The classical multivariate data analysis has been based largely on a multivariate normal distribution assumption. Therefore almost all of the work was concentrated on just location and dispersion parameters, with relatively little attention to questions of high-order characteristics. For characterization of multivariate distribution, its moments are very expressive. This fact was used to develop a test method for testing given an iid. sample of multivariate normality.\newline Most of the known procedures try to detect joint nonnormality by studying marginal normality of an observation set. Tests based on the moments of a sample directly measure joint normality ignoring marginal effects. The method introduced in this work shows, that normalized Hermite moments are able to describe a multivariate distribution in a way, that allows to discriminate them better from the normal case than another often used method (nearest distance method). To illustrate the capabilities of the normalized moments, some artificial distributions with marginal normality were tested with both methods and directly compared.