Is Median Absolute Percentage Error useless? I'm working on a project focused on pricing houses.
Looking online I see a lot of works and companies providing the performances of their model using the median instead of the mean (see for example Zillow).
And to me this makes total sense, since its a statistic less influenced by outliers (in my scenario big and particular houses).
Also since I would like to have a greater interpretability of the performance of my model I used the Absolute Percentage Error, instead of the squared error.
In the end what I ended up using is the Median Absolute Percentage Error, and with my surprise I discovered that is not a common metrics, it doesn't have a wikipedia page, few sites provide an explanation of what it is, sklearn doesn't even have it as a metrics inside the sklearn.metrics._regression.py module.
And my question is: why?
Fixing it was a no brainer, really 2 line of code and I created my own version of it and I can easily run my model with this new metric, but I was wondering if maybe I'm missing something, like if its completely useless or maybe there are other metrics that works better.
If anyone could give me some insights that would be great!

TLDR; Here is a point of view in terms of what the median/mean tell about the behaviour of the tails. The median gives little information while the mean does.

A related question is Chebychev-like inequality based on the median absolute deviation (about the median)
I answered that question while understanding the problem to be about the mean absolute deviation. The reason that I did that is because the mean makes much more sense in relation to Chebychev-like inequalities. A problem with the median is that it only relates to a single point on the distribution curve
$$\text{median}(X) = x:F(x) = 0.5$$
The median tells little about the entire distribution and what the tail of the distribution does. The median can be even zero if more than 50% of the deviation is zero.
The mean on the other hand, gives a more weighted information about the entire distribution and includes the tails.
$$\text{mean}(X) = 1- \int_0^\infty F(x) dx$$
Let's look at a few curves with median or mean equal to 1.

The red curves have an average of 1 and will be restricted to be below the black curve 1/x.
The blue curves have a median of 1 and will only pass the point (1,0.5), but further from that they can be any shape.
