I never felt that I understood the fuss about the standard
deviation, until I encountered Chebyshev's inequality. Suppose
you have a bunch of data. Clearly, the mean of the data is
somewhere in the middle, and the data are spread out more or
less far from the mean. The standard deviation says how far away
they are, in the following sense: no matter how the data are
distributed, and no matter how spread out they are, at most
$1/n^2$ of them are more than $n$ standard deviations from the
mean. For example, if the standard deviation is 2 units, then
all but at most 1/9 of the data are between $m - 6$ and $m + 6$,
where $m$ is the mean. Often the data will cluster more closely
around the mean than this, but even if you know nothing else,
you can guarantee that no more than 1/9 of the data are more
than 6 units from $m$.