In Are values in a probability density function related to standard deviation? the OP wanted to know if for a normal distribution $$ \frac{P(x-\frac\sigma{2} < X < x+\frac\sigma{2} )}{\sigma} $$ was the same as $f(x)$, and of course this is not the case.
However, it does hold true at two special values of $x$ (one on each side of the mean $\mu$).
Can somebody find those special values in terms of familiar and semi-familiar functions (for example, erf and inverse erf and and Lambert functions would be fine).
