In mathematics, the root mean square (abbrev. RMS, RMS or rms) of a set of values is the square root of the set's mean square. Given a set ${\displaystyle x_{i}}$, its RMS is denoted as either ${\displaystyle x_{\mathrm {RMS} }}$ or ${\displaystyle \mathrm {RMS} _{x}}$. The RMS is also known as the quadratic mean (denoted ${\displaystyle M_{2}}$), a special case of the generalized mean. The RMS of a continuous function is denoted ${\displaystyle f_{\mathrm {RMS} }}$ and can be defined in terms of an integral of the square of the function. In estimation theory, the root-mean-square deviation of an estimator measures how far the estimator strays from the data.

The RMS value of a set of values (or a continuous-time waveform) is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous waveform.

In the case of a set of n values ${\displaystyle \{x_{1},x_{2},\dots ,x_{n}\}}$, the RMS is

The corresponding formula for a continuous function (or waveform) f(t) defined over the interval ${\displaystyle T_{1}\leq t\leq T_{2}}$ is

and the RMS for a function over all time is

The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking the RMS of a sample consisting of equally spaced observations. Additionally, the RMS value of various waveforms can also be determined without calculus, as shown by Cartwright.

In the case of the RMS statistic of a random process, the expected value is used instead of the mean.

If the waveform is a pure sine wave, the relationships between amplitudes (peak-to-peak, peak) and RMS are fixed and known, as they are for any continuous periodic wave. However, this is not true for an arbitrary waveform, which may not be periodic or continuous. For a zero-mean sine wave, the relationship between RMS and peak-to-peak amplitude is: