Abstract: Many tests have been developed to check the normality assumption. These tests are mainly defined in two types: one is empirical distribution function test, the other is correlation and regression test. In this paper, we propose a new two-step test method based on the L_2 Wasserstein distance and the approximate distribution of i th sample order statistic. We discuss the properties of the new test method under the null hypothesis, and compare the power with other most commonly tests for four alternative groups. Finally, the new method is applied to analyse the real problem. The simulation results show that the new test method improves the efficiency in identifying asymmetric long-tailed alternatives.