RMS

RMS (or 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 a_i. 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". Measurement noise may be characterized in this manner, which might 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, which matches its RMS (RMS power), which yields the average energy transfer, given a time period lasting many cycles. (In discussions of electric power, you will see reference to the RMS as "roughly 0.707 of the peak power", but that number is specifically based upon the sine-function signal typical of AC power.)

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