### RMS

**(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 = √( ∑a_{i}²/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".
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".

(*statistics*)
**Further reading:**

http://en.wikipedia.org/wiki/Root_mean_square

**Referenced by pages:**

angular power spectrum

gravitational wave strain (h)

RMS astronomy

wavefront error (WFE)

Index