Root-Mean-Square

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

$$R(x) \equiv \sqrt{\left\langle{x^2}\right\rangle{}}$$ (1)

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

Hoehn and Niven (1985) show that

$$R(a_1+c, a_2+c, \ldots, a_n+c)<c+R(a_1, a_2, \ldots, a_n)$$ (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

References

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