Assessing Univariate and Multivariate Normality, A Guide For Non-Statisticians

Abstract

Most parametric methods rely on the assumption of normality. Results obtained from these methods are more powerful compared to their non-parametric counterparts. However for valid inference, the assumptions underlying the use of these methods should be satisfied. Many published statistical articles that make use of the assumption of normality fail to guarantee it. Hence, quite a number of published statistical results are presented with errors. As a way to reduce this, various approaches used in assessing the assumption of normality are presented and illustrated in this paper.   In assessing both univariate and multivariate normality, several methods have been proposed. In the univariate setting, the Q-Q plot, histogram, box plot, stem-and-leaf plot or dot plot are some graphical methods that can be used. Also, the properties of the normal distribution provide an alternative approach to assess normality. The Kolmogorov-Smirnov (K-S) test, Lilliefors corrected K-S test, Shapiro-Wilk test, Anderson-Darling test, Cramer-von Mises test, D'Agostino skewness test, Anscombe-Glynn kurtosis test, D'Agostino-Pearson omnibus test, and the Jarque-Bera test are also used to test for normality. However, Kolmogorov-Smirnov (K-S) test, Shapiro-Wilk test, Anderson-Darling test, and Cramer-von Mises test are widely used in practice and implemented in many statistical applications.

For multivariate normal data, marginal distribution and linear combinations should also be normal. This provides a starting point for assessing normality in the multivariate setting. A scatter plot for each pair of variables together with a Gamma plot (Chi-squared Q-Q plot) is used in assessing bivariate normality. For more than two variables, a Gamma plot can still be used to check the assumption of multivariate normality. Among the many test proposed for testing multivariate normality, Royston's and Mardia's tests are used more often and are implemented in many statistical packages.

When the normality assumption is not justifiable, techniques for non-normal data can be used. Likewise, transformation to near normality is another alternative.

Keywords: Univariate normal, Multivariate normal, Q-Q plot, Gamma plot, Kolmogorov-Smirnov test, Shapiro-Wilk test, Mardia's test, Royston's test.