The Monte Carlo code TYCHE-III has been used to calculate with a high degree of convergence the second, fourth, and sixth spatial moments of the slowing-down density distribution at the indium resonance energy, for neutrons originating from a fission source at a point in infinitely extended water and zirconium-water moderators. The effects of both inelastic scattering and anisotropy of elastic scattering in oxygen and zirconium have been included. For water, the calculations were performed using several widely available sets of data on oxygen cross sections and angular distributions. The effects on the neutron age and higher spatial moments of several fission spectra are also reported. These calculations have been compared with the moments of experimentally measured distributions of neutron flux at the indium resonance energy, after applying a suitable correction to the computed slowing-down moments. The agreement between the calculated and measured values of the neutron ages and higher moments is seen to be satisfactory.