Given a prime $p$, let $N_p$ denote the number of $x \in [1, (p-1)/2]$ such that $$x^{-1} \pmod{p} \in [1, (p-1)/2]$$ What is the limiting distribution of $$S_n := \left \{\frac{N_p}{(p-1)/2}\, : \, 1\le \text{prime} \, p \le n \right\}$$ as $n\to\infty$?
Numerical evidence suggests this quantity is distributed like a normal random variable with mean $1/2$ and standard deviation given by $f(n)$, where the graph of $f$ on $3\le n \le 300$ looks like the below data, which suggests something like $(n\log(n))^{-1}$:
