Evaluating Univariate, Bivariate, and Multivariate Normality Using Graphical Procedures.

This paper reviews graphical and non-graphical procedures for evaluating multivariate normality by guiding the reader through univariate and bivariate procedures that are necessary, but insufficient, indications of a multivariate normal distribution. A data set utilizing three dependent variables for two groups provided by George and Mallery (1999) is used to analyze histograms, stem-and-leaf plots, box-and-whisker plots, kurtosis and skewness coefficients, Q-Q plots, the Shapiro-Wilk or Kolmogorov-Smirnov statistic, and bivariate scatterplots. A procedure programmed by Thompson (1990, 1997) is used to explore multivariate normality by plotting Mahalanobis distances against derived chi-square values in a scatterplot. Evaluating Normality 3 Evaluating Univariate, Bivariate, and Multivariate Normality Using Graphical Procedures Reality is complex. Over time, researchers in the social sciences have become increasingly aware that simple univariate methods comparing an experimental group with a control group on a single dependent variable are inadequate to meet the needs of the complex phenomena that dominate educational and psychological research. In the majority of social science research, two or more dependent variables are necessary, because nearly every effect has multiple causes and nearly every cause has multiple effects. Even when studying a single construct, such as self-concept, it is often helpful to use multiple tools to measure elusive constructs (called "multi-operationalizing"). In a methodological shift that increasingly emphasizes honoring the complexity of reality, Grimm and Yarnold (1995) reported that the use of multivariate statistics in research has accelerated in the last 20 years and that it is difficult to find empirically based research articles that do not employ one or more multivariate analyses. In a comparison of the 1976 and 1992 volumes of the Journal of Consulting and Clinical Psychology (JCCP) and the Journal of Personality and Social Psychology (JPSP), Grimm and Yamold found that the use of multivariate statistics in JCCP increased from 9% to 67% in that 16 year period. For JPSP, the use of multivariate statistics increased from 16% to 57% in the same time frame. In an exhaustive study of statistical methods used in studies published in three major social science journals during a 16-year period, Emmons, Stallings, and Lane (1990) found increases for MANOVA, MANCOVA, multiple regression, and multiple correlation methods. For example, the use of MANOVA increased from 44% of the studies published from 1977 to 1981 to 54% of the studies published from 1982 to 1987. Emmons, Stallings, and Lane (1990, p. 14) concluded that "the