By the Monte Carlo method,the effect of dispersion of disc size distribution on the velocity distributions and correlations of a polydisperse granular gas with fractal size distribution is investigated in the same inelasticity.The dispersion can be described by a fractal dimension D,and the smooth hard discs are engaged in a twodimensional horizontal rectangular box,colliding inelastically with each other and driven by a homogeneous heat bath.In the steady state,the tails of the velocity distribution functions rise more significantly above a Ganssian as D increases,but the non-Ganssian velocity distribution functions do not demonstrate any apparent universal form for any value of D.The spatial velocity correlations are apparently stronger with the increase of D.The perpendicular correlations are about half the parallel correlations,and the two correlations are a power-law decay function of dimensionless distance and are of a long range.Moreover,the parallel velocity correlations of postcollisional state at contact are more than twice as large as the precollisionaJ correlations,and both of them show almost linear behaviour of the fractal dimension D.