Root mean square

, the RMS is The corresponding formula for a continuous function (or waveform) f(t) defined over the interval

Additionally, the RMS value of various waveforms can also be determined without calculus, as shown by Cartwright.

[4] 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.

[5] Alternatively, for waveforms that are perfectly positively correlated, or "in phase" with each other, their RMS values sum directly.

The RMS of an alternating electric current equals the value of constant direct current that would dissipate the same power in a resistive load.

refers to the direct current (or average) component of the signal, and

For a load of R ohms, power is given by: However, if the current is a time-varying function, I(t), this formula must be extended to reflect the fact that the current (and thus the instantaneous power) is varying over time.

Average power can also be found using the same method that in the case of a time-varying voltage, V(t), with RMS value VRMS, This equation can be used for any periodic waveform, such as a sinusoidal or sawtooth waveform, allowing us to calculate the mean power delivered into a specified load.

By taking the square root of both these equations and multiplying them together, the power is found to be: Both derivations depend on voltage and current being proportional (that is, the load, R, is purely resistive).

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 the continuous case equation above.

Since Ip is a positive constant and was to be squared within the integral: Using a trigonometric identity to eliminate squaring of trig function: but since the interval is a whole number of complete cycles (per definition of RMS), the sine terms will cancel out, leaving: A similar analysis leads to the analogous equation for sinusoidal voltage: where IP represents the peak current and VP represents the peak voltage.

Because of their usefulness in carrying out power calculations, listed voltages for power outlets (for example, 120 V in the US, or 230 V in Europe) are almost always quoted in RMS values, and not peak values.

Peak values can be calculated from RMS values from the above formula, which implies VP = VRMS × √2, assuming the source is a pure sine wave.

Thus the peak value of the mains voltage in the USA is about 120 × √2, or about 170 volts.

RMS quantities such as electric current are usually calculated over one cycle.

However, for some purposes the RMS current over a longer period is required when calculating transmission power losses.

In the physics of gas molecules, the root-mean-square speed is defined as the square root of the average squared-speed.

In physics, speed is defined as the scalar magnitude of velocity.

However, the RMS of the differences is usually the preferred measure, probably due to mathematical convention and compatibility with other formulae.

The RMS can be computed in the frequency domain, using Parseval's theorem.

Physical scientists often use the term root mean square as a synonym for standard deviation when it can be assumed the input signal has zero mean, that is, referring to the square root of the mean squared deviation of a signal from a given baseline or fit.

[8][9] This is useful for electrical engineers in calculating the "AC only" RMS of a signal.