I'll just leave a final comment: if you restrict yourself to the arithmetic mean, then you can use Cantelli's inequality to make some claims about the distance between the expectation and the median of a random variable in a way that only depends on the variance/st.dev.
On the other hand, you do not actually know the (population) expectation or (population) variance: you can only estimate them, given some samples (and, quite often, they can be undefined/unbounded).
Also, as I was trying to demonstrate in my previous comment, most "averages" are poor estimators for the expectation of a random variable (compared to the arithmetic sample mean), the same way that min(data) or max(data) are poor estimators for the expectation of a random variable, so it seems a bit "dangerous" to make such a general broad claim (again, in my humble opinion).
See: https://en.wikipedia.org/wiki/Chebyshev%27s_inequality#Cante...
On the other hand, you do not actually know the (population) expectation or (population) variance: you can only estimate them, given some samples (and, quite often, they can be undefined/unbounded).
Also, as I was trying to demonstrate in my previous comment, most "averages" are poor estimators for the expectation of a random variable (compared to the arithmetic sample mean), the same way that min(data) or max(data) are poor estimators for the expectation of a random variable, so it seems a bit "dangerous" to make such a general broad claim (again, in my humble opinion).