Root mean square error (rmse) is defined mathematically as:

where

Fi = the forecast values of the parameter in question

Oi = the corresponding verifying value (observed or analysed)

N = the number of verifying points (grid points or observations) in the verification area

RMSE is a measure of the "average" error, weighted according to the square of the error. It answers the question, "what is the average magnitude of the forecast errors?", but does not indicate the direction of the errors. Because it is a squared quantity, RMSE is influenced more strongly by large errors than by small errors. Its range is from 0 to infinity, with 0 being a perfect score.