Root mean square

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 x i {\displaystyle x_{i}} , its RMS is denoted as either x R M S {\displaystyle x_{\mathrm {RMS} }} or R M S x {\displaystyle \mathrm {RMS} _{x}} . The RMS is also known as the quadratic mean (denoted M 2 {\displaystyle M_{2}} ), a special case of the generalized mean. The RMS of a continuous function is denoted f R M S {\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.

Definition 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 { x 1 , x 2 , … , x n } {\displaystyle \{x_{1},x_{2},\dots ,x_{n}\}} , the RMS is

x RMS = 1 n ( x 1 2 + x 2 2 + ⋯ + x n 2 ) . {\displaystyle x_{\text{RMS}}={\sqrt {{\frac {1}{n}}\left({x_{1}}^{2}+{x_{2}}^{2}+\cdots +{x_{n}}^{2}\right)}}.}

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

f RMS = 1 T 2 − T 1 ∫ T 1 T 2 [ f ( t ) ] 2 d t , {\displaystyle f_{\text{RMS}}={\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{[f(t)]}^{2}\,{\rm {d}}t}}},}

and the RMS for a function over all time is

f RMS = lim T → ∞ 1 2 T ∫ − T T [ f ( t ) ] 2 d t . {\displaystyle f_{\text{RMS}}=\lim _{T\rightarrow \infty }{\sqrt {{1 \over {2T}}{\int _{-T}^{T}{[f(t)]}^{2}\,{\rm {d}}t}}}.}

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