When standardizing a set of Gaussian i.i.d. samples $\{x_i\}^N_{i=1}, X_i \sim \mathcal{N}(\mu, \sigma)$, are there analytical forms for the distributions of standardized values dependent on $N$?
Imagine $N=2$ so I have only the two samples $x_1$ and $x_2$, if I standardize these two values (i.e. subtracting sample mean $\bar{x} = \frac{1}{N}\sum_i^N x_i$ and dividing by sample standard deviation $s = \sqrt{\frac{1}{N}\sum_i^N(x_i-\bar{x})^2}$), what I will always get is $1$ and $-1$ (assume $x_1 \neq x_2$). If $N > 2$, it is not a deterministic mapping anymore. For "large enough" $N$ this should approach the Normal distribution again, but for relatively small $N$? Additionally, are there any analytical distributions for the individual standardized values $x_i$?
When plotting histograms for different $N$ it looks suspiciously like a Beta distribution with equal shape parameters $\alpha = \beta$, but in the interval $[-\sqrt{N-1},\sqrt{N-1}]$ instead of $[0,1]$.
