Because seismic anisotropy in the upper mantle is due primarily to lattice preferred orientation (LPO) of olivine crystals induced by deformation, observations of anisotropy can provide constraints on mantle deformation and flow. In this paper, we use a previously published theory for deformation-induced LPO (Ribe and Yu, 1991) to quantify the relation between seismic anisotropy and deformation for two model compositions: a pure olivine (dunite) aggregate, and a harzburgite comprising 70% olivine and 30% enstatite. Analytical and numerical solutions of the governing equations show that the predicted LPO (and hence also the seismic anisotropy) depends only on the finite strain, and not on the deformation path by which that strain is produced. The magnitude of the anisotropy depends only on the ratios c1/c2 and c2/c3, where c1>c2>c3 are the lengths of the principle axes of the finite strain ellipsoid, and the directions of maximum and minimum compressional wave velocity VP coincide with the axes c1 and c3, respectively. We calculate the percent anisotropies for compressional and shear waves propagating along the three principle axes of the finite strain ellipsoid, as functions of the axial ratios c1/c2 and c2/c3, for our two model compositions. However, inferences of the magnitude of finite strain obtained using these results are likely to be minimum estimates because our theory neglects the effects of dynamic recrystallization. Finally, we present as a byproduct of this study a new direct method for calculating the finite strain produced by an arbitrary deformation history. ¿ American Geophysical Union 1992