For a set of numbers or values of a discrete distribution , ..., , the root-mean-square (abbreviated "RMS" and sometimes called the quadratic mean), is the square root of mean of the values , namely
(1)
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(2)
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where denotes the mean of the values .
For a variate from a continuous distribution ,
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where the integrals are taken over the domain of the distribution. Similarly, for a function periodic over the interval ], the root-mean-square is defined as
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The root-mean-square is the special case of the power mean.
Hoehn and Niven (1985) show that
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for any positive constant .
Physical scientists often use the term root-mean-square as a synonym for standard deviation when they refer to the square root of the mean squared deviation of a signal from a given baseline or fit.