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Sewall Wright's population structure statistic, FST, measured among samples of world populations is often 15% or less. This would indicate that 85% of genetic variation occurs within groups while only 15% can be attributed to allele frequency differences among groups. In this paper, we show that this low value reflects strong biases that result from violating hidden assumptions that define FST. These limitations on FST are demonstrated algebraically and in the context of analyzing dinucleotide repeat allele frequencies for a set of eight loci genotyped in eight human groups and in chimpanzees. In our analyses, estimates of FST fail to identify important variation. For example, when the analysis includes only humans, FST = 0.119, but adding the chimpanzees increases it only a little, FST = 0.183. By relaxing the underlying statistical assumptions, the results for chimpanzees become consistent with common knowledge, and we see a richer pattern of human genetic diversity. Some human groups are far more diverged than would be implied by standard computations of FST, while other groups are much less diverged. We discuss the relevance of these findings to the application of biological race concepts to humans. Four different race concepts are considered: typological, population, taxonomic, and lineage. Surprisingly, a great deal of genetic variation within groups is consistent with each of these concepts. However, none of the race concepts is compatible with the patterns of variation revealed by our analyses.