The root mean square deviation (RMSD) or root mean square error (RMSE) is either one of two closely related and frequently used measures of the differences between true or predicted values on the one hand and observed values or an estimator on the other. The deviation is typically simply a differences of scalars; it can also be generalized to the vector lengths of a displacement, as in the bioinformatics concept of root mean square deviation of atomic positions.

The RMSD of a sample is the quadratic mean of the differences between the observed values and predicted ones. These deviations are called residuals when the calculations are performed over the data sample that was used for estimation (and are therefore always in reference to an estimate) and are called errors (or prediction errors) when computed out-of-sample (aka on the full set, referencing a true value rather than an estimate). The RMSD serves to aggregate the magnitudes of the errors in predictions for various data points into a single measure of predictive power. RMSD is a measure of accuracy, to compare forecasting errors of different models for a particular dataset and not between datasets, as it is scale-dependent.

RMSD is always non-negative, and a value of 0 (almost never achieved in practice) would indicate a perfect fit to the data. In general, a lower RMSD is better than a higher one. However, comparisons across different types of data would be invalid because the measure is dependent on the scale of the numbers used.

RMSD is the square root of the average of squared 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 squared error: $${\displaystyle \operatorname {RMSD} ({\hat {\theta }})={\sqrt {\operatorname {MSE} ({\hat {\theta }})}}={\sqrt {\operatorname {E} {\big (}({\hat {\theta }}-\theta )^{2}{\big )}}}.}$$

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

If X₁, ..., X_n is a sample of a population with true mean value ${\displaystyle x_{0}}$, then the RMSD of the sample is

The RMSD of predicted values ${\displaystyle {\hat {y}}_{t}}$ for times t of a regression's dependent variable ${\displaystyle y_{t},}$ with variables observed over T times, is computed for T different predictions as the square root of the mean of the squares of the deviations: