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

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.

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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.

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  • $\begingroup$ thank a lot for your time and nicely put comments. My question was only about the interpretation of error from RMSE. The first part of your answer is what I needed to clarify (I will wait to a big longer before marking as "Accepted Answer" to see if others think the same. I personally think that is it and you are right and we can not read the RMSE as we do MAE). The second part of your answer, the post you introduced, are useful facts about RMSE, but I was fully aware of those and in fact the post was in my references already (the stats.stackexchange in the third bullet point). $\endgroup$ – TwinPenguins Aug 16 '18 at 10:48

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