Dark matter particles form halos that contribute the major part of the mass of galaxy clusters. The formation of these cosmological structures have been investigated both observationally and in numerical simulations, which have confirmed the existence of a universal mass profile. However, the dynamic behaviour of dark matter in halos is not as well understood. We have used observations of 16 equilibrated galaxy clusters to show that the random velocities of dark matter particles are larger on average along the radial direction than along the tangential, and that the magnitude of this velocity anisotropy is radially varying. Our measurement implies that the collective behaviour of dark matter particles is fundamentally different from that of normal particles and the radial variation of the anisotropy velocity agrees with the predictions of numerical simulation.