Each linear estimate has two measures: the bias with respect to the actual value and the variance of the estimate noise. Both these measures must be minimal in an optimal estimate. However, if we minimize only the bias, then the variance may not be minimal. In turn, if we minimize the variance, then the bias can be large. So, there must be some other measure. This measure is the mean square error MSE = bias^2 + var which minimization gives us the optimal estimate. Then RMS = sqrt{MS}.

Simply put, RMS is a measure of the dynamics of a signal. The longer explanation is: Amplitude is the variation in the observed phenomena, e.g. pressure level, voltage, deflection. If the variations over time are oscillatory, they have positive and negative values about an average value. A sinusoidal variation centered on zero has exactly as many positive excursions as negative ones. The average value over time is zero, which is not very informative about the signal's dynamics. Squaring amplitude converts variations into power levels. Averaging these over time yields an estimate of average power. To convert back to an estimate of the average deviation, you take the square root of this average. The full length expression would be "square root of the average value of the observed deviations squared". This is sort of cumbersome, hence "RMS".

A function is a vector in a infinite dimensional space. The RMS is the euclidean distance defined as an extension of the Pythagorean theorem to intinite dimension space.

Dear Timlelt Hakima, Root Mean Square (RMS) is a mathematical transformation which let us to think about a time varying signal x(t) in terms of a constant signal. as an example RMS values of current and voltage relates ac voltage and current.

RMS level (gain) is as a powerful metric to compare average energy of time series. RMS can be applied to different windows of data. Alternatively, one may use L2 norm (Euclidian distance) to measure the similarities between two time series.