In an isotropic dissipative medium, the attenuation properties of rocks are usually specified by quality factor Q, a positive, dimensionless, real-valued, scalar quantity, independent of the direction of wave propagation. We propose a similar, scalar, but direction-dependent quality Q-factor (also called Q) for time-harmonic, homogeneous or inhomogeneous plane waves propagating in unbounded homogeneous dissipative anisotropic media. We define the Q-factor, as in isotropic viscoelastic media, as the ratio of the time-averaged complete stored energy and the dissipated energy, per unit volume. A solution of an algebraic equation of the sixth degree with complex-valued coefficients is necessary for the exact determination of Q. For weakly inhomogeneous plane waves propagating in arbitrarily anisotropic, weakly dissipative media, we simplify the exact expression for Q con-siderably using the perturbation method. The solution of the equation of the sixth degree is no longer required. We obtain a simple, explicit perturbation expression for the quality factor, denoted as . We prove that the direction-dependent is related to the attenuation coefficient α measured along a profile in the direction of the energy-velocity vector (ray direction). The quality factor does not depend on the inhomogeneity of the plane wave under consideration and thus is a convenient measure of the intrinsic dissipative properties of rocks in the ray direction. In all other directions, the quality factor is influenced by the inhomogeneity of the wave under consideration. We illustrate the peculiarities in the behavior of and its accuracy on a model of anisotropic, weakly dissipative sedimentary rock. Examples show interesting inner loops in polar diagrams of in regions of S-wave triplications.