Using flow visualization and digital image processing we have shown experimentally that convective motions and mixing of particles within vertically vibrating two-dimensional particle beds are amenable to characterization by fractal properties. The time profile of a fractal dimension presents quantitative information on the rate and degree of mixing. Thus, the fractal dimension has been used to show the variation in the rate of mixing along the bed height. Numerical studies based on molecular dynamics simulations confirmed the fractal properties of the system. The effect of bed height on the regional variations in the rate of mixing is explained in terms of kinetic energy per particle.