# Talk:Root mean square

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## German link

Should go here: http://de.wikipedia.org/wiki/Quadratisches_Mittel — Preceding unsigned comment added by 84.130.163.161 (talk) 06:20, 18 October 2011 (UTC)

## 1/N in the equation

Shouldn't rms be evaluated using the root of the summation of P(r)X where x is your value and P(r) is the probability of x?

Otherwise, aren't we assuming that the probability of each x is equal? I just learned this in my physics class today, so I am unsure about this. —Preceding unsigned comment added by 128.189.147.177 (talk) 10:34, 22 November 2007 (UTC)

It's fine, since all the $x_i$ are different. RMS does not necessarily involve probabilities, and if it did, the definition would still work, just think of it as $x_i = P_i(r) X_i$. Misho88 (talk) 02:10, 24 April 2008 (UTC)

## check parens

there is an equation with the integration of (Ip Sin(t) dt)^2. should dt be in the parentheses?

Where? There is no such thing in the page. :S

## sleeping

isn't it so that the rms is calculated as: $\frac{1}{N}\sqrt{\sum_{i=1}^{N}x_i^2}$ ?

I am sorry... I was just sleeping...

what

NO the formulae in the article is correct
root mean square means the (square) root of the mean of the square
so the mean function (which is expressed as the total over the count for discrete data) is completely inside the square root function. Plugwash 05:35, 12 Dec 2004 (UTC)
formulæ plural, formula singular

## lowercase

Why do I see rms in lowercase so much? It's an abbreviation; it should be RMS. Right? You say it as letters, not "erms", so you can't say it's the same thing as laser, turning into a word over time. - Omegatron 02:57, Jan 19, 2005 (UTC)

http://www.rane.com/par-r.html#rms
WHY!?!?! - Omegatron 21:27, Feb 26, 2005 (UTC)
According to Dennis Bohn of Rane:
"rms" is written in lower-case because that follows the general accepted practice as outlined in the Abbreviations Dictionary, 9th ed. (Ralph De Sola, Dean Stahl and Karen Kerchelich; CRC Press ISBN0-8493-8944-5, 1995) and other sources. Quoting the CRC book: "American as well as British and Canadian publishers appear to be following the trend to capitalize only those letters normally capitalized: proper nouns and important words in titles. They reserve lowercase letters for abbreviations consisting of adjectives and common nouns." This is the way it appears in the IEEE reference above (IEEE 100: The Authoritative Dictionary of IEEE Standards Terms, 7th ed. (IEEE ISBN 0-7381-2601-2, New York, 2000).). Other popular examples are "rpm," and "mph." - Omegatron 22:39, Feb 28, 2005 (UTC)
Consequently, shouldn't it be "ieee" then?
No. IEEE stands for a proper name which is always capitalized. — Omegatron 22:10, 6 November 2006 (UTC)
I dont care what IEEE says! most elec. engineers use the lower case as far as I have seen and I am one of them!!!! its very very common as far as I know. —Preceding unsigned comment added by 67.204.3.133 (talk) 19:20, 16 June 2008 (UTC)
That Abbreviations Dictionary seems to be out on a limb. Are they saying that we should write "My pc needs a new cpu and psu, some more sdram and a dvd-rw drive?"? I hope not. The examples of "rpm" and "mph" are different: they refer to units, which have a grammar all of their own. "RMS/rms/r.m.s." is somewhere between these extremes, which is why you see it in both uppercase and lowercase forms. I've just looked at several reference works and there is no clear favourite spelling, so I suspect it's just a house style thing. So what should our house style be? I think it depends on the context. As a suffix, I would write it in lowercase, as in "$E_{rms}=23\ \mathrm{V}$", but in prose I would write it in uppercase, as in "the RMS value of the waveform was 23 volts". This is how it is done in (most of) this article. --Heron 18:16, 23 June 2007 (UTC)

## Root Mean Square or Rated Minimum Sine?

I need clarification. I understood RMS to mean Rated Minimum Sine.

In the technical specs that accompanied my JVC stereo, the RMS definition appeared to be Rated Minimum Sine wave. I found one recent reference to that at the following link: http://www.advertisinglaw.com/lawsandregulations/title16-432.html.

In section 432.2 (a) it is stated:

"(a) Whenever any direct or indirect representation is made of the power output, power band or power frequency response, or distortion characteristics of sound power amplification equipment, the following disclosure shall be made clearly, conspicuously, and more prominently than any other representations or disclosures permitted under this part: The manufacturer's rated minimum sine wave continuous average power output, in watts, per channel (if the equipment is designed to amplify two or more channels simultaneously) at an impedance of 8 ohms, or, if the amplifier is not designed for an 8-ohm impedance, at the impedance for which the amplifier is primarily designed, measured with all associated channels fully driven to rated per channel power. "

I would also refer to Part B, paragraph 1, page 6, line 6 of the following document found at the following link http://www.ftc.gov/os/2000/12/amplifierrulefrn.pdf.

I don't understand. Rated power is the maximum that it can put out. That doesn't make sense to me. — Omegatron 15:41, 11 March 2006 (UTC)
RMS means root mean square. Neither of the above references says that it means rated minimum sine. The first one happens to use the terms RMS and rated minimum sine on the same web page, and the second one doesn't mention RMS anywhere. Rated minimum sine is not a meaningful phrase, it's just three adjacent words chopped out of the middle of a sentence. --Heron 16:32, 11 March 2006 (UTC)

## Peak RMS

Article states peak RMS is garbage. If the RMS is calculated for a given period, say 1s, then you can average all of the oscillations in voltage (Hz) or speaker vibration (kHz) quite well. And for each second in time you could calculate the rms (of whatever) for that 1s period. Supposing the operation goes through stages on a timescale of tens of minutes. Then it IS sensible to talk about peak RMS.

I disagree. The definition of RMS value is quite clear: it is the average of the absolute value of a signal over one period. This implies a periodic signal, hence the RMS value is independent from the time. On a non-periodic signal, one can calculate either the instantaneous or an averaged power, but this latter value needs a clear definition (what kind of averaging, over what duration?). As far as I know, the only practical interrest of the peak RMS is the overrating of audio amplifier for marketing purposes. -- CyrilB 15:25, 20 May 2006 (UTC)
Peak rms when talking about an amplifier is not garbage, because an amplifier has a variable power output (otherwise all the sounds would be the same volume). It says "Peak rms power" not referring to the a.c. component of the power, but to the maximum variable amplification it provides. Perhaps "Maximum rms power" is less contradictory. pfff... Semantics--Eh-Steve 18:58, 25 May 2006 (UTC)
RMS is a properly defined term in electrical engineering and audio engineering. 'Peak RMS' is a quantity that has not been defined. 'Maximum power' is the term you are looking for. 'RMS power' is a meaningless term invented in the audio industry. That is to say, it has not been defined, neither in this article nor anywhere else, therefore it can be interpreted differently in different situations, perhaps to mislead purchasers.

There cannot be any way that an amplifier or speaker can even have "peak rms power" (there is no way there can even be peak rms power). rms means root mean square, THE KEYWORD IS MEAN. There is peak power, max power, peak watts, and max watts but no peak rms power. rms power is simply the nominal or average power that an amplifier or speaker can handle without distorting or damaging, like overheating the amp or speaker. With car and home audio components, rms simply means "nominal power", which means it will handle the average power that it is designed to handle. Peak power means that it can handle power up to what it is rated but will distort and is not intended to be operated at those levels continuously. Just that it can handle short burst of it's max power without being damaged. And anything over the max power rating will damage them. —Preceding unsigned comment added by 205.188.116.136 (talkcontribs)

RMS power
A misnomer for average power
Peak RMS power
Given the above definition, peak RMS power is the highest average power an amplifier can put out. "Average power" by itself has nothing to do with the maximum. Average power can be measured at 1 W output for distortion measurements on a 100 W amp, for instance. The correct term would be "maximum average power" or "average power rating" or something, though.

Neither is a great term, but they both have meaning. — Omegatron 12:13, 12 October 2006 (UTC)

The problem here is the meaning of "average": The RMS value is an average over one period of the signal. This implies a periodic signal, hence a signal with a constant average. The average power is different, as it is the averaging of any signal (for example an audio signal, lets say a complete song) over a (to be defined) length of time (for example the length of the song). CyrilB 17:07, 4 November 2006 (UTC)
The RMS value does not imply a periodic waveform. You can take the RMS value of just part of a cycle, or the RMS value of the voltage of an entire song. — Omegatron 05:28, 5 November 2006 (UTC)
You're right. The definition of RMS has nothing to do with a periodical signal. But then, if we don't mention the duration over which the RMS value is calculated, then this "peak" value is meaningless. CyrilB 10:13, 5 November 2006 (UTC)

When considering the advertised output of an audio amplifier, some manufacturers, especially those in the low-end computer industry, will state a "peak maximum power output" (PMPO) that only includes a narrow part of the sine wave output near the peaks of the waveform, but the definition of rms requires that whole numbers of the entire sine wave cycle is included in the calculation such that the power output is averaged over a longer period of time. It is analogous to a car manufacturer rating the horsepower of an engine to be only measured at the instant that each cylinder is firing and not including the time between individual cyles. This would greatly inflate the resulting horsepower rating by several magnitudes. The word "peak" in the term "peak rms power" output does not refer to the peaks of the waveforms but rather to the maximum output such as when you have the volume control in the maximum position and the inputs are driven at their maximum rated level. What "peak rms power" means to me is that the power has been averaged over time using the rms of the voltages and currents during the same time periods at which time the amplifier is driven at its maximum capabilities. Stating that it is meaningless is clearly an opinion statement, as evidenced by this extended discussion; it clearly has some meaning to me and several other people. So who is going to correct the article?

I still disagree with the "peak rms power". In my opinion, we have two different things here: the RMS power of a signal, which is a value (measured or calculated) and the RMS power of an amplifier, which is a characteristic, namely the maximum RMS power the amplifier can generate. It is obvious that a 50 W amplifier will be able to operate at lower power levels, depending on the signal to amplify and the volume level. Therefore, what is called peak RMS power should simply be called RMS power (or power I can't think of any other sensible power value independant of time). CyrilB 16:57, 4 November 2006 (UTC)

an rms power rating refers to the rated normal operating characteristics of the amp, that is, the rms output that can be sustained indefinitely without overheating and/or damaging the amp. while a peak rms number by itself would be meaningless, if the rated included a duty cycle for how long and in what proportion with normal output the peak power could be maintained then there would be usefulness. That is, if the amp is rated at 50W steady 100W peak rms this doesn't mean much, but if it was rated for 50W 100W peak for peak<1s duty cycle 10:1, there is something meaningful there

## Examples

I think it could do with an example or examples to help newcomers to RMS values, maybe. They are very useful on most of the calculus pages. Jabba27 11:28, 25 May 2006 (UTC)

Absolutely. If you have an AC power supply, for example the 230-volt domestic power supply in Europe, the voltage follows a sine-wave, peaking at plus and minus 325 volts. Because this voltage has a mean (average) value of zero over one cycle, the simple average is meaningless. But the heating effect of the voltage when applied to a resistor (e.g. an electric convector) is V-squared over R, so it makes sense to average the square of the voltage over one cycle, and then take the square root. When this is done, you have a value of the equivalent DC voltage that, when applied to the same resistor, produces the same heating effect. Because of a mathematical property of a sine-wave, it is the peak voltage divided by the square-root of 2 (that is where calculus came in). This is the simplest example, applied to an AC supply with a steady voltage. Power supplies are always described in terms of RMS voltage and current. —Preceding unsigned comment added by 62.30.56.28 (talk) 12:37, 8 January 2011 (UTC)

This is an example from electrical engineering, but RMS has a wider application in mathematics. —Preceding unsigned comment added by 62.30.56.28 (talk) 12:39, 8 January 2011 (UTC)

## Small Error in the math

Corrected the Integral, the part that goes to zero has a wrong factor of -1/2*omega.

## Amplifier Power Efficiency

What is Iout in the equations? if it is the output current of the amplifier, it is an alternating quantity with a mean value of zero: therefore the second equation should be simplified, or a reference included so that the meaning can be discovered. —Preceding unsigned comment added by 62.30.56.28 (talk) 09:28, 9 January 2011 (UTC)

## Relation between SD, RMS and M

Isn't the relation between Standard Deviation (SD), Root Mean Square (RMS) and Mean (M) as

• SD^2 = N/(N-1)x(RMS^2-M^2) ?

we seem to need the factor N/(N-1).

The formulae in the article is right if you consider the population standard diviation. Calculation for samples is slightly different as the variance and standard deviation articles elaborate. Plugwash 14:05, 21 November 2006 (UTC)

Could any one put a simple proof of that (it was not clear to me at first):

$x_{\mathrm{rms}}^2 = \bar{x}^2 + \sigma_{x}^2.$

## Error in formulae???

1/N should not be squared under the root, should it? The same applies to all mean currents which otherwise would carry units of amps per sqrt(time). See also [1] —Preceding unsigned comment added by SimAnn (talkcontribs)

good catch, the bogus squares were recently put in by an anon and i have now reverted them. Plugwash 19:30, 20 November 2006 (UTC)

## Too techical?

The article seems somewhat difficult to understand at first. Even though the concept is fairly straightforward, can the application be clarified? 69.140.164.142 01:32, 7 April 2007 (UTC)

No-one else seems to agree, so I have removed the {{technical}} tag. Gandalf61 (talk) 14:07, 9 February 2009 (UTC)

## Notation

Article uses two different notations for arithmetic mean (confusing). — DIV (128.250.204.118 06:00, 8 May 2007 (UTC))

## Table of common waveforms and RMS values

There should be a table of common waveforms and RMS values based on the waveform parameters (eg Vpk), it should have atleast sine, square, rectangle, triangle, sawtooth, ramp....<add approapriate signals> Agree?ZedZzizz (talk) 05:14, 6 May 2008 (UTC)

Sounds good to me. Plugwash (talk) 14:22, 31 October 2008 (UTC)

### Rectangular wave

Shouldn't rectangular wave be $D a$ ? And further, modified square wave should be $\frac{a}{2}$ because it is half of the time either $a$ or $-a$ ? — Preceding unsigned comment added by Sinihappo (talkcontribs) 21:13, 10 October 2011 (UTC)

### rms of sawtooth and triangle wave

Afaict the sawtooth and triangle wave intuitively should have the same rms since they spend the same ammount of time at each value so I should only have to calculate for one of them

I've worked through the calculation assuming a sawtooth wave with a period 1 rising through zero. This probablly doesn't belong in the article but i've put it here so others can check I got the value right. assuming we are only interested in the interval -.5 to .5 the equation of this wave can be written as $y=2t$ so $y_{\mathrm{rms}} = \sqrt {{1 \over {0.5--0.5}} {\int_{-0.5}^{0.5} {[2x]}^2\, dt}}$
$y_{\mathrm{rms}} = \sqrt { {\int_{-0.5}^{0.5} {4x^2 dt}}}$
$y_{\mathrm{rms}} = \sqrt { \left [ \frac{4}{3}x^3 \right ]_{-0.5}^{0.5} }$
$y_{\mathrm{rms}} = \sqrt { \left [ \frac{4}{3}(0.5)^3 \right ]-\left [ \frac{4}{3}(-0.5)^3 \right ] }$
$y_{\mathrm{rms}} = {1 \over \sqrt 3}$

## rms over all time

I'm not convinced that the formula for rms over all time is correct as is. It seems to me that you could only really call it *the* rms if you could take the upper and lower limits time independently (take the integral from T1 to T2 as T1 goes to -infinity and T2 goes to infinity) and always arrive at the same limit value. Taking the limit with the upper and lower bounds of integration related in this way seems like it might lead to a different value then one might get going from -T to 2T say and taking T to infinity, depending on the function being integrated.

I suspect the function in the integral has to belong to some specific class of functions for the formula to be true.

Or am I missing something?

Ekwos (talk) 17:52, 18 June 2009 (UTC)

## Relationship to the arithmetic mean and the standard deviation

${x_{\mathrm{rms}}}^2 = \bar{x}^2 + {\sigma_{x}}^2$  ?

The above formula can be easily prooved by developping the (squared) standard deviation:

$\sigma_x^2 = \frac{1}{n} \sum ( x_i - \bar{x})^2$

$= \frac{1}{n} \sum x_i^2 - 2 \bar{x} \frac{1}{n}\sum x_i + \bar{x}^2$
$= \ \ x_{\mathrm{rms}}^2 \ - \ \, \bar{x}^2 \quad \quad \quad \quad \quad as \quad \bar{x} = \frac{1}{n} \sum x_i$

Is this proof worth being added to the main page? (I think so) If yes, please do it, or tell me how you think it should be done. —Preceding unsigned comment added by JuhSensei (talkcontribs) 14:22, 16 October 2009 (UTC)

This is well known, but the proof is not so relevant to rms, is it? I would instead find a source that can be cited, and remove most of that section. So I did that. Part of what it says still begs for a source. Dicklyon (talk) 17:43, 9 January 2011 (UTC)

## Formula changes

Can someone please check and verify the change I just made, and the change of another editor that I reverted? Dicklyon (talk) 15:39, 15 October 2011 (UTC)

### RMS for modified square wave and rectangular wave

I add now a new section.

First, Dicklyon said he reverted my change. He didn't, he changed it to something else.

Second, I made a change in talk page, but didn't add a new section, sorry for that.

Third, I calculated the integrals. Did any of you?

So I would say that rectangular wave should be $D a$.

And modified square wave should be $\frac{a}{2}$.

There is no basis for any square roots in PWM case. Roots in sine waves come from sine^2 integrals. This can be seen quite clearly when simplifying and just calculating discrete sum with eg. 10 steps per cycle.

If you want to keep the page incorrect, fine with me. But as I already spent some time, I am not going to spend any more my time with arguing with you, thank you. — Preceding unsigned comment added by Sinihappo (talkcontribs) 19:58, 15 October 2011 (UTC)

P.S.

I checked how rectangular wave was added to the article. Well, it was added without any discussion. And it is clearly wrong. When I niw try do correct it, you try to revert it, but change it to someting else than originally, after I have made an addition to the discussion page.

Could you, Dicklyon, explain, why you feel that both versions are wrong? — Preceding unsigned comment added by Sinihappo (talkcontribs) 20:14, 15 October 2011 (UTC)

P.P.S.

Sorry for not adding signature. I am a little new with this. Sinihappo (talk) 20:33, 15 October 2011 (UTC)

Thanks very much for drawing the problem to our attention, and sorry, but stuff ups happen on Wikipedia as elsewhere. I saw your change (before Dicklyon's edits) and checked it (rather quickly, on quite a small piece of paper, so I'm not guaranteeing the results), and I believe your edit was precisely correct. It's rather unexpected that some equations that have been prominently in an article like this for some time are plainly wrong, so when we see changes like yours there is a tendency to be skeptical. I see that Dicklyon posted on your talk page after my welcome, and I don't have time now to think, so I won't change the article, but will invite Dicklyon to do the simple math which I thought confirmed the accuracy of your edit. Johnuniq (talk) 07:41, 16 October 2011 (UTC)
Eerk, scratch what I said. My quick calculation must have blundered because after some thought it is clear that the article is correct. Next step would be to find a source that contradicts the article. Johnuniq (talk) 09:53, 16 October 2011 (UTC)

Sinihappo, if you look at the article's edit history, you'll see my revert of your edit, followed by what I believe is a correct correction of one formula that was wrong (I left the other as before your edit). The "modified square wave" has half the power of the square wave, and the duty cycle D wave has 1/D the power of the square wave, and the half and D need to be rooted to get the effect on the rms, yes? I did it without pencil, paper, sliderule, or matlab, so asked for someone to verify. A good source for these examples would be even better. Sinihappo, thanks for trying; your version would have been correct for the mean absolute voltage, but rms is a bit different. Dicklyon (talk) 18:46, 16 October 2011 (UTC)

Sorry, I was wrong all the time. And sorry for my arrogance, too. Sinihappo (talk) 06:42, 4 November 2011 (UTC)

Just to clarify here is the working for the "modified sinewave" case
Let T1=0 T2=1 and f=1 (integrating over a single cycle)
$RMS = \sqrt{\frac{1}{1-0}\int\limits_{0}^{1}y^2\, dt}$
$RMS = \sqrt{\int\limits_{0}^{1}y^2\, dt}$
$RMS = \sqrt{\int\limits_{0}^{0.25}y^2\, dt+\int\limits_{0.25}^{0.5}y^2\, dt+\int\limits_{0.5}^{0.75}y^2\, dt+\int\limits_{0.75}^{1}y^2\, dt}$
$RMS = \sqrt{\int\limits_{0}^{0.25}0^2\, dt+\int\limits_{0.25}^{0.5}a^2\, dt+\int\limits_{0.5}^{0.75}0^2\, dt+\int\limits_{0.75}^{1}(-a)^2\, dt}$
$RMS = \sqrt{\int\limits_{0.25}^{0.5}a^2\, dt+\int\limits_{0.75}^{1}a^2\, dt}$
$RMS = \sqrt{[ta^2]_{0.25}^{0.5}+[ta^2]_{0.75}^1}$
$RMS = \sqrt{(0.5-0.25)a^2+(1-0.75)a^2}$
$RMS = \sqrt{(0.25)a^2+(0.25)a^2}$
$RMS = \sqrt{\frac{1}{2}a^2}$
$RMS = \frac{\sqrt{1}}{\sqrt{2}}\sqrt{a^2}$
$RMS = \frac{1}{\sqrt{2}}a$
-- Plugwash (talk) 14:29, 4 November 2011 (UTC)
Thanks for verifying. Dicklyon (talk) 15:43, 4 November 2011 (UTC)

## error in formula (section "RMS in frequency domain")?

there is something wrong with the last formula in this paragraph: if the n (for the freq. domain equations) is pushed below the square root, it has to become n squared [ or vice versa ...] Herbst (talk) 21:47, 18 October 2011 (UTC)

Good point. I fixed. Dicklyon (talk) 15:42, 4 November 2011 (UTC)
Except I fixed it wrong. Should be better now. Dicklyon (talk) 04:40, 11 November 2011 (UTC)

## Relationship to 2-norm

It is easy enough to see, but for a beginner this article obscures the fact that the rms is just the 2-norm divided by root n (for an n-dimensional vector space), and thus is also a norm on the vector space. Add it as another section? 18.63.6.219 (talk) 14:22, 6 June 2012 (UTC)

True, but a varying quantity is most typically not viewed as a point in an n-dimensional vector space (though a sample of it is often handled that way). Do you have a source that talks of that relationship? Dicklyon (talk) 01:25, 14 June 2012 (UTC)

## Why is RMS usually not discussed in textbooks on statistics?

Could anyone talk a little bit about why RMS is usually not discussed in textbooks on statistics? --Roland 23:35, 13 June 2012 (UTC)

Because the mean square is so much easier to use in the statistical context, and its square root is not so important. Dicklyon (talk) 01:22, 14 June 2012 (UTC)

## Mention electrical power distribution

I think electrical power distribution/transmission should be mentioned for 2 reasons.

First, because rms voltage is the default way to indicate the line voltage in AC power distribution (ie 120V in the US is an rms value).

Second, the rms voltage between any 2 phases of an ideal 3-phase distribution system (ideal meaning perfect sine waves and phase separation of exactly 1/3 period) is equal to sqrt(3) * Vrms of one phase (phase-to-ground). The most common example I've seen is the term "120/208 V", indicating 120 Vrms phase-to-ground and 208 Vrms phase-to-phase. This term shows up in the power requirements for commercial/medical appliances and motors. This example is for the US. I think it would be appropriate for this article to explain what terms like "120/208 V" mean, as the values are directly related to the root mean square.

Unfortunately I don't have sources for this info (which is why I didn't edit the article directly), however I am an electrical engineer in the power distribution field and it is common knowledge. I think this article could benefit from this real world example of rms in action. — Preceding unsigned comment added by 74.92.43.41 (talk) 19:51, 13 March 2014 (UTC)