Nicolas Alcala and Noah Rosenberg report new theoretical results regarding the use and interpretation of population structure statistics. This work is part of a themed issue in Philosophical Transactions of the Royal Society B entitled “Celebrating 50 years since Lewontin’s apportionment of human diversity,” featuring articles about the legacy of Lewontin’s 1972 seminal paper in the fields of human genetics, statistical genetics, biomedicine, anthropology, forensics, and philosophy.