Why use Root Mean Squared Error (RMSE) instead of Mean Absolute Error (MAE)??  

Hi

I've been investigating the error generated in a calculation - I initially calculated the error as a Root Mean Normalised Squared Error.

Looking a little closer, I see the effects of squaring the error gives more weight to larger errors than smaller ones, skewing the error estimate towards the odd outlier. This is quite obvious in retrospect. 

So my question - in what instance would the Root Mean Squared Error be a more appropriate measure of error than the Mean Absolute Error? The latter seems more appropriate to me or am I missing something?

To illustrate this I have attached an example below:                 


The scatter plot shows two variables with a good correlation, 
the two histograms to the right chart the error between Y(observed )
and Y(predicted) using normalised RMSE (top) and MAE (bottom).




There are no significant outliers in this data and MAE gives a lower error than RMSE. Is there any rational, other than MAE being preferable, for using one measure of error over the other?        

This depends on your loss function. In many circumstances it makes sense to give more weight to points further away from the mean--that is, being off by 10 is more than twice as bad as being off by 5. In such cases RMSE is a more appropriate measure of error.

If being off by ten is just twice as bad as being off by 5, then MAE is more appropriate.

In any case, it doesn't make sense to compare RMSE and MAE to each other as you do in your second-to-last sentence ("MAE gives a lower error than RMSE"). MAE will never be higher than RMSE because of the way they are calculated. They only make sense in comparison to the same measure of error: you can compare RMSE for Method 1 to RMSE for Method 2, or MAE for Method 1 to MAE for Method 2, but you can't say MAE is better than RMSE for Method 1 because it's smaller.
