Denoting $\mu$ the mean ($\neq$ average), $m$ the median, $\sigma$ one standard deviation, $M$ the mode, $sgn()$ the sign function and $X$ the (random) dataset.
It's well known that
$$|\mu-m|\leq\sigma\quad (1)$$
Even though it is in general not true that any unimodal distribution satisfies either one of $$M\leq m\leq\mu\textit{ or }M\geq m\geq \mu$$ it can still be shown that the inequality
$$|\mu-m|\leq3\sigma\quad (1)$$
holds for any unimodal distribution (regardless of skew) for which the first moments exit. This is proven formally in Johnson and Rogers (1951).
EDIT: $(1)$ is a frequent textbook exercise:
\begin{eqnarray} |\mu-m| &=& |E(X-m)| \\ &\leq& E|X-m| \\ &\leq& E|X-\mu| \\ &=& E\sqrt{(X-\mu)^2} \\ &\leq& \sqrt{E(X-\mu)^2} \\ &=& \sigma \end{eqnarray}
The first equality derives from the definition of the mean, the third comes about because the median is the unique minimiser (among all $c$'s) of $E|X-c|$ and the fourth from Jensen's inequality (i.e. the definition of a convex function).
- [0]: The Moment Problem for Unimodal Distributions. N. L. Johnson and C. A. Rogers. The Annals of Mathematical Statistics, Vol. 22, No. 3 (Sep., 1951), pp. 433-439