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


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

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


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


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

Hoehn and Niven (1985) show that


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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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


Cite this as:

Weisstein, Eric W. "Root-Mean-Square." From MathWorld--A Wolfram Web Resource.

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