Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE)

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Mean absolute error (MAE)

The MAE measures the average magnitude of the errors in a set of forecasts, without considering their direction. It measures accuracy for continuous variables. The equation is given in the library references. Expressed in words, the MAE is the average over the verification sample of the absolute values of the differences between forecast and the corresponding observation. The MAE is a linear score which means that all the individual differences are weighted equally in the average.

Root mean squared error (RMSE)

The RMSE is a quadratic scoring rule which measures the average magnitude of the error. The equation for the RMSE is given in both of the references. Expressing the formula in words, the difference between forecast and corresponding observed values are each squared and then averaged over the sample. Finally, the square root of the average is taken. Since the errors are squared before they are averaged, the RMSE gives a relatively high weight to large errors. This means the RMSE is most useful when large errors are particularly undesirable.

The MAE and the RMSE can be used together to diagnose the variation in the errors in a set of forecasts. The RMSE will always be larger or equal to the MAE; the greater difference between them, the greater the variance in the individual errors in the sample. If the RMSE=MAE, then all the errors are of the same magnitude

Both the MAE and RMSE can range from 0 to ∞. They are negatively-oriented scores: Lower values are better.

You read that a set of temperature forecasts shows a MAE of 1.5 degrees and a RMSE of 2.5 degrees. What does this mean?

1. There is some variation in the magnitude of the errors 
2. Very large errors are unlikely to have occured 
3. The average difference between the forecast and the observed temperature was 1.5 degrees