Preferential attachment is a widely adopted paradigm for understanding the dynamics of social networks. Formal statistical inference, for instance GLM techniques, and model verification methods will require knowing test statistics are asymptotically normal even though node or count based network data is nothing like classical data from independently replicated experiments. We therefore study asymptotic normality of degree counts for a sequence of growing simple undirected preferential attachment graphs. The methods of proof rely on identifying martingales and then exploiting the martingale central limit theorems.