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Root-Mean-Square


For a set of n numbers or values of a discrete distribution x_i, ..., x_n, the root-mean-square (abbreviated "RMS" and sometimes called the quadratic mean), is the square root of mean of the values x_i^2, namely

x_(RMS)=sqrt((x_1^2+x_2^2+...+x_n^2)/n)
(1)
=sqrt((sum_(i=1)^(n)x_i^2)/n)
(2)
=sqrt(<x^2>),
(3)

where <x^2> denotes the mean of the values x_i^2.

For a variate X from a continuous distribution P(x),

 x_(RMS)=sqrt((int[P(x)]^2dx)/(intP(x)dx)),
(4)

where the integrals are taken over the domain of the distribution. Similarly, for a function f(t) periodic over the interval [T_1,T_2], the root-mean-square is defined as

 f_(RMS)=sqrt(1/(T_2-T_1)int_(T_1)^(T_2)[f(t)]^2dt).
(5)

The root-mean-square is the special case M_2 of the power mean.

Hoehn and Niven (1985) show that

 R(a_1+c,a_2+c,...,a_n+c)<c+R(a_1,a_2,...,a_n)
(6)

for any positive constant c.

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.


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, Variance

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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.

Referenced on Wolfram|Alpha

Root-Mean-Square

Cite this as:

Weisstein, Eric W. "Root-Mean-Square." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Root-Mean-Square.html

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