Root mean squared error (RMSE) is the square root of mean squared error.
Details
$$\mathrm{RMSE} = \sqrt{\frac{1}{n}\sum_{i = 1}^{n}(obs_i - pred_i)^2}$$
RMSE is non-negative, has the same units as the response variable, and is
zero for perfect predictions. Smaller values indicate better agreement.
Squaring means that a small number of large errors can disproportionately
increase RMSE. Therefore, when errors are skewed or asymmetric, interpret
RMSE together with mae() and inspect the error distribution rather than
relying on RMSE alone. Missing-value handling follows bias().
References
Legates, D. R. and McCabe, G. J. (1999). Evaluating the use of goodness-of-fit measures in hydrologic and hydroclimatic model validation. Water Resources Research, 35(1), 233-241. doi:10.1029/1998WR900018
Armstrong, J. S. (2001). Evaluating forecasting methods. In J. S. Armstrong (Ed.), Principles of Forecasting (pp. 443-472). Springer. doi:10.1007/978-0-306-47630-3_20
Willmott, C. J. and Matsuura, K. (2005). Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance. Climate Research, 30, 79-82. doi:10.3354/cr030079
Hodson, T. O. (2022). Root mean square error (RMSE) or mean absolute error (MAE): When to use them or not. Geoscientific Model Development, 15, 5481-5487. doi:10.5194/gmd-15-5481-2022
Examples
rmse(1:3, c(1, 3, 2))
#> [1] 0.8164966