Mean square

In mathematics and its applications, the mean square is defined as the arithmetic mean of the squares of a set of numbers or of a random variable,[1] or as the arithmetic mean of the squares of the differences between a set of numbers and a given "origin" that may not be zero (e.g. may be a mean or an assumed mean of the data).[2]

When the mean square is calculated relative to a given "target" or "correct" value, or as the mean square of differences from a sequence of correct values, it is known as mean squared error.

A typical estimate for the variance from a set of sample values ${\displaystyle x_{i}}$ uses a divisor of one less than the number ${\displaystyle n}$ of values, rather than a simple arithmetic average, and this is still called the mean square (e.g. in analysis of variance):

${\displaystyle s^{2}=\textstyle {\frac {1}{n-1}}\sum (x_{i}-{\bar {x}})^{2}}$

The second moment of a random variable, ${\displaystyle E(X^{2})}$ is also called the mean square.

The square root of a mean square is known as the root mean square (RMS or rms), and can be used as an estimate of the standard deviation of a random variable.