In my experience, there is a big
difference in when the median or the mean is the most useful tool to look at
when you want to find the central tendency. The question is what the likelihood of vast
outliers is.

A good example is height. The human
body can only grow so tall. That means that if you want to find out what an “average”
person is in height, then the arithmetic mean is useful. You take a large
enough sample, and then you can find with some large degree of certainty that
the average height of the population is close the arithmetic mean of your
sample. Most likely it will also be close to your median height, as the
standard distribution was built on such measurements.

Conversely, some measurements are
not bound. An example here is wealth. There is no biological, physical, or
chemical reason that a person cannot have all the money in the world. There are
people in the world who have so much money that most people in the world could
not imagine it. There are then also people who make that first person feel
poor. This long tail can lead to some misleading measurements. If you took the wealth
of Bill Gates and three random people off the street, the mean wealth of your
sample would be around 20 billion dollars. That may sound absurd, but some
people have so much money that they can have a heavy weight on the mean of the
whole population. That is why economists like to talk about median household
income. Currently, if I recall correctly, it is around fifty thousand dollars.
The mean household income is much more than that. We use the median here
because it gives us a truer picture of the thing being measured.