Fluid residence time is a key concept in the understanding and design of chemically reacting flows. In order to investigate how turbulent mixing affects the residence time distribution within a flow, this study examines statistics of fluid residence time from a direct numerical simulation (DNS) of a statistically stationary turbulent round jet with a jet Reynolds number of 7290. The residence time distribution in the flow is characterised by solving transport equations for the residence time of the jet fluid and for the jet fluid mass fraction. The product of the jet fluid residence time and the jet fluid mass fraction, referred to as the mass-weighted stream age, gives a quantity that has stationary statistics in the turbulent jet. Based on the observation that the statistics of the mass fraction and velocity are self-similar downstream of an initial development region, the transport equation for the jet fluid residence time is used to derive a model describing a self-similar profile for the mean of the mass-weighted stream age. The self-similar profile predicted is dependent on, but different from, the self-similar profiles for the mass fraction and the axial velocity. The DNS data confirm that the first four moments and the shape of the one-point probability density function of mass-weighted stream age are indeed self-similar, and that the model derived for the mean mass-weighted stream-age profile provides a useful approximation. Using the self-similar form of the moments and probability density functions presented it is therefore possible to estimate the local residence time distribution in a wide range of practical situations in which fluid is introduced by a high-Reynolds-number jet of fluid.