ROOT MEAN SQUARE ERROR (RMSE) VS. MEAN ABSOLUTE ERROR (MAE)

What is Root Mean Square Error (RMSE)? RMSE is a widely used metric for evaluating the performance of regression and forecasting models. It is defined as the square root of the average of the squared differences between predicted values and actual (observed) values. What is Mean Absolute Error (MAE)? MAE is another widely used regression evaluation metric, defined as the average of the absolute differences between predicted values and actual values. Although both metrics quantify the average magnitude of prediction error and are expressed in the same units as the target variable, they behave very differently in the presence of outliers, variance in the error distribution, and gradient-based optimization. This paper presents a systematic comparison of RMSE and MAE, discusses the mathematical properties and deficiencies of each metric, and provides hand-calculated numerical examples that illustrate the scenarios in which each metric is more or less informative. We show that RMSE disproportionately penalizes large errors and is therefore more sensitive to outliers, while MAE treats all errors with equal weight and can consequently mask meaningful variability in the underlying error distribution. Guidelines for choosing between the two metrics are also discussed.