We perform Monte Carlo simulations to study the two dimensional random-bond XY model on a square lattice.Two kinds of bond randomness with the coupling coefficient obeying the Gaussian or uniform distribution are discussed.It is shown that the two kinds of disorders lead to similar thermodynamic behaviors if their variances take the same value.This result implies that the variance can be chosen as a characteristic parameter to evaluate the strength of the randomness.In addition,the Berezinskii—Kosterlitz—Thouless transition temperature decreases as the variance increases and the transition can even be destroyed as long as the disorder is strong enough.