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Encyclopedia > Quadratic mean

In mathematics, root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity: it's specially useful when variates are positive and negative, ie the cases of waves. For other meanings of mathematics or math, see mathematics (disambiguation). ... A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... In science, a magnitude is the numerical size of something: see orders of magnitude. ...


It can be calculated for a series of discrete values or, for a continuously varying function. The name comes from the fact that it is the square root of the mean of the squares of the values. It is a power mean with the power t = 2. Partial plot of a function f. ... In mathematics, a square root of a number x is a number whose square (the result of multiplying the number by itself) is x. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... In algebra, the square of x is written x2 and is defined as the product of x with itself: x × x. ... A generalized mean, also known as power mean or Hölder mean, is an abstraction of the arithmetic, geometric and harmonic means. ...

Contents

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Calculating the root mean square

The rms for a collection of N values {x_1,x_2,dots,x_N} is:

x_{mathrm{rms}} = sqrt {{1 over N} sum_{i=1}^{N} x_i^2} = sqrt {{x_1^2 + x_2^2 + cdots + x_N^2} over N}

and the corresponding formula for a continuous function f(t) defined over the interval T_1 le t le T_2 (for a periodic function the interval should be a whole number of complete cycles) is:

f_{mathrm{rms}} = sqrt {{1 over {T_2-T_1}} {int_{T_1}^{T_2} {[f(t)]}^2, dt}}
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Uses

The RMS value of a function is often used in physics and electronics. For example, we may wish to calculate the power P dissipated by an electrical conductor of resistance R. It is easy to do the calculation when a constant current I flows through the conductor. It is simply, The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... The field of electronics comprises the study and use of systems that operate by controlling the flow of electrons (or other charge carriers) in devices such as thermionic valves and semiconductors. ... In physics, power (symbol: P) is the amount of work done per unit of time. ... In electricity, current refers to electric current, which is the flow of electric charge. ...

P = I2R

But what if the current is a varying function I(t)? This is where the rms value comes in. It may be trivially shown that the rms value of I(t) can be substituted for the constant current I in the above equation to give the average power dissipation:

P_mathrm{avg},! = mathrm{E}(I^2R),! (where mathrm{E}(cdot) denotes the arithmetic mean)
= Rmathrm{E}(I^2),! (R is constant so we can take it outside the average)
= I_mathrm{rms}^2R,! (by definition of RMS)

We can also show by the same method In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ...


P_mathrm{avg} = {V_mathrm{rms}^2over R},!


By square rooting both these equations and multiplying them together we get the equation


P_mathrm{avg} = V_mathrm{rms}I_mathrm{rms},!


However it is important to stress that this is based on the assumption that voltage and current are proportional (that is the load is resistive) and is not true in the general case (see AC power for more information). Usually hidden to the unaided eye, the 60Hz blinking of (non-incandescent) lighting powered by AC mains is revealed in this motion-blurred long exposure of city lights. ...


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 we define Ip to be the peak amplitude: City lights viewed in a motion blurred exposure. ... In trigonometry, an ideal sine wave is a waveform whose graph is identical to the generalized sine function y = Asin[ω(x − α)] + C, where A is the amplitude, ω is the angular frequency (2π/P where P is the wavelength), α is the phase shift, and C...


I_{mathrm{rms}} = sqrt {{1 over {T_2-T_1}} {int_{T_1}^{T_2} {(I_mathrm{p}sin(omega t)}, })^2 dt}


Since Ip is a positive real number


I_{mathrm{rms}} = I_mathrm{p}sqrt {{1 over {T_2-T_1}} {int_{T_1}^{T_2} {sin^2(omega t)}, dt}}


Using a trigonomentric identity to eliminate squareing of trig function


I_{mathrm{rms}} = I_mathrm{p}sqrt {{1 over {T_2-T_1}} {int_{T_1}^{T_2} {{1 - cos(2omega t) over 2}}, dt}}


I_{mathrm{rms}} = I_mathrm{p}sqrt {{1 over {T_2-T_1}} left [ {{t over 2} -{ sin(2omega t) over 4omega}} right ]_{T_1}^{T_2} }


but since the interval is a whole number of complete cycles (per definition of rms for a periodic function) the sin terms will cancel


I_{mathrm{rms}} = I_mathrm{p}sqrt {{1 over {T_2-T_1}} left [ {{t over 2}} right ]_{T_1}^{T_2} } = I_mathrm{p}sqrt {{1 over {T_2-T_1}} {{{T_2-T_1} over 2}} } = {I_mathrm{p} over {sqrt 2}}


The peak amplitude is half of the peak-to-peak amplitude. When the peak-to-peak amplitude is known, the same formula is applied by using half of the p-p value.


The RMS value can be calculated using equation (2) for any waveform, for example an audio or radio signal. This allows us to calculate the mean power delivered into a specified load. For this reason, listed voltages for power outlets (e.g. 110 V or 240 V) are almost always quoted in RMS values, and not peak values. now. ...


From the formula given above, we can calculate also the peak-to-peak value of the mains voltage which is approx. 310 (U.S.A) and 677 (Europe) volts respectively.


In the field of audio, mean power is often (misleadingly) referred to as RMS power. This is probably because it can be derived from the RMS voltage or RMS current. Furthermore, because RMS implies some form of averaging, expressions such as "peak RMS power", sometimes used in advertisements for audio amplifiers, are meaningless.(dubious assertion) The term sine power is used in the specification or measurement of audio amplifiers or loudspeakers. ...


In chemistry, the root mean square velocity is defined as the square root of the average velocity-squared of the molecules in a gas. The RMS velocity of an ideal gas is calculated using the following equation: Chemistry (from the Greek word χημεία (chemeia) meaning cast together or pour together) is the science of matter at the atomic to molecular scale, dealing primarily with collections of atoms (such as molecules, crystals, and metals). ... A gas is one of the four main phases of matter (after solid and liquid, and followed by plasma), that subsequently appear as a solid material is subjected to increasingly higher temperatures. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ...

{u_mathrm{rms}} = {sqrt{3RT over {M}}}

where R represents the ideal gas constant (in this case, 8.314 J/(mol⋅K)), T is the temperature of the gas in kelvins, and M is the molar mass of the compound in kilograms per mole. Molar gas constant (also known as universal gas constant, usually denoted by symbol R) is the constant occurring in the universal gas equation, i. ... The kelvin (symbol: K) is the SI unit of temperature, and is one of the seven SI base units. ... Molar mass is the mass of one mole of a chemical element or chemical compound. ...

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Relationship to the arithmetic mean and the standard deviation

If bar{x} is the arithmetic mean and σx is the standard deviation of a population then In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is defined as the square root of the variance. ...

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

Here we can see that RMS is always the same or more than the average, in that the RMS inlcudes the "error" / square deviation too.

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See also

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Least squares is a mathematical optimization technique which, when given a series of measured data, attempts to find a function which closely approximates the data (a best fit). It attempts to minimize the sum of the squares of the ordinate differences (called residuals) between points generated by the function and... A generalized mean, also known as power mean or Hölder mean, is an abstraction of the arithmetic, geometric and harmonic means. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is defined as the square root of the variance. ... Root mean square speed is a measure of the velocity of particles in a gas. ... The following table lists many specialized symbols commonly used in mathematics. ...

External links


  Results from FactBites:
 
MEAN - LoveToKnow Article on MEAN (666 words)
The adjective mean is chiefly used in the sense of average, as in mean temperature, mean birth or death rate, andc.
Mean as a substantive has the following principal applications; it is used of that quality, course of action, condition, state, andc., which is equally distant from two extremes, as in such phrases as the golden (or happy) mean.
The quadratic mean of n quantities is the square root of the arithmetical mean of their squares.
Mean - Wikipedia, the free encyclopedia (862 words)
Sample mean is often used as an estimator of the central tendency such as the population mean.
The mean is the arithmetic average of a set of values, or distribution; however, for skewed distributions, the mean is not the same as the middle value (median), or most likely (mode).
The geometric mean is an average that is useful for sets of numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean).
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