# Sample variance

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sample dispersion

A scalar characteristic of the disperson, or spread, of a sample (consisting of real numbers) relative to a fixed point $x$( called the centre of dispersion). It is numerically equal to the sum of the squares of the deviations of the values from $x$. For real-valued random variables $X _ {1} \dots X _ {n}$, the variable

$$S _ {n} ( x) = \ \sum _ {i = 1 } ^ { n } ( X _ {i} - x) ^ {2}$$

is the sample variance about $x$. The variables $X _ {1} \dots X _ {n}$ are often assumed to be independent and identically distributed in discussions about $S _ {n} ( x)$. Since, for any $x$,

$$S _ {n} ( x) = S _ {n} ( \overline{X}\; ) + n ( \overline{X}\; - x) ^ {2} \geq \ S _ {n} ( \overline{X}\; ) \equiv S _ {n} ,$$

where $\overline{X}\; = ( X _ {1} + \dots + X _ {n} )/n$, the sample variance about $x$ will be minimal when $x= \overline{X}\;$. A small value of $S _ {n}$ indicates a concentration of the sample elements about $\overline{X}\;$ and, conversely, a large value of $S _ {n}$ indicates a large scattering of the sample elements. The concept of a sample variance extends to that of a sample covariance matrix for multivariate samples.

#### References

 [1] S.S. Wilks, "Mathematical statistics" , Wiley (1962)
How to Cite This Entry:
Sample variance. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sample_variance&oldid=48610
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article