This work tests the accuracies of some common approximate methods for calculating spatially dependent neutron slowing down distributions. According to each procedure, an analytic expression for the detailed distribution of neutrons from a plane monoenergetic source in hydrogen is obtained and compared with accurate analytic solutions. Most of the latter are derived here and appear to have other far reaching potential applications. In particular, the exact value and first two lethargy derivatives of the collided angular flux at source lethargy are found in terms of elementary functions of position and angle. These results are used to show that an expression derived by McInerney for the spatial distribution of the scalar flux has, at any given position, only first-order accuracy in powers of lethargy, even though the zeroth and second spatial moments are exact at all lethargies. While the B-1 and P-1 approximations produce poor results at small lethargies, they are accurate at large values; for the errors are due primarily to high-order spatial Fourier components, and these rapidly decay with increasing lethargy. At any lethargy, a Tauberian theorem facilitates calculating the spatial derivative of the scalar flux at the source plane. This quantity is used to trace the lethargy dependence of some peculiarities of the entire spatial distributions given by the B-1 and P-1 approximations. At asymptotically large lethargies, these spatial distributions are obtained explicitly and shown to agree with a well-known accurate expression.