Averaging data is among the most fundamental of the statistician’s tools, but its implementation on non-Euclidean spaces, capturing data modalities that differ from the typical vector-valued covariates common in traditional statistical literature, often requires care. Here, we develop statistical theory to justify the use of geometric averaging methods on non-Euclidean spaces in practice. In particular, we derive properties of Wasserstein barycenters, which are the correct generalization of averages to the Wasserstein space.