An exact expression of the mean arrival time drift velocity W_m of electrons in swarms in gases is deduced by taking into account the third and higher order gradient terms in the continuity equation. Here, W_m is defined as the ratio of the difference between the two positions in the field direction to the difference between the mean arrival times of electrons at the positions. The deduced expression is:
W_m=W_r-2α_TD_L+3α_T²D₃-4α_T³D₄+…,
where W_r is the center-of-mass drift velocity, D_L is the longitudinal diffusion coefficient and D_n (n_??_3) are the higher order coefficients. α_T is the Townsend first ionization coefficient. The importance of the present expression lies in that despite that W_m is the most commonly measured drift velocity by shutter techniques, there has been no exact expression for it, and also in that recent exact treatment of electron and ion swarms has started to consider the higher order coefficients.