Empirical Moments

$\overline x = \displaystyle \frac{\displaystyle 1}{\displaystyle n}\sum\limits_{i=1}^n x_{i}$ average of $X$ sampled by $\{ x_i \}_{i=1,\ldots,n}$ [*]
$s_{XY}=\displaystyle \frac{\displaystyle 1}{\displaystyle n}\sum\limits_{i=1}^n (x_i-\overline x)(y_i-\overline y)$ empirical covariance of random variables $X$ and $Y$ sampled by $\{ x_i \}_{i=1,\ldots,n}$ and $\{ y_i \}_{i=1,\ldots,n}$ [*]
$s_{XX}=\displaystyle \frac{\displaystyle 1}{\displaystyle n}\sum\limits_{i=1}^n (x_i-\overline x)^2$ empirical variance of random variable $X$ sampled by $\{ x_i \}_{i=1,\ldots,n}$ [*]
$r_{XY} = \displaystyle \frac{\displaystyle s_{XY} }{ \sqrt{\displaystyle s_{XX}s_{YY} }}$ empirical correlation of $X$ and $Y$ [*]
${\data{S}}=\{ s_{X_{i}X_{j}}\} = x^\top\data{H} x$ empirical covariance matrix of $X_{1},\ldots, X_{p}$ or of the random vector $X=(X_{1},\ldots, X_{p})^\top$ [*], [*]
${\data{R}}=\{ r_{X_{i}X_{j}}\} = \data{D}^{-1/2}\data{S}\data{D}^{-1/2}$ empirical correlation matrix of $X_{1},\ldots, X_{p}$ or of the random vector $X=(X_{1},\ldots, X_{p})^\top$ [*], [*]