In my second post, I gave the mathematical definitions of median and arithmetic mean (average). These were pretty useless, like all mathematical definitions, because I did not explain when to use median vs. mean.
O.K., time for everyone to cringe: remember "word problems" from 4th grade mathematics? It turns out life is a word problem :-) A computer can do math calculations for you (including calculus), but computers are really bad at turning word problems into meaningful answers. A human has to decide which is the best equation to solve a word problem.