that typically occur in complex condensed-matter systems. Here, we employ a superconducting quantum bit (qubit), a transmon, to carry out an analog simulation of motional averaging, a phenomenon initially observed in nuclear magnetic resonance (NMR) spectroscopy. To realize this effect, the flux bias of the transmon is modulated by a controllable pseudo-random telegraph noise, resulting in stochastic jumping of the energy separation between two discrete values. When the jumping is faster than a dynamical threshold set by the frequency displacement of the levels, the two separated spectral lines merge into a single narrow-width, motional-averaged line. With sinusoidal modulation a complex pattern of additional sidebands is observed. We demonstrate experimentally that the modulated system remains quantum coherent, with modified transition frequencies, Rabi couplings, and dephasing rates. These results represent the first steps towards more advanced quantum simulations using artificial atoms.