# Root mean square

In mathematics, the root mean square (abbreviated RMS or rms) is a statistical measure of the magnitude of a varying quantity. It can be calculated for a series of discrete values or for a continuously varying function. The name comes from the fact that it is the square root of the mean of the squares of the values. It is a power mean with the power t=2.

The rms for a collection of N values {x1, x2, ..., xN} is:

[itex]

x_{\mathrm{rms}} = \sqrt {{1 \over N} \sum_{i=1}^{N} x_i^2} = \sqrt {{x_1^2 + x_2^2 + \cdots + x_N^2} \over N} [itex]

and the corresponding formula for a continuous function f(t) defined over the interval T1t ≤ T2 is:

[itex]

x_{\mathrm{rms}} = \sqrt {{1 \over {T_2 - T_1}} {\int_{T_1}^{T_2} {[f(t)]}^2\, dt}}. [itex]

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### Uses

The RMS value of a function is often used in physics and electronics. For example, we may wish to calculate the power P dissipated by an electrical conductor of resistance R. It is easy to do the calculation when a constant current (I) flows through the conductor. It is simply,

But what if the current is a varying function I(t)? This is where the rms value comes in. It may be shown that the rms value of I(t) can be substituted for the constant current I in the above equation to give the mean power dissipation, thus:

In the common case of alternating current, when I(t) is a sinusoidal current, as is approximately true for mains power, the rms value is easy to calculate from equation (2) above. The result is:

[itex]I_{\mathrm{rms}} = {I_p \over {\sqrt 2}}[itex]

where Ip is the peak amplitude.

The RMS value can be calculated using equation (2) for any waveform, for example an audio or radio signal. This allows us to calculate the mean power delivered into a specified load. For this reason, listed voltages for power outlets (e.g. 110V or 240V) are almost always quoted in RMS values, and not peak values.

In the field of audio, mean power is often (misleadingly) referred to as RMS power. This is probably because it can be derived from the RMS voltage or RMS current. Furthermore, because RMS implies some form of averaging, expressions such as "peak RMS power", sometimes used in advertisements for audio amplifiers, are meaningless.

## Relationship to the arithmetic mean and the standard deviation

If [itex]\bar{x}[itex] is the arithmetic mean and [itex]\sigma_{x}[itex] is the standard deviation of a population then

[itex]x_{\mathrm{rms}}^2 = \bar{x}^2 + \sigma_{x}^2.[itex]

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