The lack of knowledge of relativistic distribution functions for interacting particles introduces uncertainty into the cosmological interpretation of Doppler shifts, particularly for systems such as quasars. Here we define an average Doppler shift, reducible to the nonrelativistic form in the appropriate limit. However, even for the relativistic ideal situation (J\"uttner distribution), second-order effects yield shift corrections after averaging. Such effects should be of interest for nuclear x-ray and \ensuremath{\gamma}-ray lines.