Abstract: Data arising from neuroimaging studies often consists of a collection of sample covariance matrices describing the dependence among blood oxygen level signals measured at different locations in the brain. A natural approach to population-level inference based on these samples is to consider the arithmetic mean of these matrices. However, the nature of the data under study suggests that this is a suboptimal choice, as the arithmetic mean fails to account for the structure of the positive definite cone. Even in the absence of covariance structure, the observed covariance matrices may differ in their noise structures owing to subject-level factors (e.g., head movement in the fMRI machine), in which case a weighted average is more appropriate. In this talk, we will discuss both of these settings and present alternative choices of matrix averages that are better suited to these settings. We will demonstrate some of these techniques in an application to fMRI data.