nRMSE

nRMSE, or normalized root mean square error, is a scale-free statistic used to assess the accuracy of predictions by dividing RMSE by a normalization term. By normalizing RMSE, it facilitates comparisons across datasets or models with different units or ranges.

Calculation: Given observed values y_i and predicted values ŷ_i for i = 1..n, RMSE = sqrt( (1/n) sum (ŷ_i - y_i)^2 ). The normalized RMSE is RMSE divided by a normalization factor, such as the range (max(y) - min(y)); the standard deviation of the observed values; or the mean of the observed values. Depending on the field, one may express nRMSE as a proportion or as a percentage by multiplying by 100.

Interpretation and considerations: Lower nRMSE indicates a better fit. Because the normalization changes the scale, different definitions yield different numeric values; thus, be explicit about the normalization term. Range normalization is sensitive to outliers and to the range of data; normalization by standard deviation has its own properties; normalization by mean can be problematic if the mean is near zero.

Applications: Widely used in environmental modeling, hydrology, meteorology, and machine learning to compare forecast accuracy across datasets. It is one of several normalized error metrics and should be considered alongside other metrics such as MAE, MAPE, or R^2.