Root mean square deviation

The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) is a frequently used measure of the differences between values (sample and population values) predicted by a model or an estimator and the values actually observed. The RMSD represents the sample standard deviation of the differences between predicted values and observed values. These individual differences are called residuals when the calculations are performed over the data sample that was used for estimation, and are called prediction errors when computed out-of-sample. The RMSD serves to aggregate the magnitudes of the errors in predictions for various times into a single measure of predictive power. RMSD is a measure of accuracy, to compare forecasting errors of different models for a particular data and not between datasets, as it is scale-dependent.

Although RMSE is one of the most commonly reported measures of disagreement, some scientists misinterpret RMSD as average error, which RMSD is not. RMSD is the square root of the average of squared errors, thus RMSD confounds information concerning average error with information concerning variation in the errors. The effect of each error on RMSD is proportional to the size of the squared error thus larger errors have a disproportionately large effect on RMSD. Consequently, RMSD is sensitive to outliers.

The RMSD of an estimator ${\displaystyle {\hat {\theta }}}$ with respect to an estimated parameter ${\displaystyle \theta }$ is defined as the square root of the mean square error:

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