Three-dimensional (3-D) images of two ceramic-matrix textile composites were captured by X-ray micron-resolution computed tomography (mu CT) on a synchrotron beamline. Compared to optical images of sections, CT data reveal comprehensive geometrical information about the fiber tows; information at smaller scales, on matrix voids, individual fibers, and fiber coatings, can also be extracted but image artifacts can compromise interpretation. A statistical analysis of the shape and positioning of the fiber tows in the 3-D woven architecture is performed, based on a decomposition of the spatial variations of any geometrical characteristic of the tows into non-stochastic periodic trends and non-periodic stochastic deviations. The periodic trends are compiled by exploiting the nominal translational invariance of the textile, a process that maximizes the information content of the relatively small specimens that can be imaged at high resolution. The stochastic deviations (or geometrical defects in the textile) are summarized in terms of the standard deviation of any characteristic at a single point along the axis of a tow and correlations between the values of deviations at two different points on the same or different tows. The tow characteristics analyzed consist of the coordinates of the centroids of a tow, together with the area, aspect ratio, and orientation of its cross-section. The tabulated statistics are sufficient to calibrate a probabilistic generator (detailed elsewhere) that can create virtual specimens of any size that are individually distinct but share the statistical characteristics of the small specimens analyzed by X-ray mu CT. The data analysis presented herein forms the first step in formulating a virtual test of textile composites, by providing the statistical information required for realistic description of the textile reinforcement.