# Multivariate statistics

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Multivariate statistics is a form of statistics encompassing the simultaneous observation and analysis of more than one statistical variable. The application of multivariate statistics is multivariate analysis. Methods of bivariate statistics, for example simple linear regression and correlation, are special cases of multivariate statistics in which two variables are involved.

Multivariate statistics concerns understanding the different aims and background of each of the different forms of multivariate analysis, and how they relate to each other. The practical implementation of multivariate statistics to a particular problem may involve several types of univariate and multivariate analysis in order to understand the relationships between variables and their relevance to the actual problem being studied.

In addition, multivariate statistics is concerned with multivariate probability distributions, in terms of both:

• how these can be used to represent the distributions of observed data;
• how they can be used as part of statistical inference, particularly where several different quantities are of interest to the same analysis.

## Contents

### Types of analysis

There are many different models, each with its own type of analysis:

### Important probability distributions

There is a set of probability distributions used in multivariate analyses that play a similar role to the correposonding set of distributions that are used in univariate analysis when the normal distribution is appropriate to a dataset. These multivariate distributions are:

Further, the Inverse-Wishart distribution is important in Bayesian inference, for example in Bayesian multivariate linear regression. Additionally, Hotelling's T-square distribution is a univariate distribution, generalising Student's t-distribution, that is used in multivariate hypothesis testing.