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.

References

1. "Noise and Noise Rejection". Retrieved 6 January 2020.
2. "OECD Glossary of Statistical Terms". Retrieved 6 January 2020.