—A general procedure is suggested for calculating the upper (Voigt) and lower (Reuss) bounds of the average elastic constants of an anisotropic medium from crystallographic directions. The elastic tensors of Hooke’s law can be expanded into irreducible representations of the rotation group. The Voigt/Reuss-averaged elastic constants depend on the second and fourth moments of the distribution function rather than on the entire function used for the averaging. In this case, the distribution function depends on one angle, while the elastic constants depend on two variables. The limitations imposed by the probability theory on the moment values are investigated and used to derive general constraints on the Voigt (Reuss) bounds of elastic constants.