For a set of
numbers or values of a discrete distribution , ..., , the root-mean-square (abbreviated "RMS" and sometimes
called the quadratic mean), is the square root of
mean of the values ,
namely

where the integrals are taken over the domain of the distribution. Similarly, for a function
periodic over the interval ], the root-mean-square is defined as

(5)

The root-mean-square is the special case of the power mean.

Physical scientists often use the term root-mean-square as a synonym for standard deviation when they refer to the square root of
the mean squared deviation of a signal from a given baseline or fit.

Hoehn, L. and Niven, I. "Averages on the Move." Math. Mag.58, 151-156, 1985.Kenney, J. F. and Keeping,
E. S. "Root Mean Square." §4.15 in Mathematics
of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 59-60,
1962.