SD from Mean and Median

Question

Given the mean and median of a dataset, is it possible to compute the standard deviation of the dataset?

Answer 1

No, it is not possible to determine the standard deviation from the mean and the median. The data sets $\{-1,0,1\}$ and $\{-100,0,100\}$ have the same mean and median (both are equal to $0$, for both sets) but different standard deviations.

Answer 2

Talking about population standard deviation, it is possible to put a lower bound: the standard deviation cannot be less than the absolute difference between the median and the mean.

You can prove this by considering the conditional means of the top half of the distribution and of the bottom half of the distribution; they must each be at least the absolute difference between the median and the mean away from the mean, making the variance of the distribution at least the square of the difference between the median and the mean.

You cannot determine the standard deviation: it could be arbitrarily larger than this bound, even if the mean exists.