The prime number theorem gives a rather famous result that
$$ \lim_{n \rightarrow \infty} \frac{\pi(n)}{\text{Li}(n)} \rightarrow 1 $$
So that in some sense $\text{Li}(n)$ could be interpreted as the expected value of $\pi(n)$. From there conjectures such as riemann hypothesis hope to bound
$$ | \pi(n) - \text{Li}(n)| \le \frac{1}{8\pi} \sqrt{n} \log(n) $$
But instead of worst case bounds, we may be tempted instead to ask the question what is a function $\theta(n)$ such that
$$ \lim_{n \rightarrow \infty} \frac{|\pi(n) - \text{Li}(n)|}{\theta(n) } \ \rightarrow 1 $$
I.E. what is the "standard deviation" of the prime counting function from its expected value.
And more generally if we define $E[F]$ as function $\psi(n)$ such that $$ \lim_{n \rightarrow \infty} \frac{F(n)}{\psi(n)} \rightarrow 1 $$
then $\text{Li(n)} = E[\pi(n)]$ and $\theta(n) = \sqrt{E[\pi(n)]^2 - E[\pi(n^2)]}$. So one could instead repose these questions about statistics as looking for functions $\Gamma_k$ such that
$$ \lim_{n \rightarrow \infty} \frac{\pi(n^k)}{\Gamma_k(n)} \rightarrow \infty $$
Has anyone tabulated $\Gamma_k(n)$?