The universal scaling between the average slow relaxation/transport and the average picosecond rattling motion inside the cage of the first neighbors has been evidenced in a variety of numerical simulations and experiments. Here, we first show that the scaling does not need information concerning the arbitrarily-defined glass transition region and relies on a single characteristic length scale a21/2 which is determined even far from that region. This prompts the definition of a novel reduced rattling amplitude u21/2 which has been investigated by extensive molecular-dynamics simulations addressing the slow relaxation, the diffusivity, and the fast cage-dynamics of both components of an atomic binary mixture. States with different potential, density, and temperature are considered. It is found that if two states exhibit coinciding incoherent van Hove function on the picosecond timescale, the coincidence is observed at long times too, including the large-distance exponential decay-a signature of heterogeneous dynamics-observed when the relaxation is slow. A major result of the present study is that the correlation plot between the diffusivity of the two components of the binary mixtures and their respective reduced rattling amplitude collapse on the same master curve. This holds true also for the structural relaxation of the two components and the unique master curve coincides with the one of the average scaling. It is shown that the breakdown of the Stokes-Einstein law exhibited by the distinct atomic species of the mixture and the monomers of a chain in a polymer melt is predicted at the same reduced rattling amplitude. Finally, we evidence that the well-known temperature/density thermodynamic scaling of the transport and the relaxation of the mixture is still valid on the picosecond timescale of the rattling motion inside the cage. This provides a link between the fast dynamics and the thermodynamic scaling of the slow dynamics. © 2013 American Institute of Physics.