Previous empirical studies of population size and language change have produced equivocal results. We therefore address the question with a new set of lexical data from nearly one-half of the world's languages. We first show that relative population sizes of modern languages can be extrapolated to ancestral languages, albeit with diminishing accuracy, up to several thousand years into the past. We then test for an effect of population against the null hypothesis that the ultrametric inequality is satisfied by lexical distances among triples of related languages. The test shows mainly negligible effects of population, the exception being an apparently faster rate of change in the larger of two closely related variants. A possible explanation for the exception may be the influence on emerging standard (or cross-regional) variants from speakers who shift from different dialects to the standard. Our results strongly indicate that the sizes of speaker populations do not in and of themselves determine rates of language change. Comparison of this empirical finding with previously published computer simulations suggests that the most plausible model for language change is one in which changes propagate on a local level in a type of network in which the individuals have different degrees of connectivity.