We consider the Wasserstein distance between a sample distribution and the set of normal distributions as a measure of nonnormality. By considering the standardized version of this distance we obtain a version of Shapiro–Wilk’s test of normality. The asymptotic behavior of the statistic is studied using approximations of the quantile process by Brownian bridges. This method differs from the “ad hoc” method of de Wet and Venter and permits a similar analysis for testing other location scale families.
"Tests of goodness of fit based on the $L_2$-Wasserstein distance." Ann. Statist. 27 (4) 1230 - 1239, August 1999. https://doi.org/10.1214/aos/1017938923