Consider $\{a_1,a_2,...,a_n\}$ n points in $\mathbb{R}$. Assume their mean is $0$, their standard deviation is then given by $\sigma=\sqrt{\dfrac{1}{n}\sum_ia_i^2}$. Let $p(x)=\dfrac{1}{\sqrt{2\pi}\sigma}e^{-\frac{x^2}{2\sigma^2}}$. Then I'd like to know a lower bound for
$$F=\int_{\min_i(a)}^{\max_i (a)}p(x)dx$$
There are no assumptions on how the $a$ are distributed other than they are all distinct. My intuition tells me the lower bound is obtained when there are only two points and the corresponding $F$ that outputs (which would actually be unique regardless of the two points, since $F$ is invariant to scaling and shifting), but I don't really know how to prove that that is a lower bound.
I'd appreciate any ideas or hints on how to show this.
One can show that [when $\frac{1}{n} \sum_{i=1}^n a_i = 0$] $$\frac{1}{n} \sum_{i=1}^n a_i^2 \le \frac{1}{4} (a_{\max} - a_{\min})^2.$$ (See Popoviciu's inequality.) So $a_{\max}-a_{\min} \ge 2\sigma$, and it seems that $F$ is smallest when $a_{\max}=\sigma$ and $a_{\min}=-\sigma$.