8.3 Standard Deviation as a Norm

If we use a metric matrix \(\mathbf{D} = diag(1/n)\) then we have that the variance is given by a special type of inner product:

\[ var(\mathbf{x}) = <\mathbf{x}, \mathbf{x}>_{D} = \mathbf{x}^\mathsf{T} \mathbf{D x} \]

From this point of view, we can say that the variance of \(\mathbf{x}\) is equivalent to its squared norm when the vector space is endowed with a metric \(\mathbf{D}\). Consequently, the standard deviation is simply the length of \(\mathbf{x}\) in this particular geometric space.

\[ sd(\mathbf{x}) = \| \mathbf{x} \|_{D} \]

When looking at the standard deviation from this perspective, you can actually say that the amount of spread of a vector \(\mathbf{x}\) is actually its length (under the metric \(\mathbf{D}\)).