A special version of the moments is the set of central moments, the n-th central moment defined as:

$$ \langle (x-\langle x \rangle )^n\rangle \equiv \int\! (x-\langle x\rangle)^n p(x)\,dx $$

The zero-th and first central moments are both trivial, equal \( 1 \) and \( 0 \), respectively. But the second central moment, known as the variance of \( p \), is of particular interest. For the stochastic variable \( X \), the variance is denoted as \( \sigma^2_X \) or \( \mathrm{var}(X) \):

$$ \begin{align} \sigma^2_X\ \ =\ \ \mathrm{var}(X) & = \langle (x-\langle x\rangle)^2\rangle = \int\! (x-\langle x\rangle)^2 p(x)\,dx \tag{1}\\ & = \int\! \left(x^2 - 2 x \langle x\rangle^{2} + \langle x\rangle^2\right)p(x)\,dx \tag{2}\\ & = \langle x^2\rangle - 2 \langle x\rangle\langle x\rangle + \langle x\rangle^2 \tag{3}\\ & = \langle x^2\rangle - \langle x\rangle^2 \tag{4} \end{align} $$

The square root of the variance, \( \sigma =\sqrt{\langle (x-\langle x\rangle)^2\rangle} \) is called the standard deviation of \( p \). It is clearly just the RMS (root-mean-square) value of the deviation of the PDF from its mean value, interpreted qualitatively as the spread of \( p \) around its mean.