Astrophysics (Index)About


(root mean square, rms)
(square root of the average of some squared values)

RMS stands for root mean square, a number statistically summarizing others in the manner of the sum, mean, or median. The RMS is a type of "average" appropriate for some situations, such as to summarize a particular kind of independent values. It is short for the square root of the mean of the squares of all the numbers:

RMS = √( ∑ai²/n )

where n is the count of the values ai. For all the values over an interval (an infinite number of them), an integral (∫) is used instead of the summation:

RMS = √( (∫f(x)²dx)/s )

where s is the size of the interval over which the integral is taken.

The formula for a standard deviation incorporates an RMS (it is the RMS of some numbers' deviations from their mean value), and sometimes in astronomy literature, a standard deviation is cited as "an RMS", presumably meaning "RMS from the mean". Noise measurements may be cited in this manner, which may use the summation for a finite number of measurements. The standard deviation of a probability density function (PDF) uses the integral formula (i.e., integral of the square of the deviations from the mean value).

RMS is also used in the calculation the average power associated with a varying signal such as a varying electrical current or a varying received radio signal (using the integral calculation). For a signal repeating, such as a sine-wave (like AC electrical power), the integral can be over one complete cycle. The instantaneous power associated with a sine-wave AC signal ranges from zero to some maximum (the maximum termed the peak power), but for many purposes, the most useful value is the average power, calculated with the RMS (RMS power), which can reasonably be used to calculate the energy transferred over periods of time lasting many cycles.

Note that the phrase RMS astronomy has been used with an entirely unrelated meaning: "radio, millimeter, submillimeter".

Further reading:

Referenced by pages:
angular power spectrum
gravitational wave strain (h)
RMS astronomy
wavefront error (WFE)