Let $X_1, \ldots, X_N \in \text{Gaussian}(\mu, \sigma)$ each be a D210: Gaussian random real number such that

(i)	\begin{equation} N \in \{ 2, 3, 4, \ldots \} \end{equation}
(ii)	$X_1, \ldots, X_N$ is an D2713: Independent random collection
(iii)	\begin{equation} \mathbb{E} |X_1|^2 \in (0, \infty) \end{equation}
(iv)	\begin{equation} \mu : = \mathbb{E} X_1 \end{equation}
(v)	\begin{equation} \sigma^2 : = \text{Var} X_1 \end{equation}
(vi)	\begin{equation} S : = \left( \frac{1}{N - 1} \sum_{n = 1}^N \left( \frac{X_n - \mu}{\sigma} \right)^2 \right)^{1 / 2} \end{equation}
(vii)	$\chi \in \text{ChiSquared}(N)$ is a D212: Chi-squared random unsigned real number

Since \begin{equation} \frac{X_n - \mu}{\sigma} \sim \text{Gaussian}(0, 1) \end{equation} then this result is a special case of R5229: Bessel-corrected sample variance of independent standard gaussians is a transformed chi-squared random real number. $\square$