Consider the following problem in the large deviation theory. For constants $a_1, cdots, a_p$ the tail probability $P(M_1 > a_1, cdots, M_p > a_p)$ of the sample medians $(M_1, cdots, M_p)$ is supposed to converge to zero as sample size increases. This paper shows that this probability converges to zero exponentially fast and estimates the convergence rates of the above tail probability of the sample medians. Also compare with the rates about the sample means.