In mathematics and its applications, the root mean square of a set of numbers
x
i
{\displaystyle x_{i}}
(abbreviated as RMS, RMS or rms and denoted in formulas as either
x
R
M
S
{\displaystyle x_{\mathrm {RMS} }}
or
R
M
S
x
{\displaystyle \mathrm {RMS} _{x}}
) is defined as the square root of the mean square (the arithmetic mean of the squares) of the set.
The RMS is also known as the quadratic mean (denoted
M
2
{\displaystyle M_{2}}
) and is a particular case of the generalized mean. The RMS of a continuously varying function (denoted
f
R
M
S
{\displaystyle f_{\mathrm {RMS} }}
) can be defined in terms of an integral of the squares of the instantaneous values during a cycle.
For alternating electric current, RMS is equal to the value of the constant direct current that would produce the same power dissipation in a resistive load.
In estimation theory, the root-mean-square deviation of an estimator is a measure of the imperfection of the fit of the estimator to the data.