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root-mean-square deviation[¦rüt ¦mēn ¦skwer ‚der·ə′vā·shən]
The root-mean-square (rms) deviation of the quantities x1, x2, …, xn 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 x1, x2, … xn
In this case the rms deviation may serve as a measure of the dispersion of the system of quantities x1, x2, … xn. The more general concept of weighted rms deviation
is also used; p1, … pn in this case are called the weights corresponding to the quantities x1, … xn. The weighted rms deviation attains its least value when a is equal to the weighted mean:
(p1x1 + ..... + pnxn)/(p1 + ... + pn)
In probability theory the weighted rms deviation σx 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.