Abstract: Let X_1, X_2, \cdots, be a sequence of random variables which are independent, not necesserily identically distributed, yet have a common mean \mu. While the behavior of the distribution of T_n=\widetildeX-\mu is being investigated, the conditional distribution of D_n= 2 \sum_i=1^n\left(X_i-\barX\right) V_m, given X_i, i=1, \cdots, n, is used to approximete the distribution of the error T_n, where V_n i of D_n, i=1, \cdots, n, are random variables following Dirichlet distribution D(4,4, \cdots, 4). Let F_n be the distribution function of T_n / \sqrt\operatornamevar T_n and F_n^* the conditional distribution of D_n / \sqrt\operatornamevar^* D_n where var* D_n is the conditional variance of D_n given X_i, i =1, \cdots, n. Under certain conditions, the following result is proved in this paper: For almost all sequences x_1, x_2, \cdots, \lim _n \rightarrow \infty \sqrtn \sup _-\infty< y< \infty\left|F_n^*(y)-F_n(y)\right|=0. holds.