This paper presents numerical evidence that in quantum systems with chaotic classical dynamics, the number of scattering resonances near an energy E scales like ħ^{−(D(K_E)+1)/2} as ħ→0. Here, K_E denotes the subset of the energy surface {H=E} which stays bounded for all time under the flow generated by the classical Hamiltonian H and D(K_E) denotes its fractal dimension. Since the number of bound states in a quantum system with n degrees of freedom scales like ħ⁻ⁿ, this suggests that the quantity (D(K_E)+1)/2 represents the effective number of degrees of freedom in chaotic scattering problems. The calculations were performed using a recursive refinement technique for estimating the dimension of fractal repellors in classical Hamiltonian scattering, in conjunction with tools from modern quantum chemistry and numerical linear algebra.