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Root mean square

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Square root of the mean square

Inmathematics, theroot mean square (abbrev.RMS,RMS orrms) of aset of values is thesquare root of the set'smean square.^[1]Given a set $x_{i}$ , its RMS is denoted as either $x_{\mathrm {RMS} }$ or $\mathrm {RMS} _{x}$ . The RMS is also known as thequadratic mean (denoted $M_{2}$ ),^[2]^[3] a special case of thegeneralized mean. The RMS of a continuousfunction is denoted $f_{\mathrm {RMS} }$ and can be defined in terms of anintegral of the square of the function.Inestimation theory, theroot-mean-square deviation of an estimator measures how far the estimator strays from the data.

Definition

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The RMS value of a set of values (or acontinuous-time 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 ofn values $\{x_{1},x_{2},\dots ,x_{n}\}$ , the RMS is

x_{\text{RMS}}={\sqrt {{\frac {1}{n}}\left({x_{1}}^{2}+{x_{2}}^{2}+\cdots +{x_{n}}^{2}\right)}}.

The corresponding formula for a continuous function (or waveform)f(t) defined over the interval $T_{1}\leq t\leq T_{2}$ is

f_{\text{RMS}}={\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{[f(t)]}^{2}\,{\rm {d}}t}}},

and the RMS for a function over all time is

f_{\text{RMS}}=\lim _{T\rightarrow \infty }{\sqrt {{1 \over {2T}}{\int _{-T}^{T}{[f(t)]}^{2}\,{\rm {d}}t}}}.

The RMS over all time of aperiodic 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 sample consisting of equally spaced observations. Additionally, the RMS value of various waveforms can also be determined withoutcalculus, as shown by Cartwright.^[4]

In the case of the RMS statistic of arandom process, theexpected value is used instead of the mean.

In common waveforms

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Sine,square,triangle, andsawtooth waveforms. In each, the centerline is at 0, the positive peak is at $y=A_{1}$ and the negative peak is at $y=-A_{1}$

{\displaystyle y=A_{1}} — Sine,square,triangle, andsawtooth waveforms. In each, the centerline is at 0, the positive peak is at $y=A_{1}$ and the negative peak is at $y=-A_{1}$

A rectangular pulse wave of duty cycle D, the ratio between the pulse duration ( $\tau$ ) and the period (T); illustrated here witha = 1.

{\displaystyle \tau } — A rectangular pulse wave of duty cycle D, the ratio between the pulse duration ( $\tau$ ) and the period (T); illustrated here witha = 1.

If thewaveform is a puresine wave, the relationships between amplitudes (peak-to-peak, peak) and RMS are fixed and known, as they are for any continuousperiodic 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-peakamplitude is:

Peak-to-peak

=2{\sqrt {2}}\times {\text{RMS}}\approx 2.8\times {\text{RMS}}.

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 {3}}\times {\text{RMS}}\approx 3.5\times {\text{RMS}}.

Waveform	Variables and operators	RMS
DC	$y=A_{0}\,$	$A_{0}\,$
Sine wave	$y=A_{1}\sin(2\pi ft)\,$	${\frac {A_{1}}{\sqrt {2}}}$
Square wave	$y={\begin{cases}A_{1}&\operatorname {frac} (ft)<0.5\\-A_{1}&\operatorname {frac} (ft)>0.5\end{cases}}$	$A_{1}\,$
DC-shifted square wave	$y=A_{0}+{\begin{cases}A_{1}&\operatorname {frac} (ft)<0.5\\-A_{1}&\operatorname {frac} (ft)>0.5\end{cases}}$	${\sqrt {A_{0}^{2}+A_{1}^{2}}}\,$
Modified sine wave	$y={\begin{cases}0&\operatorname {frac} (ft)<0.25\\A_{1}&0.25<\operatorname {frac} (ft)<0.5\\0&0.5<\operatorname {frac} (ft)<0.75\\-A_{1}&\operatorname {frac} (ft)>0.75\end{cases}}$	${\frac {A_{1}}{\sqrt {2}}}$
Triangle wave	$y=\left\|2A_{1}\operatorname {frac} (ft)-A_{1}\right\|$	$A_{1} \over {\sqrt {3}}$
Sawtooth wave	$y=2A_{1}\operatorname {frac} (ft)-A_{1}\,$	$A_{1} \over {\sqrt {3}}$
Pulse wave	$y={\begin{cases}A_{1}&\operatorname {frac} (ft)<D\\0&\operatorname {frac} (ft)>D\end{cases}}$	$A_{1}{\sqrt {D}}$
Phase-to-phase sine wave	$y=A_{1}\sin(t)-A_{1}\sin \left(t-{\frac {2\pi }{3}}\right)\,$	$A_{1}{\sqrt {\frac {3}{2}}}$
where: y is displacement, t is time, f is frequency, A_i is amplitude (peak value), D is theduty cycle or the proportion of the time period (1/f) spent high, frac(r) is thefractional part ofr.

In waveform combinations

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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 areorthogonal (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).^[5]

{\text{RMS}}_{\text{Total}}={\sqrt {{\text{RMS}}_{1}^{2}+{\text{RMS}}_{2}^{2}+\cdots +{\text{RMS}}_{n}^{2}}}

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

Uses

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In electrical engineering

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Current

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The RMS of analternating electric current equals the value of constantdirect current that would dissipate the same power in aresistive load.^[1]

Voltage

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Further information:Root mean square AC voltage

A special case of RMS of waveform combinations is:^[6]

{\text{RMS}}_{\text{AC+DC}}={\sqrt {{\text{V}}_{\text{DC}}^{2}+{\text{RMS}}_{\text{AC}}^{2}}}

where ${\text{V}}_{\text{DC}}$ refers to thedirect current (or average) component of the signal, and ${\text{RMS}}_{\text{AC}}$ is thealternating current component of the signal.

Average electrical power

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Further information:AC power

Electrical engineers often need to know thepower,P, dissipated by anelectrical resistance,R. It is easy to do the calculation when there is a constantcurrent,I, through the resistance. For a load ofR ohms, power is given by:

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 theaverage power dissipated over time, which is calculated by taking the average power dissipation:

{\begin{aligned}P_{\text{Avg}}&=\left(I(t)^{2}R\right)_{\text{Avg}}&&{\text{where }}(\cdots )_{\text{Avg}}{\text{ denotes the temporal mean of a function}}\\[3pt]&=\left(I(t)^{2}\right)_{\text{Avg}}R&&{\text{(as }}R{\text{ does not vary over time, it can be factored out)}}\\[3pt]&=I_{\text{RMS}}^{2}R&&{\text{by definition of root-mean-square}}\end{aligned}}

So, the RMS value,I_RMS, of the functionI(t) is the constant current that yields the same power dissipation as the time-averaged power dissipation of the currentI(t).

Average power can also be found using the same method that in the case of a time-varyingvoltage,V(t), with RMS valueV_RMS,

P_{\text{Avg}}={V_{\text{RMS}}^{2} \over R}.

This equation can be used for any periodicwaveform, such as asinusoidal orsawtooth 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:

P_{\text{Avg}}=V_{\text{RMS}}I_{\text{RMS}}.

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 ofAC power.

In the common case ofalternating current whenI(t) is asinusoidal current, as is approximately true for mains power, the RMS value is easy to calculate from the continuous case equation above. IfI_p is defined to be the peak current, then:

I_{\text{RMS}}={\sqrt {{1 \over {T_{2}-T_{1}}}\int _{T_{1}}^{T_{2}}\left[I_{\text{p}}\sin(\omega t)\right]^{2}dt}},

wheret is time andω is theangular frequency (ω = 2π/T, whereT is the period of the wave).

SinceI_p is a positive constant and was to be squared within the integral:

I_{\text{RMS}}=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{\sin ^{2}(\omega t)}\,dt}}}.

Using atrigonometric identity to eliminate squaring of trig function:

{\begin{aligned}I_{\text{RMS}}&=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{1-\cos(2\omega t) \over 2}\,dt}}}\\[3pt]&=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}\left[{t \over 2}-{\sin(2\omega t) \over 4\omega }\right]_{T_{1}}^{T_{2}}}}\end{aligned}}

but since the interval is a whole number of complete cycles (per definition of RMS), the sine terms will cancel out, leaving:

I_{\text{RMS}}=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}\left[{t \over 2}\right]_{T_{1}}^{T_{2}}}}=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}{{T_{2}-T_{1}} \over 2}}}={I_{\text{p}} \over {\sqrt {2}}}.

A similar analysis leads to the analogous equation for sinusoidal voltage:

V_{\text{RMS}}={V_{\text{p}} \over {\sqrt {2}}},

whereI_P represents the peak current andV_P represents the peak voltage.

Because of their usefulness in carrying out power calculations, listedvoltages 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 impliesV_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 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 termRMS power is sometimes erroneously used (e.g., in the audio industry) as a synonym formean power oraverage 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, seeAudio power.

Speed

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Main article:Root-mean-square speed

In thephysics ofgas molecules, theroot-mean-square speed is defined as the square root of the average squared-speed. The RMS speed of an ideal gas iscalculated using the following equation:

v_{\text{RMS}}={\sqrt {3RT \over M}}

whereR represents thegas constant, 8.314 J/(mol·K),T is the temperature of the gas inkelvins, andM is themolar 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/h, even though the average velocity of its molecules is zero.

Error

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Main article:Root-mean-square deviation

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 of 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

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The RMS can be computed in the frequency domain, usingParseval's theorem. For a sampled signal $x[n]=x(t=nT)$ , where $T {\displaystyle T}$ is the sampling period,

\sum _{n=1}^{N}{x^{2}[n]}={\frac {1}{N}}\sum _{m=1}^{N}\left|X[m]\right|^{2},

where $X[m]=\operatorname {DFT} \{x[n]\}$ andN is the sample size, that is, the number of observations in the sample and DFT coefficients.

In this case, the RMS computed in the time domain is the same as in the frequency domain:

{\text{RMS}}\{x[n]\}={\sqrt {{\frac {1}{N}}\sum _{n}{x^{2}[n]}}}={\sqrt {{\frac {1}{N^{2}}}\sum _{m}{{\bigl |}X[m]{\bigr |}}^{2}}}={\sqrt {\sum _{m}{\left|{\frac {X[m]}{N}}\right|^{2}}}}.

Relationship to other statistics

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Notes

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^If NM =a and PM =b. AM =AM ofa andb, and radiusr = AQ = AG.
UsingPythagoras' theorem, QM² = AQ² + AM² ∴ QM = √AQ² + AM² =QM.
Using Pythagoras' theorem, AM² = AG² + GM² ∴ GM = √AM² − AG² =GM.
Usingsimilar triangles,⁠HM/GM⁠ =⁠GM/AM⁠ ∴ HM =⁠GM²/AM⁠ =HM.

References

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^^a ^b"Root-mean-square value".A Dictionary of Physics (6 ed.). Oxford University Press. 2009.ISBN 9780199233991.
^Thompson, Sylvanus P. (1965).Calculus Made Easy. Macmillan International Higher Education. p. 185.ISBN 9781349004874. Retrieved5 July 2020.^{[permanent dead link‍]}
^Jones, Alan R. (2018).Probability, Statistics and Other Frightening Stuff. Routledge. p. 48.ISBN 9781351661386. Retrieved5 July 2020.
^Cartwright, Kenneth V (Fall 2007)."Determining the Effective or RMS Voltage of Various Waveforms without Calculus"(PDF).Technology Interface.8 (1): 20 pages.
^Nastase, Adrian S."How to Derive the RMS Value of Pulse and Square Waveforms".MasteringElectronicsDesign.com. Retrieved21 January 2015.
^"Make Better AC RMS Measurements with your Digital Multimeter"(PDF).Keysight. Archived fromthe original(PDF) on 15 January 2019. Retrieved15 January 2019.
^Chris C. Bissell; David A. Chapman (1992).Digital signal transmission (2nd ed.). Cambridge University Press. p. 64.ISBN 978-0-521-42557-5.
^Weisstein, Eric W."Root-Mean-Square".MathWorld.
^"ROOT, TH1:GetRMS". Archived fromthe original on 2017-06-30. Retrieved2013-07-18.

External links

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v t e Machine learning evaluation metrics
Regression	MSE MAE sMAPE MAPE MASE MSPE RMS RMSE/RMSD R² MDA MAD
Classification	F-score P4 Accuracy Precision Recall Kappa MCC AUC ROC Sensitivity and specificity Logarithmic Loss
Clustering	Silhouette Calinski-Harabasz index Davies-Bouldin Dunn index Hopkins statistic Jaccard index Rand index Similarity measure SMC SimHash
Ranking	MRR NDCG AP
Computer Vision	PSNR SSIM IoU
NLP	Perplexity BLEU
Deep Learning Related Metrics	Inception score FID
Recommender system	Coverage Intra-list Similarity
Similarity	Cosine similarity Euclidean distance Pearson correlation coefficient
Confusion matrix