root-mean-square deviation

Aliases
  • RMSE
  • rmse
  • RMSD
  • rmsd
  • root-mean-square error
  • root-mean-square deviation
  • L2 norm
Image of Author
Austin Gatlin
October 16, 2023 (last updated April 4, 2025)

https://en.wikipedia.org/wiki/Root-mean-square_deviation

Given a list of actual values , , and a list of predicted values , . Deviation, or error is . Squared deviation is . Mean squared deviation is . Root mean square deviation is, finally:

This value can be normalized in a variety of ways, knowns a NRMSE or NRMSD.

RMSD vs MAE

MAE, or mean absolute error is . I find this to be the "most intuitive" approach to estimating error. The pros and cons of RMSE vs MAE are discussed below.

From Wikipedia: Mean absolute error: Related measures (emphasis my own).

MAE is not identical to root-mean square error (RMSE), although some researchers report and interpret it that way. The MAE is conceptually simpler and also easier to interpret than RMSE: it is simply the average absolute vertical or horizontal distance between each point in a scatter plot and the Y=X line. In other words, MAE is the average absolute difference between X and Y. Furthermore, each error contributes to MAE in proportion to the absolute value of the error. This is in contrast to RMSE which involves squaring the differences, so that a few large differences will increase the RMSE to a greater degree than the MAE.

This represents the pros and cons well. RMSE gets larger the larger the deviations. This might be desirable, but one large outlier might skew the interpretation, when compared to MAE.

AI/ML

In Artificial Intelligence and deep learning, RMSD is also called L2 norm, and MAE is called L1 norm.