Mean square

In mathematics and its applications, the mean square is normally defined as the arithmetic mean of the squares of a set of numbers or of a random variable.^[1]

It may also be defined as the arithmetic mean of the squares of the deviations between a set of numbers and a reference value (e.g., may be a mean or an assumed mean of the data),^[2] in which case it may be known as mean square deviation. When the reference value is the assumed true value, the result is known as mean squared error.

A typical estimate for the sample variance from a set of sample values ${\displaystyle x_{i}}$ uses a divisor of the number of values minus one, n-1, rather than n as in a simple quadratic mean, 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 when the random variable is zero-mean.