A multivariate normal distribution is a generalization of the bell-shaped normal distribution to situations involving two or more continuous variables. Each individual variable follows a normal distribution, and the entire set of variables follows a joint distribution that includes not just their individual means and variances, but also their linear relationships through covariance or correlation. For a distribution to be multivariate normal, any linear combination of the variables must also be normally distributed.
This type of distribution is fundamental in multivariate statistics because it allows analysts to model systems where variables are not independent but related in a predictable, linear way. It underpins methods like multivariate regression, discriminant analysis, and the estimation of confidence regions in multiple dimensions.
For example, consider modeling students’ verbal and math test scores. Both sets of scores are often approximately normally distributed, and they also tend to be correlated—students who do well in one often do well in the other. A multivariate normal distribution can capture both the individual score patterns and the strength of their relationship.
In the contour plot below, the elliptical shape of the contours is a hallmark of a multivariate normal distribution for bivariate relationships. The ellipses represent regions of equal probability density, and their smooth, symmetric appearance reflects the linear relationships between the variables. The orientation of the ellipses shows the direction of correlation, and their elongation reflects its strength—the more stretched the ellipse, the stronger the correlation.
If the data were not multivariate normal, the contours would likely deviate from this elliptical form. For example, they might appear asymmetric, skewed, lumpy, or show multiple peaks (multimodal). Such irregular shapes would indicate that the relationship between the variables is not fully captured by a multivariate normal model and may suggest nonlinear patterns, outliers, or heavy tails.