rootmeansquare

Root mean square (RMS) is the square root of the mean of the squares of a set of values or of a continuous function. It provides a measure of the magnitude of a varying quantity and is widely used when signs of values would cancel under a simple average.

For a finite set x1, x2, ..., xN, the RMS is defined as sqrt((1/N) ∑ xi^2). For a continuous function x(t) over an interval [a, b], the RMS is sqrt((1/(b−a)) ∫_a^b x(t)^2 dt). If the function is periodic with period T, the RMS over one period is sqrt((1/T) ∫_0^T x(t)^2 dt).

The RMS is related to energy and to the mean and variance of a data set. The energy of a signal is the integral (or sum) of its squared values: ∫ x^2 dt (or ∑ xi^2). The RMS^2 equals energy per unit time for a finite interval, i.e., energy divided by the duration. In probabilistic terms, RMS^2 = μ^2 + σ^2, where μ is the mean and σ^2 is the variance; in particular, for zero-mean data RMS equals the standard deviation.

Applications of RMS span science and engineering. In electrical engineering, the RMS value of an alternating current or voltage corresponds to the DC value that would produce the same heating effect in a resistor: P = V_rms^2 / R or I_rms^2 R. In statistics and data analysis, RMS quantifies the overall magnitude of a set of numbers and is used in various normalization and error-measure calculations. The term is often denoted as x_rms or X_rms.