The most popular method for modeling count data is Poisson regression. When data display over-dispersion, thereby deeming Poisson regression inadequate, typically negative-binomial regression is instead used. We show that count data that appear to be equi-dispersed or over-dispersed may actually stem from a mixture of populations with different dispersion levels. To detect and model such a mixture, we introduce a generalization of the Conway-Maxwell-Poisson (COM-Poisson) regression that allows for group-level dispersion. We illustrate mixed dispersion effects and the proposed methodology via semi-authentic data.