With the help of a tensor method, we investigate the evolution properties of the state of polarization of an electromagnetic Gaussian Schell-model beam propagating through a gradient-index (GRIN) fiber. We find that the Stokes parameters and the polarization ellipse exhibit periodicity. The initial beam parameters affect the values of the Stokes parameters and the parameters of the polarization ellipse. Furthermore, based on the second-order moments of the Wigner distribution function, the explicit expression for the propagation factor (known as the M(2) factor) in the GRIN fiber is derived. It is shown that the M(2) factor remains invariant on propagation and is determined only by the initial beam parameters.