True variance

In statistics, the term "true variance" is often used to refer to the unobservable variance of a whole finite population, as distinguished from an observable statistic based on a sample. Suppose a number, such as a person's height or income or age or cholesterol level, is assigned to every member of a population of "n" individuals. Let "x"_"i" be the number assigned to the "i"th individual, for "i" = 1, ..., "n". Then the variance is

:$sigma^2=\{1\; over\; n\}sum\_\{i=1\}^n\; (x\_i-overline\{x\})^2,quadquadquad(1)$

where

:$mu=overline\{x\}=\{x\_1+cdots+x\_n\; over\; n\}$

is the population mean. If "x"_"i" were the "i"th member of a random sample rather than of the whole population, then one sometimes uses the same function seen in (1) above as an estimate of the "true variance" or "population variance" σ². But sometimes one replaces "n" with "n" − 1, or "n" + 1 or otherwise alters the expression (1), in order to estimate σ². In particular, using "n" − 1 makes the estimator unbiased, and in some often considered contexts, using "n" + 1 minimizes the mean squared error of estimation.

Statisticians do not normally use Greek letters μ and σ for estimates based on samples, but only for (often) unobservable characteristics of whole populations. Because the "true" or "population" variance uses the denominator 1/"n" rather than 1/("n" − 1), it is conventional among those concerned with computation sometimes to call the expression (1), with the denominator 1/"n", the "true variance" without regard to whether it is an estimate or a characteristic of whole population or a random sample.

References

* Andrae, von (1872). Über die Bestimmung des wahrscheinlichen Fehlers durch die gegebenen Differenzen vom gleich genauen Beobachtungen einer Unbekannten. "Astronomische Nachrichten", vol. 84.

* Helmert, F.R. (1876). Die Berechnung des wahrscheinlichen Beobachtungsfehlers aus den ersten Potenzen der Differenzen gleichgenauer director Beobachtungen. "Astronomische Nachrichten", vol. 88.

* Kendall, M. (1943). "The Advanced Theory of Statistics". In Stuart, A., & Ord, J.K. (1987) "Kendall’s Advanced Theory of Statistics", 5th Ed. London: Griffin.
* Press, W. H., Teukolsky, S.A., Vetterling, W.T., & Flannery, B.P. (1992, 2^nd Ed.). "Numerical Recipes". Cambridge, MA: Cambridge University Press.

*Krus, D.J., & Ceurvorst, R.W. (1979) Dominance, information, and hierarchical scaling of variance space. Applied Psychological Measurement, 3, 515-527 [http://www.visualstatistics.net/Scaling/Dominance%20-%20Information/Dominance-Information.htm (Request reprint).]

*Shannon. C. E., & Weaver, W. (1949) The mathematical theory of communication. Urbana: University of Illinois Press

ee also

* Variance

External links

* [http://www.visualstatistics.net/Scaling/Meaning%20of%20Variance/Variance%20and%20differences%20between%20values%20of%20a%20variable.htm Variance and the differences between values of a variable]