# Root-Mean-Square Deviation

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## root-mean-square deviation

[¦rüt ¦mēn ¦skwer ‚der·ə′vā·shən]*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Root-Mean-Square Deviation

The root-mean-square (rms) deviation of the quantities *x*_{1}, *x*_{2}, …, *x*_{n} from *a* is the square root of the expression

The rms deviation has its least value when *a* = *x̅*, where *x̅* is the arithmetic mean of the quantities *x*_{1}, *x*_{2}, … *x*_{n}

In this case the rms deviation may serve as a measure of the dispersion of the system of quantities *x*_{1}, *x*_{2}, … *x*_{n}. The more general concept of weighted rms deviation

is also used; *p*_{1}, … *p*_{n} in this case are called the weights corresponding to the quantities *x*_{1}, … *x*_{n}. The weighted rms deviation attains its least value when *a* is equal to the weighted mean:

(*p _{1}x_{1} + ..... + p_{n}x_{n})/(p_{1} + ... + p_{n}*)

In probability theory the weighted rms deviation σ_{x} of a random variable *X* (from its mathe matical expectation) is the square root of the variance , and is called the standard deviation of *X*.

The standard deviation is used as a measure of the quality of statistical estimates and in this case is called the standard error.