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

			(1)
			(2)
			(3)

where  denotes the mean of the values .

For a variate  from a continuous distribution ,

(4)

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.

Hoehn and Niven (1985) show that

(6)

for any positive constant .

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.

REFERENCES:

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.