Quantitative diffusion tensor imaging (DTI) has become the major imaging modality to study properties of white matter and the geometry of fiber tracts of the human brain. Clinical studies mostly focus on regional statistics of fractional anisotropy (FA) and mean diffusivity (MD) derived from tensors. Existing analysis techniques do not sufficiently take into account that the measurements are tensors, and thus require proper interpolation and statistics of tensors, and that regions of interest are fiber tracts with complex spatial geometry. We propose a new framework for quantitative tract-oriented DTI analysis that systematically includes tensor interpolation and averaging, using nonlinear Riemannian symmetric space. A new measure of tensor anisotropy, called geodesic anisotropy (GA) is applied and compared with FA. As a result, tracts of interest are represented by the geometry of the medial spine attributed with tensor statistics (average and variance) calculated within cross-sections. Feasibility of our approach is demonstrated on various fiber tracts of a single data set. A validation study, based on six repeated scans of the same subject, assesses the reproducibility of this new DTI data analysis framework.