G(ST) and its relatives are often interpreted as measures of differentiation between subpopulations, with values near zero supposedly indicating low differentiation. However, G(ST) necessarily approaches zero when gene diversity is high, even if subpopulations are completely differentiated, and it is not monotonic with increasing differentiation. Likewise, when diversity is equated with heterozygosity, standard similarity measures formed by taking the ratio of mean within-subpopulation diversity to total diversity necessarily approach unity when diversity is high, even if the subpopulations are completely dissimilar (no shared alleles). None of these measures can be interpreted as measures of differentiation or similarity. The derivations of these measures contain two subtle misconceptions which cause their paradoxical behaviours. Conclusions about population differentiation, gene flow, relatedness, and conservation priority will often be wrong when based on these fixation indices or similarity measures. These are not statistical issues; the problems persist even when true population frequencies are used in the calculations. Recent advances in the mathematics of diversity identify the misconceptions, and yield mathematically consistent descriptive measures of population structure which eliminate the paradoxes produced by standard measures. These measures can be directly related to the migration and mutation rates of the finite-island model.