Denote $\mu$ the mean ($\neq$ average), $m$ the median, $\sigma$ the standard deviation and $M$ the mode. Finally, let $X$ be the sample, a realization of a continuous unimodal distribution $F$ for which the first two moments exit.
It's well known that
$$|\mu-m|\leq\sigma\quad (1)$$
This 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).
Even though it is in general not true (Abadir, 2005) that any unimodal distribution must satisfy either one of $$M\leq m\leq\mu\textit{ or }M\geq m\geq \mu\quad (2)$$ it can still be shown that the inequality
$$|\mu-M|\leq3\sigma\quad (3)$$
holds for any unimodal distribution (regardless of skew). This is proven formally in Johnson and Rogers (1951) though the proof depends on many auxiliary lemma's that are hard to fit here. Go see the original paper.
At any rate, putting $(1)$ and $(3)$ together yields:
$$|M-\mu|\leq 3|\mu-m|$$
This could be extended to hold for the original claim (i.e. without the absolute values), provided that you restrict yourself to the class of distributions for which $(2)$ holds (among skewed distributions, the Beta, Log-normal and Gamma are three that satisfy $(2)$). A sufficient condition for a distribution $F$ to satisfy $(2)$ is given in [2]. If $F$:
$$F(m−x)+F(m+x)\geq 1 \text{ for all }x$$
then $\mu\leq m\leq M$
- [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
- [1]: The Mean-Median-Mode Inequality: Counterexamples Karim M. Abadir Econometric Theory, Vol. 21, No. 2 (Apr., 2005), pp. 477-482
- [2]: W. R. van Zwet, Mean, median, mode II, Statist. Neerlandica, 33 (1979), pp. 1--5.