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The corresponding formula for a continuous function (or waveform) f(t) ... the root-mean-square speed is defined as the square root of the average squared-speed.
In fluid dynamics, normalized root mean square deviation (NRMSD), coefficient of variation (CV), and percent RMS are used to quantify the uniformity of flow behavior such as velocity profile, temperature distribution, or gas species concentration. The value is compared to industry standards to optimize the design of flow and thermal equipment ...
In mathematics, the QM-AM-GM-HM inequalities, also known as the mean inequality chain, state the relationship between the harmonic mean, geometric mean, arithmetic mean, and quadratic mean (also known as root mean square). Suppose that ,, …, are positive real numbers. Then
When a dynamical system fluctuates about some well-defined average position, the RMSD from the average over time can be referred to as the RMSF or root mean square fluctuation. The size of this fluctuation can be measured, for example using Mössbauer spectroscopy or nuclear magnetic resonance, and can provide important physical information.
The second moment of a random variable, () is also called the mean square. The square root of a mean square is known as the root mean square (RMS or rms), and can be used as an estimate of the standard deviation of a random variable when the random variable is zero-mean.
A geometric construction of the quadratic mean and the Pythagorean means (of two numbers a and b). Harmonic mean denoted by H, geometric by G, arithmetic by A and quadratic mean (also known as root mean square) denoted by Q. Comparison of the arithmetic, geometric and harmonic means of a pair of numbers.
Some formulas in these fields are expressed using the squared coefficient of variation, often abbreviated SCV. In modeling, a variation of the CV is the CV(RMSD). Essentially the CV(RMSD) replaces the standard deviation term with the Root Mean Square Deviation (RMSD).
The geometric mean of the three numbers is the cube root of their product, for example with numbers , , and , the geometric mean is = =. The geometric mean is useful whenever the quantities to be averaged combine multiplicatively, such as population growth rates or interest rates of a financial investment.