A dynamic model of a one-dimensional granular gas with quasi Gaussian size distribution is presented, in which the rods are subject to inelastic mutual collisions and thermalized by a viscosity heat bath. The dispersive degree of the rod size distribution can be measured by the standard deviation sigma at the same mean value mu. By Monte Carlo simulations, the effect of the dispersion of the quasi Gaussian size distribution on dynamic behavior of the system is investigated in the same inelasticity case for the first time. When the typical relaxation time tau of the driving Brownian process is longer than the mean collision time tau(c), the average energy of the system decays exponentially with time towards a stable asymptotic value, and the energy relaxation time tau(B) to a nonequilibrium steady state becomes shorter with increasing values of sigma. In the steady state, as or increases, the velocity distribution deviates more obviously from the Gaussian one, such as the higher kurtosis and the fatter tails of the velocity distribution functions. Furthermore, with the increase of sigma, the rods cluster more significantly, and the spatial correlations of density and velocities become stronger. (c) 2008 Elsevier B.V. All rights reserved.