Root mean square

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In mathematics, root mean square (abbreviated RMS or rms), also known as the quadratic mean, 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 <math>t=2</math>.

1 Calculating the root mean square
2 Uses
3 Relationship to the arithmetic mean and the standard deviation
4 See also
5 External links

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Calculating the root mean square

The rms for a collection of <math>N</math> values <math>\{x_1,x_2,\dots,x_N\}</math> is:

<math>

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} </math>

and the corresponding formula for a continuous function <math>f(t)</math> defined over the interval <math>T_1 \le t \le T_2</math> (for a periodic function the interval should be a whole number of complete cycles) is:

<math>

f_{\mathrm{rms}} = \sqrt {{1 \over {T_2-T_1}} {\int_{T_1}^{T_2} {[f(t)]}^2\, dt}} </math>

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Uses

(in this section <math>\mathrm{E}(x)</math> means the mean of x)

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

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

<math>P_\mathrm{avg}\,\!</math>	<math>= \mathrm{E}(I^2R)\,\!</math>
	<math>= R\mathrm{E}(I^2)\,\!</math> (R is constant so we can take it outside the average)
	<math>= I_\mathrm{rms}^2R\,\!</math> (by definition of RMS)

We can also show by the same method

<math>P_\mathrm{avg} = {V_\mathrm{rms}^2\over R}\,\!</math>

By square rooting both these equations and multiplying them together we get the equation

<math>P_\mathrm{avg} = V_\mathrm{rms}I_\mathrm{rms}\,\!</math>

However it is important to stress that this is based on the assumption that voltage and current are proportional (that is the load is resistive) and is NOT true in the general case (see AC power for more information).

In the common case of alternating current, when <math>I(t)</math> 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:

<math>I_{\mathrm{rms}} = {I_\mathrm{p} \over {\sqrt 2}}</math>

where <math>I_\mathrm{p}</math> 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. 110 V or 240 V) 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.

In chemistry, the root mean square velocity is defined as the square root of the average velocity-squared of the molecules in a gas. The RMS velocity of a gas is calculated using the following equation:

<math>{u_\mathrm{rms}} = {\sqrt{3RT \over {M}}}</math>

where <math>R</math> represents the ideal gas constant (in this case, 8.314 J/(mol⋅K)), <math>T</math> is the temperature of the gas in kelvins, and <math>M</math> is the molar mass of the compound in kilograms per mole.

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