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 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.
See alsoArithmetic-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 Explore with Wolfram|Alpha ReferencesHoehn, L. and Niven, I. "Averages on the Move." Math. Mag. 58, 151-156, 1985.Kenney, J. F. and Keeping, E. S. "Root Mean Square." ยง4.15 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 59-60, 1962. Referenced on Wolfram|AlphaRoot-Mean-Square Cite this as:Weisstein, Eric W. "Root-Mean-Square." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Root-Mean-Square.html
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