Probability distribution · BEGINNER LESSON
Multivariate normal
A multivariate normal distribution is a probability model for a p-dimensional random vector whose joint density has a specific bell-shaped quadratic form with a mean vector and a variance-covariance matrix. In the nondegenerate case, it describes continuous vector observations in which the whole distribution is determined by the mean and covariance.
What it is
The multivariate normal is the extension of the normal distribution from one variable to several variables measured together as a vector. Here, a vector is just a list of numbers such as X1, X2, ..., Xp. The model uses a mean vector, which gives the average of each variable, and a variance-covariance matrix, which records spread and pairwise association among variables. [0,2]
[1] 6.5.4.2. The Multivariate Normal Distribution [3] MULTIVARIATE NORMAL DISTRIBUTIONS
How it is defined
NIST gives the standard density form for a p-dimensional random vector X. The formula contains the mean vector m and the variance-covariance matrix Σ. MIT presents equivalent ways to define the same idea: by the joint density, by a linear transformation of independent standard normal variables, or by the property that every linear combination a^T X is normal. These views agree for multivariate normal distributions. [0,2]
[1] 6.5.4.2. The Multivariate Normal Distribution [3] MULTIVARIATE NORMAL DISTRIBUTIONS
Mechanism and meaning
The quadratic term (X − m)' Σ^-1 (X − m) measures how far a point is from the center after accounting for correlations. Mahalanobis distance is a related distance measure that adjusts for correlation among variables. This is why multivariate normality is not just about each variable separately: the joint pattern matters. Under the MIT account, the distribution of X is fully determined by its mean and covariance, and if the covariance matrix is diagonal, the components are uncorrelated; in the multivariate normal case, that also means they are independent. [1,2]
[2] Statistical Assumptions of Substantive Analyses Across the... [3] MULTIVARIATE NORMAL DISTRIBUTIONS
How people check it in practice
In applied statistics, multivariate analyses such as repeated measures ANOVA, MANOVA, MANCOVA, multiple linear regression, and canonical correlation often assume multivariate normality. The review source says this assumption can be checked graphically with a plot of Mahalanobis distances against chi-square values, or with Mardia’s coefficient of multivariate kurtosis. If the plot is roughly straight, the data are treated as multivariate normal; if the critical ratio for Mardia’s kurtosis is below 1.96, that is taken as support at the 0.05 level. [1]
[2] Statistical Assumptions of Substantive Analyses Across the...
Limitations and cautions
The normal model may fail for skewed or strongly kurtotic data, and violations can affect significance tests in regression and other linear methods. The review also notes that homogeneity of variance-covariance is a separate assumption for some group analyses, and unequal sample sizes can make violations more serious. MIT also shows that some normal vectors are degenerate, meaning probability may lie on a lower-dimensional subspace, so not every multivariate normal has a full joint density. [1,2]
[2] Statistical Assumptions of Substantive Analyses Across the... [3] MULTIVARIATE NORMAL DISTRIBUTIONS
SEE THE IDEA
How the multivariate normal model is built
Record several variables together as X = (X1, ..., Xp).
Set the center and describe spread plus correlations.
Use the centered vector with Σ^-1 to measure multivariate distance.
A joint normal-shaped density or an equivalent normal construction.
Each step adds structure: data vector -> center/spread description -> correlation-adjusted distance -> full multivariate normal model.
The model starts with several variables measured together and ends with a joint probability rule for their combined behavior.
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Sources and further reading
This general lesson was generated from public source material. It explains a concept; it does not interpret a private paper or provide medical advice.