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The normalized root mean squared error (NRMSE), also called a scatter index, is a statistical error indicator defined as [1].

Where Oi are observed values and Si are simulated values.

It can also be calculated as RMSE/range or RMSE/mean. Which formula you use depends on your data and the purpose for calculating it.

Disadvantages of NRSME

Root mean square error can be used to compare different models. However, RMSE doesn’t perform well if comparing models fits for different response variables or if the response variable is standardized, log-transformed, or otherwise modified. To overcome these issues, the NRMSE is used instead [2]. One downside is that the NRMSE will lose the units associated with the response variable.

Generally speaking, lower NRMSE values indicate less residual variance for a model. However, the indicator is not always reliable for finding the best model, especially for small samples. To overcome this problem, some authors have suggested corrected indicators (e.g., [3]).


[1] Mentaschi, L. et al. Why NRMSE is not completely reliable for forecast/hindcast model test performances. Geophysical Research Abstracts Vol. 15, EGU2013-7059, 2013. Retrieved June 18, 2022 from:
[2] Saskia (2019). How to normalize the RMSE. Retrieved June 18, 2021 from:
[3] Hanna, S.R. and Heinold, D.W. 1985 Development and Application of a Simple Method for Evaluating Air Quality, API Pub. No. 4409, Washington, D.C.

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