Summary: In this paper, we consider the multivariate normality test based on the sample measures of multivariate skewness and kurtosis defined by M. S. Srivastava [“A measure of skewness and kurtosis and a graphical method for assessing multivariate normality”, Stat. Probab. Lett. 2, No. 5, 263–267 (1984; doi:10.1016/0167-7152(84)90062-2)]. K. Koizumi et al. [J. Stat., Adv. Theory Appl. 1, No. 2, 207–220 (2009; Zbl 1173.62047)] proposed test statistics \(M_1\) and \(M_2\) using Srivastava’s sample skewness and kurtosis, which are asymptotically distributed as \(\chi^2\)-distribution. We propose a new test statistic \(M_3\) by taking account of the variance of \(M_2\) under the normality. In order to evaluate the accuracy of the proposed test statistic, the numerical results by a Monte Carlo simulation for some selected values of parameters are presented.