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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and its applications, the root mean square of a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
of the mean square (the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The co ...
of the squares) of the set. The RMS is also known as the quadratic mean (denoted M_2) and is a particular case of the generalized mean. The RMS of a continuously varying function (denoted f_\mathrm) can be defined in terms of an
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
of the squares of the instantaneous values during a cycle. For alternating electric current, RMS is equal to the value of the constant
direct current Direct current (DC) is one-directional flow of electric charge. An electrochemical cell is a prime example of DC power. Direct current may flow through a conductor such as a wire, but can also flow through semiconductors, insulators, or ev ...
that would produce the same power dissipation in a resistive load. In
estimation theory Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their valu ...
, the root-mean-square deviation of an estimator is a measure of the imperfection of the fit of the estimator to the data.


Definition

The RMS value of a set of values (or a continuous-time
waveform In electronics, acoustics, and related fields, the waveform of a signal is the shape of its graph as a function of time, independent of its time and magnitude scales and of any displacement in time.David Crecraft, David Gorham, ''Electro ...
) 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 physics, the RMS current value can also be defined as the "value of the direct current that dissipates the same power in a resistor." In the case of a set of ''n'' values \, the RMS is : x_\text = \sqrt. The corresponding formula for a continuous function (or waveform) ''f''(''t'') defined over the interval T_1 \le t \le T_2 is : f_\text = \sqrt , and the RMS for a function over all time is : f_\text = \lim_ \sqrt . The RMS over all time of a
periodic function A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to des ...
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 sample consisting of equally spaced observations. Additionally, the RMS value of various waveforms can also be determined without
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, as shown by Cartwright. In the case of the RMS statistic of a random process, the
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
is used instead of the mean.


In common waveforms

If the
waveform In electronics, acoustics, and related fields, the waveform of a signal is the shape of its graph as a function of time, independent of its time and magnitude scales and of any displacement in time.David Crecraft, David Gorham, ''Electro ...
is a pure
sine wave A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ...
, the relationships between amplitudes (peak-to-peak, peak) and RMS are fixed and known, as they are for any continuous periodic wave. However, this is not true for an arbitrary waveform, which may not be periodic or continuous. For a zero-mean sine wave, the relationship between RMS and peak-to-peak
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
is: :''Peak-to-peak'' = 2 \sqrt \times \text \approx 2.8 \times \text. For other waveforms, the relationships are not the same as they are for sine waves. For example, for either a triangular or sawtooth wave :''Peak-to-peak'' = 2 \sqrt \times \text \approx 3.5 \times \text.


In waveform combinations

Waveforms made by summing known simple waveforms have an RMS value that is the root of the sum of squares of the component RMS values, if the component waveforms are
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
(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). :\text_\text =\sqrt Alternatively, for waveforms that are perfectly positively correlated, or "in phase" with each other, their RMS values sum directly.


Uses


In electrical engineering


Voltage

A special case of RMS of waveform combinations is: :\text_\text = \sqrt where \text_\text refers to the
direct current Direct current (DC) is one-directional flow of electric charge. An electrochemical cell is a prime example of DC power. Direct current may flow through a conductor such as a wire, but can also flow through semiconductors, insulators, or ev ...
(or average) component of the signal, and \text_\text is the
alternating current Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in whic ...
component of the signal.


Average electrical power

Electrical engineers often need to know the power, ''P'', dissipated by an
electrical resistance The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallel ...
, ''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: :P = I^2 R. 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. 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: :\begin P_ &= \left( I(t)^2R \right)_ &&\text \left( \cdots \right)_ \text \\ pt&= \left( I(t)^2 \right)_ R &&\text R \text \\ pt&= I_\text^2R &&\text \end So, the RMS value, ''I''RMS, of the function ''I''(''t'') is the constant current that yields the same power dissipation as the time-averaged power dissipation of the current ''I''(''t''). Average power can also be found using the same method that in the case of a time-varying
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
, ''V''(''t''), with RMS value ''V''RMS, :P_\text = . This equation can be used for any periodic
waveform In electronics, acoustics, and related fields, the waveform of a signal is the shape of its graph as a function of time, independent of its time and magnitude scales and of any displacement in time.David Crecraft, David Gorham, ''Electro ...
, such as a
sinusoidal A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often i ...
or
sawtooth wave The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. It is so named based on its resemblance to the teeth of a plain-toothed saw with a zero rake angle. A single sawtooth, or an intermittently triggered sawtooth, is called ...
form, 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: :P_\text = V_\text I_\text. Both derivations depend on voltage and current being proportional (that is, the load, ''R'', is purely resistive). Reactive loads (that is, 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 Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in whic ...
when ''I''(''t'') is a
sinusoidal A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often i ...
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: :I_\text = \sqrt, where ''t'' is time and ''ω'' is the
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
(''ω'' = 2/''T'', where ''T'' is the period of the wave). Since ''I''p is a positive constant: :I_\text = I_\text \sqrt. Using a trigonometric identity to eliminate squaring of trig function: :\begin I_\text &= I_\text \sqrt \\ pt &= I_\text \sqrt \end but since the interval is a whole number of complete cycles (per definition of RMS), the sine terms will cancel out, leaving: :I_\text = I_\text \sqrt = I_\text \sqrt = . A similar analysis leads to the analogous equation for sinusoidal voltage: :V_\text = , where ''I''P represents the peak current and ''V''P represents the peak voltage. Because of their usefulness in carrying out power calculations, listed
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
s for power outlets (for example, 120V in the US, or 230V 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 ''V'' = ''V''RMS × , assuming the source is a pure sine wave. Thus the peak value of the mains voltage in the USA is about 120 × , or about 170 volts. The peak-to-peak voltage, being double this, is about 340 volts. A similar calculation indicates that the peak mains voltage in Europe is about 325 volts, and the peak-to-peak mains voltage, 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 24-hour day represents an average current of 5 amps, but an RMS current of 7.07 amps, in the long term. The term ''RMS power'' is sometimes erroneously 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.


Speed

In the
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
of gas molecules, the root-mean-square speed is defined as the square root of the average squared-speed. The RMS speed of an ideal gas is calculated using the following equation: :v_\text = \sqrt where ''R'' represents the gas constant, 8.314 J/(mol·K), ''T'' is the temperature of the gas in
kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and ...
s, and ''M'' is the molar mass of the gas in kilograms per mole. In physics, speed is defined as the scalar magnitude of velocity. For a stationary gas, the average speed of its molecules can be in the order of thousands of km/hr, even though the average velocity of its molecules is zero.


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 absolute values of the pairwise differences could be a useful measure of the variability of the differences. However, the RMS of the differences is usually the preferred measure, probably due to mathematical convention and compatibility with other formulae.


In frequency domain

The RMS can be computed in the frequency domain, using Parseval's theorem. For a sampled signal x = x(t=nT), where T is the sampling period, :\sum_^N = \frac\sum_^N \left, X \^2, where X = \operatorname\ and ''N'' is the sample size, that is, the number of observations in the sample and FFT coefficients. In this case, the RMS computed in the time domain is the same as in the frequency domain: : \text\ = \sqrt = \sqrt = \sqrt.


Relationship to other statistics

If \bar is the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The co ...
and \sigma_x is the standard deviation of a
population Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction usi ...
or a
waveform In electronics, acoustics, and related fields, the waveform of a signal is the shape of its graph as a function of time, independent of its time and magnitude scales and of any displacement in time.David Crecraft, David Gorham, ''Electro ...
, then: :x_\text^2 = \overline^2 + \sigma_x^2 = \overline. 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 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. This is useful for electrical engineers in calculating the "AC only" RMS of a signal. Standard deviation being the RMS of a signal's variation about the mean, rather than about 0, the
DC component DC, D.C., D/C, Dc, or dc may refer to: Places * Washington, D.C. (District of Columbia), the capital and the federal territory of the United States * Bogotá, Distrito Capital, the capital city of Colombia * Dubai City, as distinct from t ...
is removed (that is, RMS(signal) = stdev(signal) if the mean signal is 0).


See also

*
Average rectified value In electrical engineering, the average rectified value (ARV) of a quantity is the average of its absolute value. The average of a symmetric alternating value is zero and it is therefore not useful to characterize it. Thus the easiest way to determ ...
(ARV) * Central moment * Geometric mean * L2 norm *
Least squares The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the re ...
* List of mathematical symbols *
Mean squared displacement In statistical mechanics, the mean squared displacement (MSD, also mean square displacement, average squared displacement, or mean square fluctuation) is a measure of the deviation of the position of a particle with respect to a reference positio ...
*
True RMS converter For the measurement of an alternating current the signal is often converted into a direct current of equivalent value, the root mean square (RMS). Simple instrumentation and signal converters carry out this conversion by filtering the signal in ...


References

{{Reflist


External links


A case for why RMS is a misnomer when applied to audio power
Means Statistical deviation and dispersion it:Valore efficace