08-11-2021-1927 - Variance (various def)

The variance of a random variable XX, with mean EX=μXEX=μX, is defined as
Var(X)=E[(X−μX)2].

https://www.probabilitycourse.com/chapter3/3_2_4_variance.php

In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by , , , , or . [1] [2]
https://en.wikipedia.org/wiki/Variance


Variance

All Law Statistics (Field Genetics)

the fact or quality of being different, divergent, or inconsistent.

the state or fact of disagreeing or quarreling.

Law a discrepancy between two statements or documents.

Oxford Variant

https://languages.oup.com/google-dictionary-en/

Probability and Statistics > Moments >

History and Terminology > Notation >

History and Terminology > Wolfram Language Commands >

Interactive Entries > Interactive Demonstrations >

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

(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)

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

(6)

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

			(7)
			(8)
			(9)
			(10)
			(11)
			(12)

For multiple variables, the variance is given using the definition of covariance,

			(13)
			(14)
			(15)
			(16)
			(17)

A linear sum has a similar form:

			(18)
			(19)
			(20)

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.

Referenced on Wolfram|Alpha: Variance

CITE THIS AS:

Weisstein, Eric W. "Variance." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Variance.html

https://mathworld.wolfram.com/Variance.html

variance

 noun

Save Word

To save this word, you'll need to log in.

Log In  

var·​i·​ance |  \ ˈver-ē-ən(t)s   \

Definition of variance

1: the fact, quality, or state of being variable or variant : DIFFERENCE, VARIATIONyearly variance in crops

2: the fact or state of being in disagreement : DISSENSION, DISPUTE

3: a disagreement between two parts of the same legal proceeding that must be consonant

4: a license to do some act contrary to the usual rulea zoning variance

5: the square of the standard deviation

at variance

: not in harmony or agreement

https://www.merriam-webster.com/dictionary/variance

Genetic variance is a concept outlined by the English biologist and statistician Ronald Fisher in his fundamental theorem of natural selection which he outlined in his 1930 book The Genetical Theory of Natural Selection which postulates that the rate of change of biological fitness can be calculated by the genetic variance of the fitness itself.^[1] Fisher tried to give a statistical formula about how the change of fitness in a population can be attributed to changes in the allele frequency. Fisher made no restrictive assumptions in his formula concerning fitness parameters, mate choices or the number of alleles and loci involved.^[2]

https://en.wikipedia.org/wiki/Genetic_variance