Diffusion Tensor Imaging (DTI) is a new form of Magnetic Resonance Imaging (MRI) that is revolutionizing brain research as it allows insight into the structure of the white matter. As opposed to standard MRI, DTI measurements at each volume element are not scalars but three-dimensional ellipsoids. This is a new form of data for which standard statistical methods do not apply. In this work, I propose a new probability model for random ellipsoids based on normal random matrix theory, and derive estimation and testing tools for the ellipsoids’ eigenvalues and eigenvectors. In addition, by framing the problem of comparing images as a multiple comparisons problem, interesting spatial locations are found using false discovery rate inference combined with a new form of the empirical null for one-sided tests. These methods are shown in the context of a DTI study of reading ability in children.