Mardias Multivariate Normality Test
- Tests whether a multivariate sample is consistent with a multivariate normal distribution.
- Compares the sample covariance matrix to the theoretical covariance matrix for a normal population.
- If the sample and theoretical covariance matrices differ significantly, the population is judged not normally distributed.
Definition
Section titled “Definition”Mardia’s multivariate normality test is a statistical test, named after Kanti Mardia (developed in the 1970s), used to determine whether a sample of multivariate data comes from a normally distributed population by comparing the sample covariance matrix with the theoretical covariance matrix of a normally distributed population.
Explanation
Section titled “Explanation”A population is normally distributed when most values cluster around a central range and remaining values are symmetrically distributed, producing a bell-shaped curve for a single variable. For multivariate data (more than one variable), multivariate normality extends this concept across variables.
Mardia’s test assesses multivariate normality by comparing the sample covariance matrix, which contains variances and covariances of the sample variables, with the theoretical covariance matrix of a normally distributed population that has the same means and variances as the sample. If the two matrices are similar, the population is treated as normally distributed; if they differ significantly, the population is treated as not normally distributed.
Examples
Section titled “Examples”Single-variable example: heights
Section titled “Single-variable example: heights”Suppose we have a sample of 10 people and measure each person’s height. If the plotted heights form a bell-shaped curve, we can conclude that the population of heights is normally distributed.
Multivariate example: heights and weights
Section titled “Multivariate example: heights and weights”Suppose we have a sample of 10 people and measure each person’s height and weight. We calculate the sample covariance matrix for height and weight and compare it with the theoretical covariance matrix of a normally distributed population with the same means and variances. If the sample covariance matrix is similar to the theoretical matrix, we conclude that the population of heights and weights is normally distributed; if they differ significantly, we conclude it is not.
Related terms
Section titled “Related terms”- Covariance matrix
- Normal distribution