Root-mean-square

Root-mean-square (RMS) is the square root of the mean of the squares of the set. It has broad applications in engineering, including in electrical engineering and in statistical learning (root-mean-square error) where it measures the quality of fit of a model.

$$f_{RMS}=\sqrt{\frac{1}{n}(x_1^2+\cdots +x_n^2)}=\sqrt{\frac{1}{n}{\int }_0^n{f}^2(t)dt}$$

Electrical engineering

The RMS value is a measure of a signal of the average power due to that signal. We define the RMS current and voltage from the above definition, so that:

$$P_{avg}=R{I}_{RMS}^2=\frac{V_{RMS}^2}{R}=V_{RMS}{I}_{RMS}$$

Especially for a sinusoidal signal (for example, $v(t)=V_A\cos (\omega t)$):

$$V_{RMS}=\frac{V_A}{\sqrt{2}};I_{RMS}=\frac{I_A}{\sqrt{2}}$$

And because the RMS value is proportional to the amplitude of the signal, the RMS voltage is a useful measure of the signal strength.