The computationally expensive nature of molecular dynamics simulation limits the access to length (nanometer) and time scales (nanosecond) that are orders of magnitude smaller than the experiment it models. This limitation warrants a careful estimation of statistical uncertainty associated with the properties calculated from these simulations. The assumption that a simulation is long enough so that the ergodic hypothesis applies is often invoked in the literature for the computation of properties of interest from a single molecular dynamics simulation. Here, we demonstrate that making this assumption without validation results in poor estimates of the self-diffusion coefficient from a single molecular dynamics simulation of Lennard-Jones fluid. This problem is shown to be even more severe when the diffusion coefficient of macromolecules is calculated from a single molecular dynamics simulation. We have shown that conducting multiple independent simulations is necessary to obtain reliable estimates of diffusion coefficients and their associated statistical uncertainties. We show that even a “routine” calculation of the self-diffusion coefficient for a Lennard-Jones fluid, as determined from a linear fit of the mean squared displacement of particles as a function of time, violates the key assumptions of linear regression. A rigorous approach for addressing these issues is presented.