Ratio of median to mean Is the ratio of the median to the mean of a distribution used for any descriptors e.g. measure of skewness? 
 A: Not that I know of, but there's a related concept: the Nonparametric skew. Let $\mu$ be the mean $\sigma$ the standard deviation and $\nu$ the median. The Nonparametric skew is
$$S = \frac{\mu-\nu}{\sigma}$$
Yours is
$$S^* = \mu/\nu=(S \sigma + \nu)/\nu=S\sigma/\nu+1$$
And also
$$S = (S^*-1)/(\sigma/\nu)$$
A: So glad you asked this question:
In my undergrad thesis I did an econometric hypothesis test and demonstrated a relationship between (lagged) income inequality and economic growth. I used the Gini index as a measure of income inequality -- conventional.
But my advisor was most impressed with an appendix I wrote where I demonstrated that the Gini index is non-linear and the ratio of median to mean might be a better candidate for use in linear regression models. 
For a typical left-skewed distribution, the Gini might be around 0.45.  As we move toward a more normalized or equal distribution, the Gini index would drop. If, hypothetically, we kept moving toward a right-skewed distribution, the Gini index would again rise. Hence non-linear.
In contrast the ratio of median to mean would be linear; if we start with an extremely left-skewed distribution and move thru a normal distribution and continue toward a more right-skewed distribution, the ratio of median to mean continues to increase from Mm < 1.0 to Mm = 1.0 to Mm > 1.0.
As whuber points out above, the use of this descriptive stat is constrained to distributions whose values are always positive. So this is compelling as a descriptor of an income distribution. :D
