Root mean square
In statistics, the root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical measure defined as the square root of the arithmetic mean of the squares of a set of numbers.^{[1]}
RMS can also be calculated for a continuously varying function.
In physics it is a characteristic of a continuously varying quantity, such as a cyclically alternating electric current, obtained by taking the mean of the squares of the instantaneous values during a cycle. It is equal to the value of the direct current that would produce the same power dissipation in a resistive load.^{[1]} This is a result of Joule's first law, which states that the power in resistive load is proportional to the square of the current (and, as a consequence of Ohm's law, also to the square of the voltage).
In econometrics the root mean square error of an estimator is a measure of the imperfection of the fit of the estimator to the data.
The root mean square is a particular case of the generalized mean, with exponent 2.
Contents
Definition
The RMS value of a set of values (or a continuoustime 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 , the RMS
The corresponding formula for a continuous function (or waveform) f(t) defined over the interval is
and the RMS for a function over all time is
The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking the RMS of a series of equally spaced samples. Additionally, the RMS value of various waveforms can also be determined without calculus, as shown by Cartwright.^{[2]}
In the case of the RMS statistic of a random process, the expected value is used instead of the mean.
RMS of common waveforms
Waveform  Equation  RMS 

DC, constant  
Sine wave  
Square wave  
DCshifted square wave  
Modified sine wave  
Triangle wave  
Sawtooth wave  
Pulse train  
Phasetophase voltage  
Notes: t is time f is frequency a is amplitude (peak value) D is the duty cycle or the percent(%) spent high of the period (1/f) {r} is the fractional part of r 
Waveforms made by summing known simple waveforms have an RMS that is the root of the sum of squares of the component RMS values, if the component waveforms are orthogonal (that is, if the average of the product of one simple waveform with another is zero for all pairs other than a waveform times itself).^{[3]}
A special case of this, particularly helpful in electrical engineering, is
where refers to the DC component of the signal and is the AC component of the signal.
Uses
The RMS value of a function is often used in physics and electrical engineering.
Average electrical power
Electrical engineers often need to know the power, P, dissipated by an electrical resistance, R. It is easy to do the calculation when there is a constant current, I, through the resistance. For a load of R ohms, power is defined simply as:
However, if the current is a timevarying function, I(t), this formula must be extended to reflect the fact that the current (and thus the instantaneous power) is varying over time. If the function is periodic (such as household AC power), it is still meaningful to discuss the average power dissipated over time, which is calculated by taking the average power dissipation:

(where denotes the mean of a function) (as R does not vary over time, it can be factored out) (by definition of RMS)
So, the RMS value, I_{RMS}, of the function I(t) is the constant current that yields the same power dissipation as the timeaveraged power dissipation of the current I(t).
Average power can also be found using the same method that in the case of a timevarying voltage, V(t), with RMS value V_{RMS},
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 (i.e., the load, R, is purely resistive). Reactive loads (i.e., loads capable of not just dissipating energy but also storing it) are discussed under the topic of AC power.
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. If I_{p} is defined to be the peak current, then:
where t is time and ω is the angular frequency (ω = 2π/T, where T is the period of the wave).
Since I_{p} is a positive constant:
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 sin terms will cancel out, leaving:
A similar analysis leads to the analogous equation for sinusoidal voltage:
Where I_{P} represents the peak current and V_{P} represents the peak voltage.
Because of their usefulness in carrying out power calculations, listed voltages for power outlets, e.g. 120 V (USA) or 230 V (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 V_{p} = V_{RMS} × √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. The peaktopeak voltage, being twice this, is about 340 volts. A similar calculation indicates that the peaktopeak mains voltage in Europe is about 650 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. The same principle applies, and (for example) a current of 10 amps used for 12 hours each day represents an RMS current of 5 amps in the long term.
The term "RMS power" is sometimes used in the audio industry as a synonym for "mean power" or "average power" (it is proportional to the square of the RMS voltage or RMS current in a resistive load). For a discussion of audio power measurements and their shortcomings, see Audio power.
Rootmeansquare speed
In the physics of gas molecules, the rootmeansquare speed is defined as the square root of the average squaredspeed. The RMS speed of an ideal gas is calculated using the following equation:
where R represents the ideal gas constant, 8.314 J/(mol·K), T is the temperature of the gas in kelvins, and M is the molar mass of the gas in kilograms. The generally accepted terminology for speed as compared to velocity is that the former is the scalar magnitude of the latter. Therefore, although the average speed is between zero and the RMS speed, the average velocity for a stationary gas is zero.
Rootmeansquare error
When two data sets—one set from theoretical prediction and the other from actual measurement of some physical variable, for instance—are compared, the RMS of the pairwise differences of the two data sets can serve as a measure how far on average the error is from 0.
The mean of the pairwise differences does not measure the variability of the difference, and the variability as indicated by the standard deviation is around the mean instead of 0. Therefore, the RMS of the differences is a meaningful measure of the error.
RMS in frequency domain
The RMS can be computed in the frequency domain, using Parseval's theorem. For a sampled signal , where is the sampling period,
 ,
where and N is number of samples and FFT coefficients.
In this case, the RMS computed in the time domain is the same as in the frequency domain:
Relationship to the arithmetic mean and the standard deviation
If is the arithmetic mean and is the standard deviation of a population or a waveform then:^{[4]}
From this it is clear that the RMS value is always greater than or equal to the average, in that the RMS includes the "error" / square deviation as well.
Physical scientists often use the term "root mean square" as a synonym for standard deviation when referring to the square root of the mean squared deviation of a signal from a given baseline or fit.^{[5]} ^{[6]} This is useful for electrical engineers in calculating the "AC only" RMS of a signal. Standard deviation being the root mean square of a signal's variation about the mean, rather than about 0, the DC component is removed (i.e. RMS(signal) = Stdev(signal) if the mean signal is 0).
See also
 Central moment
 Geometric mean
 L2 norm
 Least squares
 Mean squared displacement
 Mean squared error
 Root mean square deviation
 Rootmeansquare speed
 Table of mathematical symbols
 True RMS converter
References
 ↑ ^{1.0} ^{1.1} A Dictionary of Physics (6 ed.). Oxford University Press. 2009. ISBN 9780199233991.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Cartwright, Kenneth V (Fall 2007). "Determining the Effective or RMS Voltage of Various Waveforms without Calculus" (PDF). Technology Interface. 8 (1): 20 pages.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms". MasteringElectronicsDesign.com. Retrieved 21 January 2015.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Chris C. Bissell and David A. Chapman (1992). Digital signal transmission (2nd ed.). Cambridge University Press. p. 64. ISBN 9780521425575.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ "RootMeanSquare".<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ "ROOT, TH1:GetRMS".<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>