The complete spectrum of the endstates -naked singularities, or blackholes -of gravitational collapse is analyzed for a wide class of N -dimensional spacetimes in spherical symmetry, which includes and generalizes the dust solutions and the case of vanishing radial stresses. The final fate of the collapse is shown to be fully determined by the local behavior of a single scalar function and by the dimension N of the spacetime. In particular, the "critical" behavior of the N = 4 spacetimes, where a sort of phase transition from black hole to naked singularity can occur, is still present if N = 5 but does not occur if N > 5, independently from the initial data of the collapse. Physically, the results turn out to be related to the kinematical properties of the considered solutions.