A systematic study of a series of Mo_1-xW_xS ₂ layered mixed crystals, with 0 ≤ x ≤ 1, grown by the chemical-vapor transport method were carried out by using Raman scattering measurements. The peaks of the two dominant first-order Raman-active modes, A_1g and E2g1, and several second-order bands have been observed in the range of 200-1000 cm^-1. The peaks corresponding to A_1g mode show one-mode type behavior while the peaks of E2g1 mode demonstrate two-mode type behavior for the entire series. The results can be explained on the basis of the atomic displacements for each mode. For A_1g mode only sulfur atoms vibrate and this give rise to a one-mode type behavior for the mixed crystals. For E2g1 mode metal atoms also vibrate as well as sulfur atoms, the mass difference of the vibrating Mo and W cations causes the two-mode type behavior of E2g1 mode. In addition, the observation of largest asymmetry and broadening of A_1g mode for Mo_0.5W_0.5S ₂ has been attributed to random alloy scattering.