# 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 good measure of accuracy, but only to compare forecasting errors of different models for a particular variable and not between variables, as it is scale-dependent.[1]

## Formula

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

$\operatorname{RMSD}(\hat{\theta}) = \sqrt{\operatorname{MSE}(\hat{\theta})} = \sqrt{\operatorname{E}((\hat{\theta}-\theta)^2)}.$

For an unbiased estimator, the RMSD is the square root of the variance, known as the standard error.

The RMSD of predicted values $\hat y_t$ for times t of a regression's dependent variable $y$ is computed for n different predictions as the square root of the mean of the squares of the deviations:

$\operatorname{RMSD}=\sqrt{\frac{\sum_{t=1}^n (\hat y_t - y)^2}{n}}.$

In some disciplines, the RMSD is used to compare differences between two things that may vary, neither of which is accepted as the "standard". For example, when measuring the average difference between two time series $x_{1,t}$ and $x_{2,t}$, the formula becomes

$\operatorname{RMSD}= \sqrt{\frac{\sum_{t=1}^n (x_{1,t} - x_{2,t})^2}{n}}.$

## Normalized root-mean-square deviation

Normalizing the RMSD facilitates the comparison between datasets or models with different scales. Though there is no consistent means of normalization in the literature, the range of the measured data defined as the maximum value minus the minimum value is a common choice:[2]

$\mathrm{NRMSD} = \frac{\mathrm{RMSD}}{y_\max -y_\min}$

This value is commonly referred to as the normalized root-mean-square deviation or error (NRMSD or NRMSE), and often expressed as a percentage, where lower values indicate less residual variance.

Another common choice is to normalize by the mean value of the measurements:[3]

$\mathrm{CV(RMSE)} = \frac {\mathrm{RMSE}}{\bar y}.$

When the RMSD is normalized by the mean measured value, is usually called coefficient of variation of the RMSE, CV(RMSE). It is analogous to the coefficient of variation with the RMSE taking the place of the standard deviation.