Akin to the ordinary standard deviation (using the mean), the standard devation from the median is a measure of how much the data in a set differ from the median.
For each of the n elements in the data set, calculate the difference between the value and the median. Call this δ.
The standard deviation from the median = (√ Σ( δ2 ) ) / n
As with the 'standard' standard deviation (i.e. from the mean), you can make this the ‘population’ standard deviation from the median by replacing the n with ( n - 1 ). This is somewhat more honest for small sets, even though it is larger.
Speaking of larger standard deviations, the standard deviation from the median is always higher than the standard deviation from the mean for a given set (unless the mean equals the median, in which case they're the same). This does not invalidate the use of the median compared to the mean, however (see Median for the reasons).
One can extend this concept to any other statistical estimate (e.g. interquartile mean, jackknife mean), by replacing the differences appropriately. | 
