Interpreting the Root Mean Squared Error (RMSE)! I read all about pros and cons of RMSE vs. other absolute errors namely mean absolute error (MAE). See the the following references:


MAE and RMSE — Which Metric is Better?
What's the bottom line? How to compare models
Or this nice blogpost, or this question in stats.stackexchange containing interesting responses, and this one in datascience.stackexchange


Still I can not get my head around something about RMSE:

Scenario: Let's say we have a regressor for predicting house prices with a MAE of 20.5\$ and a RMSE of 24.5\$. Based on MAE, I can certainly interpret that the average difference between the predicted and the actual price is 20.5\$. How can I interpret RMSE? Can we still safely say the predicted and the actual price are off by 24.5\$ at the same time base on RMSE (upper-bound of prediction error)?

In the first medium post, it says: 


  RMSE does not describe average error alone and has other implications
  that are more difficult to tease out and understand.


It confuses me a little. And I could not find any reliable reference to also clearly state that one can safely interpret RSME as one does MAE. Is RMSE is simply a only mathematically more convenient for optimization etc., and we are better off with MAE for the interpretation?

Any detailed explanation is highly appreciated.
How can I interpret RMSE?


RMSE is exactly what's defined. $24.5 is the square root of the average of squared differences between your prediction and your actual observation. Taking squared differences is more common than absolute difference in statistics, as you might have learnt from the classical linear regression.


  It confuses me a little. And I could not find any reliable reference
  to also clearly state that one can safely interpret RSME as one does
  MAE. Is RMSE is simply a only mathematically more convenient for
  optimization etc., and we are better off with MAE for the
  interpretation?


I think this post should help you. I'll answer your questions directly:


RMSE is easier mathematically and also practically. Have you heard of derivative? The derivative for MAE is undefined when your prediction is equal to observation but well defined everywhere for RMSE. In machine learning, a well defined gradient function is generally better.
Both RMSE and MAE are useful, but they are two very different metrics. In regression, it's generally about choosing between linear regression and quantile regression. They are two very different models!
As stated in the link, if you don't want your residuals affect your model too much, MAE could be better. Otherwise, if your data set is well defined (not many residuals), RMSE could be better. 


There is no right or wrong on which one is better. Think like two different algorithms for doing modelling.