7.5 Centroid

So far we’ve considered the mean of a single variable \(X\), which gives us one measure of center or measure of a typical individual. But what about the mean in a multivariate sense? If we have more than one variable, say \(p\) variables \(X_1, \dots, X_p\), is there an extension of the notion of typical individual?

If we have several variables or vectors of the same size, like in a \(n \times p\) matrix \(\mathbf{X}\), we can get the mean vector also known as centroid. The row vector of means of \(\mathbf{X}\) is denoted by:

\[ \mathbf{\bar{x}} = \frac{1}{n} \mathbf{1}_{n}^\mathsf{T} \mathbf{X} \]

and it will be a vector containing the means of of all variables:

\[ \mathbf{\bar{x}} = (\bar{x}_1, \bar{x}_2, \dots, \bar{x}_p) \]