findings in light of recent results from neutral community theory, and in the context of estimating biodiversity.