Variance - Wikipedia, The Free Encyclopedia

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VarianceFrom Wikipedia, the free encyclopedia

In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of aprobability distribution, describing how far the numbers lie from the mean (expected value). In particular, the variance is one of the momentof a distribution. In that context, it forms part of a systematic approach to distinguishing between probability distributions. While other suchapproaches have been developed, those based on moments are advantageous in terms of mathematical and computational simplicity.

The variance is a parameter describing in part either the actual probability distribution of an observed population of numbers, or the

theoretical probability distribution of a sample (a not-fully-observed population) of numbers. In the latter case a sample of data from such adistribution can be used to construct an estimate of its variance: in the simplest cases this estimate can be the sample variance.

Contents

1 Basic discussion1.1 Examples1.2 Units of measurement1.3 Estimating the variance

2 Definition

2.1 Continuous random variable2.2 Discrete random variable

3 Examples3.1 Exponential distribution3.2 Fair die

4 Properties4.1 Basic properties4.2 Sum of uncorrelated variables (Bienaym formula)4.3 Product of independent variables4.4 Sum of correlated variables4.5 Weighted sum of variables4.6 Decomposition4.7 Formulae for the variance4.8 Calculation from the CDF4.9 Characteristic property4.10 Matrix notation for the variance of a linear combination

5 Approximating the variance of a function6 Population variance and sample variance

6.1 Distribution of the sample variance6.2 Samuelson's inequality6.3 Relations with the harmonic and arithmetic means

7 Generalizations8 Tests of equality of variances

9 History10 Moment of inertia11 See also12 Notes

Basic discussion

Examples
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The variance of a random variable or distribution is the expectation, or mean, of the squared deviation of that variable from its expected valuor mean. Thus the variance is a measure of the amount of variation of the values of that variable, taking account of all possible values andtheir probabilities or weightings (not just the extremes which give the range).

For example, a perfect six-sided die, when thrown, has expected value of

Its expected absolute deviationthe mean of the equally likely absolute deviations from the meanis

But its expectedsquareddeviationits variance (the mean of the equally likely squared deviations)is

As another example, if a coin is tossed twice, the number of heads is: 0 with probability 0.25, 1 with probability 0.5 and 2 with probability0.25. Thus the expected value of the number of heads is:

and the variance is:

Units of measurement

Unlike expected absolute deviation, the variance of a variable has units that are the square of the units of the variable itself. For example, avariable measured in inches will have a variance measured in square inches. For this reason, describing data sets via their standard deviationor root mean square deviation is often preferred over using the variance. In the dice example the standard deviation is 2.9 1.7, slightlylarger than the expected absolute deviation of 1.5.

The standard deviation and the expected absolute deviation can both be used as an indicator of the "spread" of a distribution. The standarddeviation is more amenable to algebraic manipulation than the expected absolute deviation, and, together with variance and its generalizationcovariance, is used frequently in theoretical statistics; however the expected absolute deviation tends to be more robust as it is less sensitiveto outliers arising from measurement anomalies or an unduly heavy-tailed distribution.

Estimating the variance

Real-world distributions such as the distribution of yesterday's rain throughout the day are typically not fully known, unlike the behavior ofperfect dice or an ideal distribution such as the normal distribution, because it is impractical to account for every raindrop. Instead oneestimates the mean and variance of the whole distribution as the computed mean and variance of a sample ofn observations drawn suitablyrandomly from the whole sample space, in this example the set of all measurements of yesterday's rainfall in all available rain gauges.

This method of estimation is close to optimal, with the caveat that it underestimates the variance by a factor of ( n 1) / n. (For example,when n = 1 the variance of a single observation is obviously zero regardless of the true variance). This gives a bias which should be correctefor when n is small by multiplying by n / (n 1). If the mean is determined in some other way than from the same samples used to estimatethe variance then this bias does not arise and the variance can safely be estimated as that of the samples.

To illustrate the relation between the population variance and the sample variance, suppose that in the (not entirely observed) population ofnumerical values, the value 1 occurs 1/3 of the time, the value 2 occurs 1/3 of the time, and the value 4 occurs 1/3 of the time. The populatiomean is (1/3)[1 + 2 + 4] = 7/3. The equally likely deviations from the population mean are 1 7/3, 2 7/3, and 4 7/3. The populationvariance the expected squared deviation from the mean 7/3 is (1/3)[(4/3)2 + (1/3)2 + (5/3)2] = 14/9. Now suppose for the sake a simple example that we take a very small sample ofn = 2 observations, and consider the nine equally likely possibilities for the set ofnumbers within that sample: (1, 1), (1, 2), (1,4), (2, 1), (2,2), (2, 4), (4,1), (4, 2), and (4, 4). For these nine possible samples, the samplevariance of the two numbers is respectively 0, 1/4, 9/4, 1/4, 0, 4/4, 9/4, 4/4, and 0. With our plan to observe two values, we could end up
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computing any of these sample variances (and indeed if we hypothetically could observe a pair of numbers many times, we would computeeach of these sample variances 1/9 of the time). So the expected value, over all possible samples that might be drawn from the population, the computed sample variance is (1/9)[0 + 1/4 + 9/4 + 1/4 + 0 + 4/4 + 9/4 + 4/4 + 0] = 7/9. This value of 7/9 for the expected value of ousample variance computation is a substantial underestimate of the true population variance, which we computed as 14/9, because our sampsize of just two observations was so small. But if we adjust for this downward bias by multiplying our computed sample variance, whicheverit may be, by n/(n 1) = 2/(2 1) = 2, then our estimate of the population variance would be any one of 0, 1/2, 9/2, 1/2, 0, 4/2, 9/2, 4/2,and 0. The average of these is indeed the correct population variance of 14/9, so on average over all possible samples we would have thecorrect estimate of the population variance.

The variance of a real-valued random variable is its second central moment, and it also happens to be its second cumulant. Just as somedistributions do not have a mean, some do not have a variance. The mean exists whenever the variance exists, but the converse is notnecessarily true.

Definition

If a random variableXhas the expected value (mean) = E[X], then the variance ofXis the covariance ofXwith itself, given by:

That is, the variance is the expected value of the squared difference between the variable's realization and the variable's mean. This definitioencompasses random variables that are discrete, continuous, neither, or mixed. From the corresponding expression for covariance, it can bexpanded:

A mnemonic for the above expression is "mean of square minus square of mean". The variance of random variableXis typically designatedas Var(X), , or simply 2 (pronounced "sigma squared").

Continuous random variable

If the random variableXis continuous with probability density functionf(x), then the variance equals the second central moment, given by

where is the expected value,

and where the integrals are definite integrals taken forx ranging over the range ofX.

If a continuous distribution does not have an expected value, as is the case for the Cauchy distribution, it does not have a variance either.Many other distributions for which the expected value does exist also do not have a finite variance because the integral in the variancedefinition diverges. An example is a Pareto distribution whose index ksatisfies 1 < k 2.

Discrete random variable

If the random variableXis discrete with probability mass functionx1p1, ...,xnpn, then
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where is the expected value, i.e.

.

(When such a discrete weighted variance is specified by weights whose sum is not 1, then one divides by the sum of the weights.) That is, it the expected value of the square of the deviation ofXfrom its own mean. In plain language, it can be expressed as "The mean of the squareof the deviations of the data points from the average". It is thus the mean squared deviation.

The variance of a set ofn equally likely values can be written as

where 1/n is the probability of each value.

The variance of a set ofn equally likely values can be expressed, without directly referring to the mean, in terms of squared deviations of allpoints from each other, as

Examples

Exponential distribution

The exponential distribution with parameter is a continuous distribution whose support is the semi-infinite interval [0,). Its probabilitydensity function is given by:

and it has expected value = 1. Therefore the variance is equal to:

So for an exponentially distributed random variable 2 = 2.

Fair die

A six-sided fair die can be modelled with a discrete random variable with outcomes 1 through 6, each with equal probability . The expectevalue is (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5. Therefore the variance can be computed to be:

The general formula for the variance of the outcomeXof a die of n sides is:
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PropertiesBasic properties

Variance is non-negative because the squares are positive or zero.

The variance of a constant random variable is zero, and if the variance of a variable in a data set is 0, then all the entries have the same valu

Variance is invariant with respect to changes in a location parameter. That is, if a constant is added to all values of the variable, the varianceunchanged.

If all values are scaled by a constant, the variance is scaled by the square of that constant.

The variance of a sum of two random variables is given by:

In general we have for the sum of random variables:

These results lead to the variance of a linear combination as:

The variance of a finite sum ofuncorrelatedrandom variables is equal to the sum of their variances. This stems from the above identity andthe fact that for uncorrelated variables the covariance is zero; that is, if
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then

Sum of uncorrelated variables (Bienaym formula)

See also: Sum of normally distributed random variables

One reason for the use of the variance in preference to other measures of dispersion is that the variance of the sum (or the difference) ofuncorrelated random variables is the sum of their variances:

This statement is called the Bienaym formula.[1] and was discovered in 1853.[citation needed] It is often made with the stronger conditionthat the variables are independent, but uncorrelatedness suffices. So if all the variables have the same variance 2, then, since division by na linear transformation, this formula immediately implies that the variance of their mean is

That is, the variance of the mean decreases when n increases. This formula for the variance of the mean is used in the definition of thestandard error of the sample mean, which is used in the central limit theorem.

Product of independent variables

If two variables X and Y are independent, the variance of their product is given by[2][3]

Sum of correlated variables

In general, if the variables are correlated, then the variance of their sum is the sum of their covariances:

(Note: This by definition includes the variance of each variable, since Cov(Xi,Xi) = Var(Xi).)

Here Cov is the covariance, which is zero for independent random variables (if it exists). The formula states that the variance of a sum isequal to the sum of all elements in the covariance matrix of the components. This formula is used in the theory of Cronbach's alpha in classictest theory.

So if the variables have equal variance 2 and the average correlation of distinct variables is, then the variance of their mean is

This implies that the variance of the mean increases with the average of the correlations. Moreover, if the variables have unit variance, forexample if they are standardized, then this simplifies to
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This formula is used in the SpearmanBrown prediction formula of classical test theory. This converges to ifn goes to infinity, provided ththe average correlation remains constant or converges too. So for the variance of the mean of standardized variables with equal correlationsor converging average correlation we have

Therefore, the variance of the mean of a large number of standardized variables is approximately equal to their average correlation. Thismakes clear that the sample mean of correlated variables does generally not converge to the population mean, even though the Law of largenumbers states that the sample mean will converge for independent variables.

Weighted sum of variables

The scaling property and the Bienaym formula, along with this property from the covariance page: Cov(aX, bY) = ab Cov(X, Y) jointlyimply that

This implies that in a weighted sum of variables, the variable with the largest weight will have a disproportionally large weight in the variance

of the total. For example, ifXand Yare uncorrelated and the weight ofXis two times the weight ofY, then the weight of the variance ofXwill be four times the weight of the variance ofY.

The expression above can be extended to a weighted sum of multiple variables:

Decomposition

The general formula for variance decomposition or the law of total variance is: If and are two random variables and the variance ofexists, then

Here, is the conditional expectation of given , and is the conditional variance of given . (A more intuitexplanation is that given a particular value of , then follows a distribution with mean and variance . The abovformula tells how to find based on the distributions of these two quantities when is allowed to vary.) This formula is oftenapplied in analysis of variance, where the corresponding formula is

here refers to the Mean of the Squares. It is also used in linear regression analysis, where the corresponding formula is

This can also be derived from the additivity of variances, since the total (observed) score is the sum of the predicted score and the errorscore, where the latter two are uncorrelated.

Similar decompositions are possible for the sum of squared deviations (sum of squares, ):

Formulae for the variance
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Main article: algebraic formula for the variance

A formula often used for deriving the variance of a theoretical distribution is as follows:

This will be useful when it is possible to derive formulae for the expected value and for the expected value of the square.

This formula is also sometimes used in connection with the sample variance. While useful for hand calculations, it is not advised for computecalculations as it suffers from catastrophic cancellation if the two components of the equation are similar in magnitude and floating point

arithmetic is used.[citation needed] This is discussed below.

Calculation from the CDF

The population variance for a non-negative random variable can be expressed in terms of the cumulative distribution functionFusing

whereH(u) = 1 F(u) is the right tail function. This expression can be used to calculate the variance in situations where the CDF, but not thdensity, can be conveniently expressed.

Characteristic property

The second moment of a random variable attains the minimum value when taken around the first moment (i.e., mean) of the random variablei.e. . Conversely, if a continuous function satisfies

for all random variablesX, then it is necessarily of the form , where a > 0. This also holds in the multidimensional

case.[4]

Matrix notation for the variance of a linear combination

Let's define as a column vector of n random variables , and c as a column vector of N scalars . Thereforeis a linear combination of these random variables, where denotes the transpose of vector . Let also be the variance-covariance matr

of the vector X. The variance of is given by:[5]

Approximating the variance of a function

The delta method uses second-order Taylor expansions to approximate the variance of a function of one or more random variables: seeTaylor expansions for the moments of functions of random variables. For example, the approximate variance of a function of one variable isgiven by

provided thatfis twice differentiable and that the mean and variance ofXare finite.

Population variance and sample variance

See also: Unbiased estimation of standard deviation

In general, thepopulation variance of afinite population of sizeNis given by
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where

is the population mean, and

In many practical situations, the true variance of a population is not known a priori and must be computed somehow. When dealing withextremely large populations, it is not possible to count every object in the population.

A common task is to estimate the variance of a population from a sample.[6] We take a sample with replacement ofn valuesy1, ...,yn fromthe population, where n


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Distribution and cumulative distribution ofs2/2, for various values of = n-1, when

theyi are independent normally distributed

while, in contrast,

The use of the term n 1 is called Bessel's correction, and it is also used in sample covariance and the sample standard deviation (the squaroot of variance). The square root is a concave function and thus introduces negative bias (by Jensen's inequality), which depends on thedistribution, and thus the corrected sample standard deviation (using Bessel's correction) is biased. The unbiased estimation of standarddeviation is a technically involved problem, though for the normal distribution using the term n 1.5 yields an almost unbiased estimator.

The unbiased sample variance is a U-statistic for the function (x1,x2) = (x1 x2)2/2, meaning that it is obtained by averaging a 2-sample

statistic over 2-element subsets of the population.

Distribution of the sample variance

Being a function of random variables, the sample variance is itself a random variable, and it isnatural to study its distribution. In the case thatyi are independent observations from a normal

distribution, Cochran's theorem shows thats2 follows a scaled chi-squared distribution:[8]

As a direct consequence, it follows that

and[9]

If theyi are independent and identically distributed, but not necessarily normally distributed,

then[10]

where is the excess kurtosis of the distribution and4 is the fourth moment about the mean.

If the conditions of the law of large numbers hold for the squared observations, s2 is a consistent estimator of2.[citation needed]. One cansee indeed that the variance of the estimator tends asymptotically to zero.

Samuelson's inequality
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Samuelson's inequality is a result that states, given that the sample mean and variance have been calculated from a particular sample, boundon the values that individual values in the sample can take.[11] Values must lie within the limits m s (n 1)1/2 .

Relations with the harmonic and arithmetic means

It has been shown[12] that for a sample of real numbers that

whereMis the maximum of the sample,A is the arithmetic mean,His the harmonic mean of the sample and Varis the variance of thesample.

This bound has been improved on and it is known that variance is bounded by

where m is the minimum of the sample.[13]

Generalizations

If is a vector-valued random variable, with values in , and thought of as a column vector, then the natural generalization of variance i, where and is the transpose of , and so is a row vector. This variance is a positive sem

definite square matrix, commonly referred to as the covariance matrix.

If is a complex-valued random variable, with values in , then its variance is , where is the conjugate

transpose of . This variance is also a positive semi-definite square matrix.

Tests of equality of variances

Testing for the equality of two or more variances is difficult. The F test and chi square tests are both sensitive to non normality and are notrecommended for this purpose.

Several non parametric tests have been proposed: these include the Barton-David-Ansari-Fruend-Siegel-Tukey test, the Capon test, Moodtest, the Klotz test and the Sukhatme test. The Sukhatme test applies to two variances and requires that both medians be known and equal zero. The Mood, Klotz, Capon and Barton-David-Ansari-Fruend-Siegel-Tukey tests also apply to two variances. They allow the median tbe unknown but do require that the two medians are equal.

The Lehman test is a parametric test of two variances. Of this test there are several variants known. Other tests of the equality of variancesinclude the Box test, the Box-Anderson test and the Moses test.

Resampling methods, which include the bootstrap and the jackknife, may be used to test the equality of variances.

History

The term variance was first introduced by Ronald Fisher in his 1918 paperThe Correlation Between Relatives on the Supposition ofendelian Inheritance:[14]

The great body of available statistics show us that the deviations of a human measurement from its mean follow very closely theNormal Law of Errors, and, therefore, that the variability may be uniformly measured by the standard deviation correspondingto the square root of the mean square error. When there are two independent causes of variability capable of producing in an
http://en.wikipedia.org/wiki/Mean_square_errorhttp://en.wikipedia.org/wiki/Square_roothttp://en.wikipedia.org/wiki/Standard_deviationhttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Biometryhttp://en.wikipedia.org/wiki/Variance#cite_note-14http://en.wikipedia.org/wiki/The_Correlation_Between_Relatives_on_the_Supposition_of_Mendelian_Inheritancehttp://en.wikipedia.org/wiki/Ronald_Fisherhttp://en.wikipedia.org/wiki/Resampling_(statistics)http://en.wikipedia.org/wiki/Bootstraphttp://en.wikipedia.org/w/index.php?title=Moses_test&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Box-Anderson_test&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Box_test&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Lehman_test&action=edit&redlink=1http://en.wikipedia.org/wiki/Medianhttp://en.wikipedia.org/w/index.php?title=Sukhatme_test&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Klotz_test&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Mood_test&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Capon_test&action=edit&redlink=1http://en.wikipedia.org/wiki/Chi_square_testhttp://en.wikipedia.org/wiki/F_testhttp://en.wikipedia.org/wiki/Conjugate_transposehttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Covariance_matrixhttp://en.wikipedia.org/wiki/Positive_definite_matrixhttp://en.wikipedia.org/wiki/Vector_spacehttp://en.wikipedia.org/wiki/Variance#cite_note-Sharma2008-13http://en.wikipedia.org/wiki/Harmonic_meanhttp://en.wikipedia.org/wiki/Variance#cite_note-12http://en.wikipedia.org/wiki/Variance#cite_note-11http://en.wikipedia.org/wiki/Samuelson%27s_inequality


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otherwise uniform population distributions with standard deviations and , it is found that the distribution, when both causes

act together, has a standard deviation . It is therefore desirable in analysing the causes of variability to deal with the

square of the standard deviation as the measure of variability. We shall term this quantity the Variance...

Moment of inertia

The variance of a probability distribution is analogous to the moment of inertia in classical mechanics of a corresponding mass distributionalong a line, with respect to rotation about its center of mass.[citation needed] It is because of this analogy that such things as the variance ar

called moments of probability distributions.[citation needed] The covariance matrix is related to the moment of inertia tensor for multivariatedistributions. The moment of inertia of a cloud ofn points with a covariance matrix of is given by[citation needed]

This difference between moment of inertia in physics and in statistics is clear for points that are gathered along a line. Suppose many pointsare close to thex axis and distributed along it. The covariance matrix might look like

That is, there is the most variance in thex direction. However, physicists would consider this to have a low moment aboutthex axis so themoment-of-inertia tensor is

See also

Algorithms for calculating varianceAverage absolute deviation

BhatiaDavis inequalityCommon-method varianceCorrelationCovarianceChebyshev's inequalityDistance varianceEstimation of covariance matricesExplained varianceHomoscedasticityMean absolute errorMean difference

Mean preserving spreadPooled variancePopoviciu's inequality on variancesQualitative variationSample mean and covarianceSemivarianceSkewnessTaylor's lawWeighted sample variance

Notes
http://en.wikipedia.org/wiki/Weighted_mean#Weighted_sample_variancehttp://en.wikipedia.org/wiki/Taylor%27s_lawhttp://en.wikipedia.org/wiki/Skewnesshttp://en.wikipedia.org/wiki/Semivariancehttp://en.wikipedia.org/wiki/Sample_mean_and_covariancehttp://en.wikipedia.org/wiki/Qualitative_variationhttp://en.wikipedia.org/wiki/Popoviciu%27s_inequality_on_varianceshttp://en.wikipedia.org/wiki/Pooled_variancehttp://en.wikipedia.org/wiki/Mean_preserving_spreadhttp://en.wikipedia.org/wiki/Mean_differencehttp://en.wikipedia.org/wiki/Mean_absolute_errorhttp://en.wikipedia.org/wiki/Homoscedasticityhttp://en.wikipedia.org/wiki/Explained_variancehttp://en.wikipedia.org/wiki/Estimation_of_covariance_matriceshttp://en.wikipedia.org/wiki/Distance_variancehttp://en.wikipedia.org/wiki/Chebyshev%27s_inequalityhttp://en.wikipedia.org/wiki/Covariancehttp://en.wikipedia.org/wiki/Correlationhttp://en.wikipedia.org/wiki/Common-method_variancehttp://en.wikipedia.org/wiki/Bhatia%E2%80%93Davis_inequalityhttp://en.wikipedia.org/wiki/Average_absolute_deviationhttp://en.wikipedia.org/wiki/Algorithms_for_calculating_variancehttp://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/Moment_of_inertia_tensorhttp://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/Probability_distributionhttp://en.wikipedia.org/wiki/Moment_(mathematics)http://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Moment_of_inertia


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1. ^ Loeve, M. (1977) "Probability Theory", Graduate Texts in Mathematics, Volume 45, 4th edition, Springer-Verlag, p. 12.2. ^ Goodman, Leo A., "On the exact variance of products,"Journal of the American Statistical Association, December 1960, 708713.3. ^ Goodman, Leo A., "The variance of the product of K random variables," Journal of the American Statistical Association, March 1962, 544. ^ Kagan, A.; Shepp, L. A. (1998). "Why the variance?". Statistics & Probability Letters38 (4): 329. doi:10.1016/S0167-7152(98)00041-8

(http://dx.doi.org/10.1016%2FS0167-7152%2898%2900041-8).5. ^ Johnson, Richard; Wichern, Dean (2001),Applied Multiv ariate Statistical Analysis, Prentice Hall, p. 76, ISBN 0-13-187715-16. ^ Navidi, William (2006) Statistics for Engineers and Scientists, McGraw-Hill, pg 14.7. ^ Montgomery, D. C. and Runger, G. C. (1994) Applied statistics and probability f or engineers, page 201. John Wiley & Sons New York8. ^ Knight K. (2000),Mathematical Statistics, Chapman and Hall, New York. (proposition 2.11)

9. ^ Casella and Berger (2002) Statistical Inference, Example 7.3.3, p. 33110. ^ Neter, Wasserman, and Kutner (1990) Applied Linear Statistical Models, 3rd edition, pp. 622-62311. ^ Samuelson, Paul (1968)"How Deviant Can You Be?", Journal of the American Statistical Association, 63, number 324 (December, 1968)

pp. 15221525 JSTOR 2285901 (http://ww w.jstor.org/stable/2285901)12. ^ Mercer A McD (2000) Bounds for A-G, A-H, G-H, and a family of inequalities of Ky Fans type, using a general method. J Math Anal App

243, 16317313. ^ Sharma R (2008) Some more inequalities for arithmetic mean, harmonic mean and variance. J Math Inequalities 2 (1) 10911414. ^ Ronald Fisher (1918) The c orrelation between relatives on the supposition of Mendelian Inheritance

(http://digital.library.adelaide.edu.au/dspace/bitstream/2440/15097/1/9.pdf)

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