We consider the task of estimating the mean of a population of positive definite or positive semi-definite matrices. Such estimation tasks arise naturally, for example, in the analysis of neuroscience data and in network analysis. We compare two natural estimators for use in this setting. The first is to take as our population estimate the arithmetic mean of the observed covariance matrices. The second approach, which has received far less attention in the literature, is to compute the Fréchet mean of the sample matrices, in which the mean is taken with respect to the geometry of the positive definite cone. Empirically, neither of these two sample means dominates the other in recovering the population mean. We provide a theoretical explanation for this behavior, as well as a comparison of these two methods as applied to networks derived from fMRI data.