Neglecting local solute dispersion, we prove that under realistic conditions the ensemble moments of the local concentration satisfy a mean convection dispersion equation (CDE) for both conservative and instantaneously adsorbing solutes, provided that the solute velocity field is stochastically stationary. Thus the same algorithm can be used to model the ensemble mean concentration and its uncertainty. For uniform initial conditions the solution of the mean CDE contains all the information about the local concentration probability distribution function, which is not Gaussian. The governing equation for k-point ensemble moments of the concentration is similar in form to the mean CDE, with the same effective dispersion coefficient and convective velocity. The special case of an impulse initial condition is used to illustrate that volume averaging can significantly change the concentration coefficient of variation except at the plume front. The persistence in the longitudinal direction of concentration moments and concentration k-point moments is caused by large longitudinal megadispersion. Finally, it is suggested that ''ergodicity'' should be interpreted operationally in terms of an acceptably small coefficient of variation of the concentration field so that this concept can be applied to experimental field data.