Perturbation analysis of Markov chains provides bounds on the e_ect a change in the Markov transition matrix has on the corresponding stationary distribution. This paper compares and analyzes bounds found in the literature and provides a new bound. In our analysis, we provide for the first time an analysis on the relative error of bounds. Specically, we show that condition number bounds have a non-vanishing relative error as the size of the perturbation tends to zero. Our new perturbation bound will have the desirable feature that the relative error vanishes as the size of the perturbation tends to zero. We discuss a series of examples to illustrate applicability of the various bounds. For example, we address the question on how the bounds behave as the size of the system grows.