A partial description of the joint distribution of the data is provided here. Three aspects of the data are of importance, the first two of which you should already be familiar with from univariate statistics. These are:

1. Central Tendency: What is a typical value for each variable?
2. Dispersion: How far apart are the individual observations from a central value for a given variable?
3. Association: This might (or might not!) be a new measure for you. When more than one variable are studied together, how does each variable relate to the remaining variables? How are the variables simultaneously related to one another? Are they positively or negatively related?

Statistics, as a subject matter, is the science and art of using sample information to make generalizations about populations.

Here are the Notations that will be used:

\(X_{ij}\) = Observation for variable j in subject i .

\(p\) = Number of variables

\(n\) = Number of subjects

In the example to come, we'll have data on 737 people (subjects) and 5 nutritional outcomes (variables). So,

\(p\) = 5 variables

\(n\) = 737 subjects

In multivariate statistics we will always be working with vectors of observations. So in this case we are going to arrange the data for the p variables on each subject into a vector. In the expression below, \(\textbf{X}_i\) is the vector of observations for the \(i^{th}\) subject, \(i\) = 1 to \(n\) (737). Therefore, the data for the \(j^{th}\) variable will be located in the \(j^{th}\) element of this subject's vector, \(j\) = 1 to \(p\) (5).

\[\mathbf{X}_i = \left(\begin{array}{l}X_{i1}\\X_{i2}\\ \vdots \\ X_{ip}\end{array}\right)\]