Root mean square

Geometric construction to find arithmetic media (A), quadratic (Q), geometric (G) and harmonic (H) of two numbers a and b.

In mathematics, the root mean square, root mean square value, root mean square, or RMS (from English root mean square) is a statistical measure of the magnitude of a variable quantity. It can be calculated for a series of discrete values or for a continuous variable mathematical function, for example helical movement. The name derives from the fact that it is the square root of the arithmetic mean of the squares of the values.

Sometimes the variable takes positive and negative values, as occurs, for example, in measurement errors. In such a case, one may be interested in obtaining an average that does not capture the effects of the sign. This problem is solved by means of the so-called square mean. It consists of squaring all the observations (thus the negative signs disappear), then obtaining their arithmetic mean and finally extracting the square root of said mean to return to the original unit of measurement. The standard deviation is a root mean square.

Other statistical means are the weighted mean, generalized mean, harmonic mean.

Definition

The root mean square for a collection of N values {x₁, x₂,... x_n} of a discrete variable x, is given by the formula (1):

xRMS=1N␡ ␡ i=1Nxi2=x12+x22+ +xN2N(1){displaystyle x_{mathrm {RMS} }={sqrt {{1 over N}sum _{i=1}{N}x_{i}{2}}}}}}{sqrt {{x_{x}{1⁄2} +x{2}{2}{2}{2} +cdots +x_{n}{n}{c}{c}}}{x1⁄2}}}{x1⁄2}}}{x1⁄2}}}{x1⁄2}}}{x1⁄2}}}}{x1⁄2}}}{x1⁄2}}}{x1⁄2}}}{x1⁄2}}}{x1⁄2}}}}}}{x1⁄2} }{x1⁄2 } } } } } } } } } } } } } } } } } } } } } } } }

For a function of continuous variable f(t) defined over the interval T₁ ≤ t ≤ T₂ is given by the expression:

xrms=1T2− − T1∫ ∫ T1T2[chuckles]f(t)]2dt.(2){displaystyle x_{mathrm {rms} }={sqrt {{1 over {T_{2}-T_{1}}}{int _{T_{1}}}}{T_{2}}}{[f(t)]}{2}{2}}{,dt}}}}.

Property

There is an order relationship of the means obtained from the same collection of values

H≤ G ≤ A ≤ Q where H is the harmonic mean; G, geometric mean; A, arithmetic mean; Q, Quadratic mean

Applications

Effective value of an alternating current

Generally, the effective value is used in physics and engineering, although it has other uses.

Mean square velocity of a gas

In physics, the root mean square of the speed of a gas is defined as the square root of the mean of the squares of the speeds of the molecules of a gas. The RMS velocity of an ideal gas is calculated using the following equation:

vRMS=3kTM{displaystyle {v_{mathrm {RMS}}}}={sqrt {3kT over {M}}}}

where k{displaystyle k} represents the Boltzmann constant (in this case, 1.3806503*10^-23J/K), T is the gas temperature in kelvins, and M is the mass of gas, measured in kilograms.