Abstract: Let X_1, X_2, \cdots be a sequence of random variables which are independently and identically distributed with a common variance \mu_2. While the behavior of the distribution of T_n=\frac1n \sum_i=1^n\left(X_i-\barX\right)^2-\mu_2 is being investigated, the conditional distribution of D_n=\sum_i=1^n V_n\left(X_i-\sum_i=1^n V_n i X_i\right)^2-\frac1n \sum_i=1^n\left(X_i-\barX\right)^2, given X_i, i=1, \cdots, n, is used to approximate the distribution of error T_n where V_n k of D_n, i=1,2, \cdots, n, are random variables following Dirichlet distribution D(4,4, \cdots, 4). Let F_n be distribution function of T_n / \sqrt\operatornameVar^2 T_n and F_n^* the conditional distribation of D_n \sqrt\operatornameVar^* D_n where \operatornameVar^* D_n is the conditional variance of D_n given X_i i-1,2, \cdots, n. Under certain conditions, the following results is proved in this paper: For almost all sequences X_1, X_2, \cdots. i) \sqrtn D_n \xrightarrow\mathscrL^n N\left(0, \mu_4-\mu_2^2\right). where \mu_4=E\left(X_1-\mu\right)^4 \text and \mu=E X_1. ii) \sqrtn \sup _-\infty< y< \infty\left|F_n^*(y)-F_n(y)\right|=O(1) iii) \lim _n \rightarrow \infty \sup _-\infty< y< \infty\left|F_n^*(y)-F_n(y)\right|=0 At last in this paper the random weighing method is applied to sample sarveys.