Sample variance

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)