The erodibility of a grain on a rough bed is controlled by, among other factors, its relative projection above the mean bed, its exposure relative to upstream grains, and its friction angle. Here we report direct measurements of friction angles, grain projection and exposure, and small-scale topographic structure on a variety of water-worked mixed-grain sediment surfaces. Using a simple analytical model of the force balance on individual grains, we calculate the distribution of critical shear stress for idealized spherical grains on the measured bed topography. The friction angle, projection, and exposure of single grain sizes vary widely from point to point within a given bed surface; the variability within a single surface often exceeds the difference between the mean values of disparate surfaces. As a result, the critical shear stress for a given grain size on a sediment surface is characterized by a probability distribution, rather than a single value. On a given bed, the crtitical shear stress distributions of different grain sizes have similar lower bounds, but above their lower tails they diverge rapidly, with smaller grains having substantially higher median critical shear stresses. Large numbers of fines, trapp.ed within pockets on the bed or shielded by upstream grains, are effectively lost to the flow. Our calculations suggest that critical shear stress, as conventionally measured, is defined by the most erodible grains, entrained during transient shear stress excursions associated with the turbulent flow; this implies a physical basis for the indeterminacy of initial motion. These observations suggest that transport rate/shear stress relationships may be controlled, in part, by the increasing numbers of grains that become available for entrainment as mean shear stress increases. They also suggest that bed textures and grain size distributions may be controlled, within the constraints of an imposed shear stress and sediment supply regime, by the influence of each size fraction on the erodibility of other grain sizes present on the bed.