Recently, some randomized iterative methods are proposed to solve large-scale factorised linear systems. In this paper, we present two randomized average block iterative methods which still take advantage of the factored form and need not perform the entire matrix. The new methods are pseudoinverse-free and can be implemented for parallel computation. Furthermore, we analyze their convergence behaviors and obtain the exponential convergence rate. Finally, some numerical examples are carried out to show the effectiveness of our new methods.