A multivariate observation is typically represented as a vector. So, if m variables Y₁, ...,Y_m have been measured on each experimental unit, the m-dimensional observation consisting of m measurements may be represented by Y = (Y₁, ..., Y_m)^T. The entire experiment then typically consists of a large number of such m-dimensional vectors. For example, the shapes of 3D scans of faces may be represented by vectors S_i = (x_i1, y_i1, z_i1, x_i2, y_i2,z_i2, ..., x_im, y_im, z_im)^T or (p_i1, p_i2, ..., p_im)^T where p_ij = (x_ij, y_ij, z_ij) and m is the number of vertices being used in the representation of the face. ( NB notice that vertices must be in one-to-one correspondence for statistical analysis of the shape (and texture) vectors to make sense).

Usually these m-dimensional vectors are considered to be random vectors meaning that their elements, or coordinates, are random variables. An experimental unit given a certain treatment results in a random vector drawn from an m-dimensional distribution representing the (hypothetical) population of experimental units given this treatment. Important characteristics of such a multivariate distribution are the population means, variances and covariances.

The mean, E(Y), of the random vector Y is represented by the vector consisting of the means of the individual variables, that is

E(Y) = ( E(Y₁), ..., E(Y_m))^T ,

so that the mean face shape M, given n faces (in 1-1 correspondence) is

M = =

The variance, Var(Y_i), of a random variable Y_i in a multivariate distribution is defined to be the mean of the square of the deviation of Y_i from the mean of ith variables i.e.

Var(Y_i) = E( ( Y_i - E(Y_i))² )

It may therefore be looked upon as a measure of dispersion. Variance of a random vector is defined as for the mean i.e.

Var(Y) = ( Var(Y₁), ..., Var(Y_m))

The covariance, Cov(Y_i, Y_j), of two random variables Y_i and Y_j is defined as

Cov(Y_i, Y_j) = E( (Y_i - E(Y_i))( Y_j - E(Y_j)) )

Notice that Var(Y_i) = Cov(Y_i, Y_i)

The covariance matrix (also known as variance or variance-covariance matrix)of the m-dimensional random vector Y is defined as the m x m matrix whose ijth entry is Cov(Y_i, Y_j), and is therefore symmetric as Cov(Y_i, Y_j) = Cov(Y_j, Y_i). This matrix finds great use in principal components analysis.

As covariance is often difficult to interpret, it is often standardised by dividing by the product of the standard deviations of the two variables to give the correlation coefficient

Cor(Y_i, ,Y_j) =