# Explanation of Root Mean Squared **URL:** \ **Category:** Factual Questions\ **Created:** [June 9, 2003, 3:53pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654 "2003-06-09T15:53:37Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![McLeo](https://avatars.discourse-cdn.com/v4/letter/m/91b2a8/32.png) [@McLeo](https://boards.straightdope.com/u/McLeo)\ **Post date:** [June 9, 2003, 3:53pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/1 "2003-06-09T15:53:37Z")
I know that when we have 120 V coming out of the outlet, it’s 120 V rms, but I don’t exactly understand how they came up with that. IIRC the amplitude of the wave is greater than 120, and rms is the average voltage coming out which is the amplitude divided by root 2. But why is the average not in the dead center of the wave? And from first year chemistry I vaguely recall that the speed of the atoms is calculated at a rms speed. Can someone explain to me the origin, definition, and the applications of RMS? Enlighten me. ---
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [June 9, 2003, 4:22pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/2 "2003-06-09T16:22:07Z")
120 V (rms) is the equivalent in power to 120V DC. The power is proportional to the square of the voltage so the mean voltage is not the effective voltage. You have to square it, get the mean and then the root. ---
**Author:** ![scr4](https://avatars.discourse-cdn.com/v4/letter/s/59ef9b/32.png) [@scr4](https://boards.straightdope.com/u/scr4)\ **Post date:** [June 9, 2003, 4:36pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/3 "2003-06-09T16:36:57Z")
Since the 120v from the outlet is AC (alternating current), it alternates between positive and negative voltage. So the average is always zero. RMS is often used for values which average to zero, but when you want to know how far from zero it is typically. ---
**Author:** ![Balthisar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/balthisar/32/2825_2.png) [@Balthisar](https://boards.straightdope.com/u/Balthisar)\ **Post date:** [June 9, 2003, 6:03pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/4 "2003-06-09T18:03:22Z")
It has to do with the shape of the sin wave. The amount of electricity isn’t strictly the amplitude of the wave, but the amplitude _and_ the shape of the wave, or better yet, the area of the wave. In the case of normal, generator-produced AC power without any type of rectification going on, you’re dealing with a sine. So that 170V peak voltage (340V peak-to-peak!) only occupies as much surface area (graphically speaking) as a 120V DC signal, given the same pulse width. If you have alternating DC (i.e., a wave-shaped, amplitude modulated DC that “looks” like AC), the same maths apply. Change the shape of the waveform, and the RMS value goes out the window, although the same maths are used for power calculations. Two good examples are SCR- or IGBT-driven power controllers. In the case of an SCR, you get “part” of the sine wave, and you can use the firing angle with RMS to figure out the real potential. IGBTs are big transistors, they can shape a power wave according to just about any input signal you give them. Unless you’re driving them as a sine wave, you’ll need to figure out a different way to calculate potential. This can be done using calculus to determine the graphed area of the wave. ---
**Author:** ![Crafter\_Man](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/crafter_man/32/458_2.png) [@Crafter\_Man](https://boards.straightdope.com/u/Crafter_Man)\ **Post date:** [June 9, 2003, 7:06pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/5 "2003-06-09T19:06:37Z")
What others have said. Here’s my 2-cents. As stated by **sailor** , think of RMS voltage as the “equivalent” DC voltage. 120 V RMS is therefore equivalent to 120 VDC when connected to a resistive load. This makes some assumptions, but you get the idea. “Average voltage” is a bit more confusing. The average voltage at your outlet, as pointed out by **scr4** , is actually zero. This is because the positive and negative swings (which are otherwise identical) mathematically cancel. But this isn’t very helpful. So we have another average – the average of the _half wave_. Why do we do this? Because many loads don’t care about polarity; for many things, the power dissipated during the negative swing is identical to the power dissipated during the positive swing. As an example, take two light bulbs[sup]1[/sup]. Bulb “A” has 120 VAC on it; this means there’s a positive half wave, followed by a negative half wave, followed by a positive half wave, etc etc. Bulb “B” has the same waveform except all half waves are positive, i.e. there’s a positive half wave, followed by a positive half wave, followed by a positive half wave, etc etc. These two bulbs will have the same brightness and dissipate the same power. Why? Because a light bulb doesn’t care about current direction or voltage polarity. Because of this, it is easier to analyze the circuit as if all half waves were positive. And when we do this, we get a non-zero average voltage. In the case of a sine wave, the average voltage is 0.637 \* amplitude. [sup]1[/sup] [sub]To keep things simple, I’m assuming they’re purely resistive.[/sub] ---
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [June 9, 2003, 7:10pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/6 "2003-06-09T19:10:09Z")
> [@](#): > > \*Originally posted by scr4 \* > \*\*Since the 120v from the outlet is AC (alternating current), it alternates between positive and negative voltage. So the average is always zero. RMS is often used for values which average to zero, but when you want to know how far from zero it is typically. \*\* You obviously do not understand the concept and _I_ don’t understand why you bother posting, especially since someone had already posted something better. It has nothing to do with the average being zero or any other number. It has to do with the fact that instant power is proportional to the square of instant voltage so the average _power_ is proportional to the average of the integral of the square of the voltage. So, to obtain the equivalent voltage you would take the square root of that. I don’t know why people feel compelled to post when they don’t know the answer. (I believe there was a pit rant recently) > [@](#): > > \*Originally posted by Balthisar \* > \*\*It has to do with the shape of the sin wave. The amount of electricity isn’t strictly the amplitude of the wave, but the amplitude _and_ the shape of the wave, or better yet, the area of the wave. In the case of normal, generator-produced AC power without any type of rectification going on, you’re dealing with a sine. So that 170V peak voltage (340V peak-to-peak!) only occupies as much surface area (graphically speaking) as a 120V DC signal, given the same pulse width. If you have alternating DC (i.e., a wave-shaped, amplitude modulated DC that “looks” like AC), the same maths apply. > > Change the shape of the waveform, and the RMS value goes out the window, although the same maths are used for power calculations. Two good examples are SCR- or IGBT-driven power controllers. In the case of an SCR, you get “part” of the sine wave, and you can use the firing angle with RMS to figure out the real potential. IGBTs are big transistors, they can shape a power wave according to just about any input signal you give them. Unless you’re driving them as a sine wave, you’ll need to figure out a different way to calculate potential. This can be done using calculus to determine the graphed area of the wave. \*\* Also wrong. The RMS value of any wave is still the RMS value of that wave regardless of the shape (although the formula will be different). The RMS value of _any_ wave represents the equivalent DC voltage. When applied to a resistor two RMS signals of equal voltage will produce the same amount of heat. ---
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [June 9, 2003, 8:01pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/7 "2003-06-09T20:01:05Z")
Examples: A DC source which gives 10 V has an average value of 10 V and an RMS value of 10 V. A source which gives out a square wave which is 10 V 50% of the time and 0 V the other 50% has an average voltage of 5 V but an RMS voltage of SQRT(.5_0^2 + .5_10^2)= 7 Volts. If you connect a resistor to this source it will produce the same heat as if it were connected to a 7 Volt source, ---
**Author:** ![Crafter\_Man](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/crafter_man/32/458_2.png) [@Crafter\_Man](https://boards.straightdope.com/u/Crafter_Man)\ **Post date:** [June 9, 2003, 8:05pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/8 "2003-06-09T20:05:06Z")
> [@](#): > > \*Originally posted by sailor \* > **I don’t know why people feel compelled to post when they don’t know the answer.** Was that really necessary, **sailor**? IMO **scr4** has been a valuable contributor to these electrical/electronic threads over the years, and he’s almost always 100% accurate in his posts. While I might agree his response in this particular thread is a bit unclear, I don’t have any doubt that he fully understands the concept of rms and average voltage. ---
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [June 9, 2003, 8:20pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/9 "2003-06-09T20:20:20Z")
I did not mean to be mean but the response is clearly misleading. > [@](#): > > RMS is often used for values which average to zero, but when you want to know how far from zero it is typically has _nothing_ to do with RMS which, as I said, is a measurement of the equivalent power. I would not have said anything if his was the first post but I had already posted a better reply and his only served to confuse, not to clarify the issue. But, as I said, no offense was intended. ---
**Author:** ![Balthisar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/balthisar/32/2825_2.png) [@Balthisar](https://boards.straightdope.com/u/Balthisar)\ **Post date:** [June 10, 2003, 10:42am UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/10 "2003-06-10T10:42:04Z")
I’ll freely admit I contributed to a little confusion by not distinguishing between “RMS” and “quadratic mean.” They’re the exact same thing and mean exactly the same thing. But when referring to electricity we so often are accustomed to calculating RMS using sqrt(2) we sometimes use two terms that mean the same thing in two different contexts. So I’ll rectify that just so **sailor** doesn’t come gunning for me, too. 🙂 RMS is an average (“root _mean_ square”) and applies to any signal – AC or DC (but it kind of pointless in non-AM DC). In power control and home systems and most industrial systems we’re dealing with sine waves, so it’s common to equate “RMS” with “square root of 2” or its inverse. When we start changing the wave shape, the mean is still the RMS, only you can no longer automatically assume you’re dealing with the sqrt(2). For example, to calculate the RMS of a non-AM DC signal, the factor is 1, or it’s inverse 1/1. We still call this the RMS value, despite the fact we’re not using the typical sqrt(2). ---
**Author:** ![scr4](https://avatars.discourse-cdn.com/v4/letter/s/59ef9b/32.png) [@scr4](https://boards.straightdope.com/u/scr4)\ **Post date:** [June 10, 2003, 12:28pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/11 "2003-06-10T12:28:10Z")
> [@](#): > > \*Originally posted by sailor \* > **I would not have said anything if his was the first post but I had already posted a better reply and his only served to confuse, not to clarify the issue.** I apologize for the hasty over-simplification, but I was attempting to answer the second part of the OP, namely why RMS is a common concept in science. Your first response (or anybody else’s) did not address this. ---
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [June 10, 2003, 7:06pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/12 "2003-06-10T19:06:29Z")
> [@](#): > > \*Originally posted by scr4 \* > I apologize for the hasty over-simplification, but I was attempting to answer the second part of the OP, namely why RMS is a common concept in science. Your first response (or anybody else’s) did not address this. Except that I believe I _had_ addressed it. It is a measurement of the power contained by the signal: > [@](#): > > \*Originally posted by sailor \* > 120 V (rms) is the equivalent in power to 120V DC. The power is proportional to the square of the voltage so the mean voltage is not the effective voltage. You have to square it, get the mean and then the root. and not, as you said, anything to do with: > [@](#): > > \*Originally posted by scr4 \* > RMS is often used for values which average to zero, but when you want to know how far from zero it is typically. I believe that is plainly wrong. It has \*nothing to do with the average of the signal and everything to do with the power which is proportional to the square. RMS is a _very_ simple concept and it is defined as the **square root of the mean (over time) of the square**. Period. Forget about sine or no sine. The RMS value of _any_ signal represents the power the signal contains and to say it is used primarily for sine waves is nonsense. The concept is universal and used with any wave form where we need to know the power. Electrical signals have different shapes and depending on the shape the characteristics of the signal will vary. Two signals can have the same RMS value and very different peak values. Two signals can have the same peak values and very different RMS values. It is essential to understand the different concepts if you are to work with electrical systems. I already gave some examples. Here is another one: Linear audio amplifiers are limited by the peak to peak signal they can output and by the power they can output but, generally speaking, they can output a sine wave to the specified load without overloading. That means they can safely output 0.7 RMS of the peak voltage. Speakers, on the other hand, cannot generally withstand that kind of signal without damage. The type of signal (human voice, music) fed to speakers has an RMS value which is only a very small fraction of the peak value. A speaker is limited by the RMS value it can take and not really by the peak value. If you feed a sine signal to an amplifier and blast it into the speakers you can easily damage them. So, we have a situation where the amplifier is limited by design much more by the peak value while the speakers are limited by the RMS value and this depends on the type of signal and it is necessary to understand these things. Analog multimeters essentially measure _average_ voltage. Those which have AC scales are graduated assuming the signal is a sine wave but will give false readings if the signal has a different mean/rms value. Very few meters will measure RMS directly. RMS value is used _all the time_ in all sorts of electronics. It is used for audio as I have just shown. It is used in switching power converters, radio modulation, etc. It is an essential characteristic of _any_ electrical signal. And, again, it is obtained _always_ by taking the **square root of the mean (over time) of the square of the signal**. I have had to calculate RMS values of all sorts of waves. Most recently I calculated the RMS value of a 220VAC, 50 Hz sine wave after being chopped by a triac dimmer as a function of when the triac was fired. The first column is time in milliseconds into the half cycle when the triac is triggered, the second voltage is the voltage across the triac at this point and the last colum is the RMS value of the resulting (chopped) wave. If your dimmer fires after 6.25 ms your resulting wave would have an RMS value of 113 V so you could power a standard 110V lightbulb. But this same wave will have a peak value of 287 volts and will overcharge and damage a device designed for 110V which has a rectifier and capacitor filter as the peak value of a sine 110 V sine is about 155 V. Depending on the use the peak or the RMS values may come into play. ```auto ms V on V RMS 0.00 0.0 220 0.25 24.4 220 0.50 48.7 220 0.75 72.6 220 1.00 96.1 219 1.25 119.0 219 1.50 141.2 218 1.75 162.5 216 2.00 182.8 215 2.25 202.0 212 2.50 219.9 210 2.75 236.5 207 3.00 251.6 203 3.25 265.2 199 3.50 277.1 194 3.75 287.3 189 4.00 295.8 183 4.25 302.4 177 4.50 307.2 170 4.75 310.0 163 5.00 311.0 156 5.25 310.0 148 5.50 307.2 139 5.75 302.4 131 6.00 295.8 122 6.25 287.3 113 6.50 277.1 103 6.75 265.2 94 7.00 251.6 85 7.25 236.5 76 7.50 219.9 66 7.75 202.0 57 8.00 182.8 49 8.25 162.5 40 8.50 141.2 32 8.75 119.0 25 9.00 96.1 18 9.25 72.6 12 9.50 48.7 6 9.75 24.4 2 10.00 0.0 0 ``` ---
**Author:** ![chukhung](https://avatars.discourse-cdn.com/v4/letter/c/e79b87/32.png) [@chukhung](https://boards.straightdope.com/u/chukhung)\ **Post date:** [June 10, 2003, 7:26pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/13 "2003-06-10T19:26:42Z")
Isn’t the concept of RMS often used in statistics (e.g., in regression analysis)? From my long-ago statistics class, I dimly remember RMS error being used as “goodness of fit” diagnostic for a regression line. ---
**Author:** ![scr4](https://avatars.discourse-cdn.com/v4/letter/s/59ef9b/32.png) [@scr4](https://boards.straightdope.com/u/scr4)\ **Post date:** [June 10, 2003, 7:27pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/14 "2003-06-10T19:27:19Z")
> [@](#): > > \*Originally posted by sailor \* > **Except that I believe I _had_ addressed it. It is a measurement of the power contained by the signal** I believe the OP was asking a more general question, not limited to voltages of electrical signals. The speed of atoms was given as an example. For that example, I believe the RMS is relevant because kinetic energy is proportional to the square of velocity. Therefore RMS speed gives the speed that corresponds to the average speed. But of course my understanding is, as **sailor** pointed out, very limited and possibly completeley incorrect so I’d appreciate it if others could correct me. ---
**Author:** ![scr4](https://avatars.discourse-cdn.com/v4/letter/s/59ef9b/32.png) [@scr4](https://boards.straightdope.com/u/scr4)\ **Post date:** [June 10, 2003, 7:28pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/15 "2003-06-10T19:28:48Z")
That should be “…RMS speed gives the speed that corresponds to the average **kinetic energy**.” Sorry. ---
**Author:** ![scr4](https://avatars.discourse-cdn.com/v4/letter/s/59ef9b/32.png) [@scr4](https://boards.straightdope.com/u/scr4)\ **Post date:** [June 10, 2003, 7:32pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/16 "2003-06-10T19:32:48Z")
> [@](#): > > \*Originally posted by sailor \* > **I believe that is plainly wrong.** I already apologized for over-simplifying - by that I mean I simplified it to a point where it is no longer correct. No need to rub it in. ☹ ---
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [June 10, 2003, 8:07pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/17 "2003-06-10T20:07:49Z")
I did not mean to “rub it in” and I am sorry if I came across that way. Just wanting to make sure we get the concept right. And yes, the concept is useful anywhere where it is meaningful. If I have a turbine and 50% of the molecules hit the blade with a speed V1 and 50% with a speed V2, the equivalent speed is not the mean of the two but the RMS because the energy is proportional to the square of the speed. Wind at 20 knots 50% of the time carries more energy than wind at 10 knots 100% of the time ---
**Author:** ![Crafter\_Man](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/crafter_man/32/458_2.png) [@Crafter\_Man](https://boards.straightdope.com/u/Crafter_Man)\ **Post date:** [June 10, 2003, 8:20pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/18 "2003-06-10T20:20:55Z")
Some nit picks. Couldn’t help it. 😉 > [@](#): > > _Originally posted by sailor \* > \*\* The RMS value of any signal represents the power the signal contains…_\* I guess I’m bothered by the word “contains.” It is possible for a signal to have a non-zero RMS value yet not “contain” any power. As an example, the outlet next to my desk is at 120 VAC RMS, yet the power is zero because nothing is plugged in. RMS is really an indication of potential energy or power. > [@](#): > > \*\*RMS … is defined as the **square root of the mean (over time) of the square. Period.** This is true, but I’d like to add something: RMS in itself is meaningless unless the boundary conditions are defined. For **periodic signals** , RMS is usually computed over one period. For **noise** , it is usually computed over a fixed time record. For **non-noise & non-periodic signals** , computed the RMS doesn’t have a lot of value, at least usually. ---
**Author:** ![Balthisar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/balthisar/32/2825_2.png) [@Balthisar](https://boards.straightdope.com/u/Balthisar)\ **Post date:** [June 10, 2003, 9:53pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/19 "2003-06-10T21:53:04Z")
> [@](#): > > \*Originally posted by sailor \* > \*\*RMS is a _very_ simple concept and it is defined as the **square root of the mean (over time) of the square**. Period. Forget about sine or no sine. The RMS value of _any_ signal represents the power the signal contains and to say it is used primarily for sine waves is nonsense. \*\* Just so that **sailor** ’s comment doesn’t look contrary to mine above, I’ll point out that the commonly used “sqrt(2)” (~1.414) is used to calculate RMS for sine waves. In beginner electricity classes where RMS is taught the sqrt(2) is taught. It somehow becomes entangled that RMS=sqrt(2). I’m only trying to point out the same thing as **sailor** , that everything has an RMS, and as **sailor** said earlier, “the formula will be different.” The formula being, of course, sqrt(2), which only applies to sine shaped waves. ---
**Author:** ![casdave](https://avatars.discourse-cdn.com/v4/letter/c/c6cbf5/32.png) [@casdave](https://boards.straightdope.com/u/casdave)\ **Post date:** [June 10, 2003, 9:56pm UTC](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654/20 "2003-06-10T21:56:54Z")
The classic sq rt value only works for a single phase electricity, as used in typical domestic supplies. When you start to look at 3 phase or polyphase systems you need to look at differant roots, the calculations are very similar and the principle is the same, just that there are more terms in the maths to deal with. All that EE’s are trying to do is to get some idea of power, voltage or current, RMS as described previously is simplistic especially in AC systems as there are other circuit conditions that may have a dramatic effect, such as power factor. RMS does not exclusively mean power, it is a mathematical term describing a method of calculation. The term RMS can be applied to anything that can be represented by a waveform, but most usually periodic waveforms. There are many periodic waveforms in engineering, in physics, and the OP mentions something in chemistry, which I had no idea of. Obviously depending upon the quantity in question, RMS values are going to mean differant things. The reason that RMS is used is that not all periodic waveforms are symetrical, so trying to get an average over half a cycle will not work. By using RMS what you effectively do is to flip(reflect would be a better word to use in this context) all the part of the waveform on the other side of the zero volts baseline up so that it is all on the same side, from which your average can be taken. It is possible to have a sinewave that never crosses zero too, so to use the half sinewave as an average can only happen under specific conditions. Which I guess is partly what **Crafter** is saying in the last part of his post. [Next page](https://boards.straightdope.com/t/explanation-of-root-mean-squared/180654.md?page=2)