http://mathworld.wolfram.com/Variance.html
Variance
For a single variate having a distribution
with known population mean
, the population variance
, commonly also written
, is defined as
|
(1)
|
where is the population mean and
denotes the expectation value of
. For a discrete distribution with
possible values of
, the population variance is therefore
|
(2)
|
whereas for a continuous distribution, it is given by
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(3)
|
The variance is therefore equal to the second central moment .
Note that some care is needed in interpreting as a variance, since the symbol
is also commonly used as a parameter related to but not equivalent to the square root of the variance, for example in the log normal distribution, Maxwell distribution, and Rayleigh distribution.
If the underlying distribution is not known, then the sample variance may be computed as
|
(4)
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where is the sample mean.
Note that the sample variance defined above is not an unbiased estimator for the population variance
. In order to obtain an unbiased estimator for
, it is necessary to instead define a "bias-corrected sample variance"
|
(5)
|
The distinction between and
is a common source of confusion, and extreme care should be exercised when consulting the literature to determine which convention is in use, especially since the uninformative notation
is commonly used for both. The bias-corrected sample variance
for a list of data is implemented as Variance[list].
The square root of the variance is known as the standard deviation.
The reason that gives a biased estimator of the population variance is that two free parameters
and
are actually being estimated from the data itself. In such cases, it is appropriate to use a Student's t-distribution instead of a normal distribution as a model since, very loosely speaking, Student's t-distribution is the "best" that can be done without knowing
.
Formally, in order to estimate the population variance from a sample of
elements with a priori unknown mean (i.e., the mean is estimated from the sample itself), we need an unbiased estimator for
. This is given by the k-statistic
, where
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(6)
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and is the sample variance uncorrected for bias.
It turns out that the quantity has a chi-squared distribution.
For set of data , the variance of the data obtained by a linear transformation is given by
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(7)
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(8)
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(9)
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(10)
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(11)
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(12)
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For multiple variables, the variance is given using the definition of covariance,
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(13)
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(14)
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(15)
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(16)
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(17)
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A linear sum has a similar form:
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(18)
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(19)
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(20)
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These equations can be expressed using the covariance matrix.
SEE ALSO: Central Moment, Charlier's Check, Covariance, Covariance Matrix, Error Propagation, k-Statistic, Mean, Moment, Raw Moment, Sample Variance, Sample Variance Computation, Sample Variance Distribution, Sigma, Standard Error, Statistical Correlation
REFERENCES:
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.
Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 144-145, 1984.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Moments of a Distribution: Mean, Variance, Skewness, and So Forth." §14.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 604-609, 1992.
Roberts, M. J. and Riccardo, R. A Student's Guide to Analysis of Variance. London: Routledge, 1999.