TOPICS
Root-Mean-Square
For a set of
numbers or values of a discrete distribution
, ...,
, the root-mean-square (abbreviated "RMS" and sometimes called the quadratic mean), is thesquare root of mean of the values
, namely
 |  |  | (1) |
 |  |  | (2) |
 |  |  | (3) |
where
denotes the mean of the values
.
For avariate
from a continuous distribution
,
![x_(RMS)=sqrt((int[P(x)]^2dx)/(intP(x)dx)),](/image.pl?url=https%3a%2f%2fmathworld.wolfram.com%2fimages%2fequations%2fRoot-Mean-Square%2fNumberedEquation1.svg&f=jpg&w=240) | (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
![f_(RMS)=sqrt(1/(T_2-T_1)int_(T_1)^(T_2)[f(t)]^2dt).](/image.pl?url=https%3a%2f%2fmathworld.wolfram.com%2fimages%2fequations%2fRoot-Mean-Square%2fNumberedEquation2.svg&f=jpg&w=240) | (5) |
The root-mean-square is the special case
of thepower mean.
Hoehn and Niven (1985) show that
 | (6) |
for anypositive constant
.
Physical scientists often use the term root-mean-square as a synonym forstandard deviation when they refer to thesquare root of the mean squared deviation of a signal from a given baseline or fit.
See also
Arithmetic-Geometric Mean,Arithmetic-Harmonic Mean,Geometric Mean,Harmonic Mean,Harmonic-Geometric Mean,Mean,Mean Square Displacement,Power Mean,Pythagorean Means,Standard Deviation,Statistical Median,VarianceExplore with Wolfram|Alpha
More things to try:
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 inMathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 59-60, 1962.Referenced on Wolfram|Alpha
Root-Mean-SquareCite this as:
Weisstein, Eric W. "Root-Mean-Square."FromMathWorld--A Wolfram Web Resource.https://mathworld.wolfram.com/Root-Mean-Square.html