The root mean square is a type of mean.

Given real numbers \(a_1\), \(a_2\), …, \(a_n\), the root mean square (often abbreviated to RMS) is obtained by calculating the arithmetic mean of the squares of \(a_1\), …, \(a_n\), and then taking the square root of this: \[\sqrt{\frac{a_1^2+a_2^2+\cdots+a_n^2}{n}}\]

It is useful when trying to measure the average “size” of numbers, where their sign is unimportant, as the squaring makes all of the numbers non-negative.

The most common case of using the root mean square is when calculating the standard deviation of a set of numbers \(x_1\), …, \(x_n\). The standard deviation is the root mean square of the deviations of these numbers from the mean, that is, the root mean square of \((x_1-\bar x)\), …, \((x_n-\bar x)\), where \(\bar x\) is the mean of \(x_1\), …, \(x_n\), so \[\text{standard deviation}=\sqrt{\frac{(x_1-\bar x)^2+(x_2-\bar x)^2+\cdots+(x_n-\bar x)^2}{n}}.\]

The root mean square can also be used for continuous functions, with integration replacing summation. If the function \(f(x)\) is defined for \(a\le x\le b\), then the root mean square value of \(f(x)\) over this interval is \[\sqrt{\frac{1}{b-a}\int_a^b (f(x))^2\,dx}.\]

The root mean square is an example of a power mean.