Root-Mean-Square Deviation

The root-mean-square (rms) deviation of the quantities x₁, x₂, …, 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 x₁, x₂, … x_n

In this case the rms deviation may serve as a measure of the dispersion of the system of quantities x₁, x₂, … x_n. The more general concept of weighted rms deviation

is also used; p₁, … p_n in this case are called the weights corresponding to the quantities x₁, … x_n. The weighted rms deviation attains its least value when a is equal to the weighted mean:

(p₁x₁ + ..... + p_nx_n)/(p₁ + ... + 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.