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The root-mean-square (RMS) of a variate $x$, sometimes called the Quadratic Mean, is the Square Root of the mean squared value of $x$:

$\displaystyle R(x)$ $\textstyle \equiv$ $\displaystyle \sqrt{\left\langle{x^2}\right\rangle{}}$ (1)
  $\textstyle =$ $\displaystyle \left\{\begin{array}{ll}\sqrt{\sum_{i=1}^n x_i^2\over n} & \mbox{...
...x\over \int P(x)\,dx} & \mbox{for a continuous distribution.}\end{array}\right.$ (2)

Hoehn and Niven (1985) show that
R(a_1+c, a_2+c, \ldots, a_n+c)<c+R(a_1, a_2, \ldots, a_n)
\end{displaymath} (3)

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, Generalized Mean, Geometric Mean, Harmonic Mean, Harmonic-Geometric Mean, Mean, Median (Statistics), Standard Deviation, Variance


Hoehn, L. and Niven, I. ``Averages on the Move.'' Math. Mag. 58, 151-156, 1985.

© 1996-9 Eric W. Weisstein