Interpreting the Root Mean Squared Error (RMSE)!

Question

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.

Answer 1

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.

Answer 2

Root Mean Square Error (RMSE) is the standard deviation of the residuals (prediction errors). Residuals are a measure of how far from the regression line data points are; RMSE is a measure of how to spread out these residuals are. In other words, it tells you how concentrated the data is around the line of best fit.

One way to assess how well a regression model fits a dataset is to calculate the root mean square error, which is a metric that tells us the average distance between the predicted values from the model and the actual values in the dataset.

The lower the RMSE, the better a given model is able to "fit" a dataset

Find the detailed explanation on How to Interpret Root Mean Square Error (RMSE) with examples.