Generalization of standard deviation? Standard deviation is often taken from a one-dimensional list of numbers. However, how would one evaluate standard deviation of a set of points of $N$-dimensional coordinates?
 A: It's defined as $\sqrt{|\Sigma|}$ where $|\Sigma|$ is the determinant of the variance-covariance matrix, a $N\times N$ matrix with the single variances on the main diagonal and covariances elsewhere
A: Suppose $X$ is a random point in $\mathbb R^{n\times1}.$ Then $\mu = \operatorname E(X) \in \mathbb R^{n\times1}$ is defined componentwise and then
$$
\Sigma = \operatorname{var}(X) = \operatorname E\big((X-\mu)(X-\mu)^\top\big) \in \mathbb R^{n\times n}
$$
is an $n\times n$ nonnegative-definite symmetric matrix.
It can be shown via the (finite-dimensional version of the) spectral theorem that $\Sigma$ has a nonnegative-definite symmetric square root $\Sigma^{1/2}$ (definitely not defined componentwise).
If $A\in\mathbb R^{k\times n}$ then
$$
\operatorname{var}(X) = A\Sigma A^\top \in\mathbb R^{k\times k}.
$$
From this it follows that if $\operatorname{var}(Y) = I_n$ (the $n\times n$ identity matrix) and $\Sigma$ is any nonnegative-definite symmetric real matrix, then $\Sigma^{1/2} Y$ has variance $\Sigma,$ and therefore every nonnegative-definite symmetric real matrix is a variance.
If $\Sigma$ is invertible then it can be shown that a random vector whose density is
$$
\frac 1 {\sqrt{2\pi}^{\,n}} \frac 1 {\big(\det\Sigma^{1/2}\big)} \exp\left( - \frac 1 2 (x-\mu)^\top \Sigma^{-1} (X-\mu) \right)
$$
has expected value $\mu$ and variance $\Sigma.$
