 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 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.

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.

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