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In Probability Theory and Statistics , the variance of a Random Variable (or somewhat more precisely, of a Probability Distribution ) is a measure of its Statistical Dispersion , indicating how its possible values are spread around the Expected Value . While the expected value shows the location of the distribution, the variance indicates the variability of the values. A more understandable measure is the square root of the variance, called the Standard Deviation . As its name implies it gives in a standard form an indication of the usual deviations from the mean. The variance of a Real -valued random variable is its second Central Moment , and it also happens to be its second Cumulant . ELEMENTARY DESCRIPTION The ''variance'' of a list of numbers expresses how large the differences between the list elements are. It can be defined in several ways such as the following algorithm: compute the difference between each possible pair of numbers; square the differences; compute the mean of these squares; divide this by 2. The resulting value is the variance. The squaring is done to treat negative and positive differences alike - they need to add up, rather than cancel each other. In principle, this can be done by taking the absolute values (i.e., just dropping the signs), but squaring is more convenient for mathematicians, as the squared function is differentiable for all real numbers, and the absolute value is non-differentiable at zero. The variance increases as the differences between the numbers increase. Hence, it is a measure of dispersion. The same result could have been obtained using another process, which is the second definition: Compute the mean; subtract the mean from each number (the outcomes are called "deviations"); square the deviations; take the mean of these squares. This will have the same outcome as the first definition, with less work. The variance increases if the differences between the numbers and mean increases. Hence, the variance can also be viewed as a measure for size of the deviations from the mean. That is, it says how far away the numbers are from their mean. If the variance is small, then most numbers are close to the mean. If the first definition is considered again with an example, then it becomes clear that something special happens. Suppose the numbers are simply 1, 2, 3, 4. The differences can be arranged in a table: ''1'' ''2'' ''3'' ''4'' ''1'' 0 1 2 3 ''2'' −1 0 1 2 ''3'' −2 −1 0 1 ''4'' −3 −2 −1 0 The squared differences are 0 1 4 9 1 0 1 4 4 1 0 1 9 4 1 0 So the variance is 0.5 × (0 + 1 + ... + 1 + 0) / 16 = 1.25. However, there are zeros on the diagonal. The diagonal always contains zeros, because a number subtracted from itself yields zero. So it could be argued that the diagonal should not be counted when computing the mean of the squares. That is, the sum of squares should be divided not by 16, but by 12. If that is done, then the variance would be 0.5 × (0 + 1 + ... + 1 + 0) /12 = 1.667. In general, if the number of elements is ''n'', then the number of off-diagonal squares is ''n''(''n'' − 1), so this example can now be generalized into a third definition of the variance: it is half of the sum of the squares of the pairwise differences divided by the number of distinct pairs, ''n''(''n'' − 1). If the second definition is modified such that it produces the same result, then the fourth definition is obtained: The sum of the Squared Deviations from the mean, divided by ''n'' − 1. Definitions 1 and 2 give always the same result, and result of definitions 3 and 4 is also the same but slightly different from that of 1 and 2. The variance according to the definitions 3 or 4 is sometimes called the 'unbiased estimate'. The definition of variance applies to a list of numbers, or more generally a variable, rather than to a set of numbers. The implication of this is that if a number occurs 20 times in the list, then it should be entered 20 times in the computation. All the above definitions are valid for a finite population, where each element has an equal weight. However, statisticians also want to deal with infinite populations, and for this reason the more formal definition below has been developed. It corresponds to the second definition above, but it is more general in that it also applies to infinite populations and continuous variables. DEFINITION If μ = E(''X'') is the Expected Value (mean) of the random variable ''X'', then the variance is :$\backslash operatorname\{Var\}(X)\; =\; \backslash operatorname\{E\}(\; (\; X\; -\; \backslash mu\; )\; ^\; 2\; ).\backslash ,$ If the random variable is Discrete with Probability Mass Function ''p''2, ..., ''p''''n'', this is equivalent to :$\backslash sum\_\{i=1\}^n\; (x\_i\; -\; \backslash mu)^2\; p\_i\backslash ,.$ (Note: this variance should be divided by the sum of weights in the case of a discrete Weighted Variance .) That is, it is the expected value of the Square Of The Deviation of ''X'' from its own mean. In plain language, it can be expressed as "The average of the square of the distance of each data point from the mean". It is thus the ''mean squared deviation''. The variance of random variable ''X'' is typically designated as Var(''X''), $\backslash scriptstyle\backslash sigma\_X^2$, or simply σ2. Note that the above definition can be used for both Discrete and Continuous random variables. Of all the points about which squared deviations could have been calculated, it is fairly easy to prove that using the mean produces the minimum value for the sum (and average) of squared deviations. Many distributions, such as the Cauchy Distribution , do not have a variance because the relevant integral diverges. In particular, if a distribution does not have an expected value, it does not have a variance either. The converse is not true: there are distributions for which the expected value exists, but the variance does not. PROPERTIES, INTRODUCTION # Variance is non-negative because the squares are positive or zero. # If all values of a random variable are equal, then its variance is 0. For example, the variance of 2, 2, 2, 2 is 0. # In a finite population or sample, if some elements of the variable are unequal, then the variance is larger than 0. For example, the variance of 2, 2, 2, 3 is larger than 0. The variance of −1, −2, −3 is also positive. # In a finite population, if the list is extended with a number that is equal to the mean, then the variance decreases unless it was 0. For example, the variance of 1, 2, 3 is smaller than the variance of 1, 3. # The unit of variance is the square of the unit of observation. For example, the variance of a set of heights measured in centimeters will be given in square centimeters. This fact is inconvenient and has motivated many statisticians to use instead the square root of the variance, known as the Standard Deviation , as a summary of dispersion. # Scaling: ## Adding a constant: If a constant number is added to all values of the variable, then the variance does not change. For example, the variance of 1, 2, 3, 4 is 1.25 and the variance of 11, 12, 13, 14 is also 1.25. ## Multiplying by a constant: If the values of the variable are multiplied by a constant number, then the variance is multiplied by the square of the constant. For example, the variance of 1, 2, 3, 4 is 1.25 and the variance of 10, 20, 30, 40 is 125. In this example, the values of the variable were multiplied by 10 and then the variance is multiplied by 100. This is related to property 5. ## Properties 6.1 (adding a constant) and 6.2 (multiplying by a constant) jointly determine what happens with the variance after a scale transformation of the values: If ''a'' and ''b'' are constant numbers, and the variables ''X'' and ''Y'' are related by ''Y'' = ''aX'' + ''b'', then Var(''Y'') = ''a''2 Var(''X''). For example, suppose that the temperatures on several days have been measured in degrees Celsius and that the variance was 10. Suppose that the temperatures are converted to degrees Fahrenheit . These two temperature scales are related by the equation °F = 1.8 × °C + 32. So with ''a'' = 1.8 and ''b'' = 32 we obtain that the variance of the list of converted temperatures will be 1.8 × 1.8 × 10 = 32.4. # Chebyshev's Inequality : The fraction of values of which the distance from the mean is larger than or equal to some positive number ''a'', is at most Var(''X'')/''a''2. Although there are many other mathematical formulas that involve the variance, this one is especially important because it illuminates the role of the variance as a measure of dispersion. According to this inequality, the variance of a variable provides information about the percentage of values that lie far away from the mean. For example, if you know that the variance of a variable is 10, and you want to know how many values are at least 5 units away from the mean, then Chebyshev's inequality implies that this is at most 10 / (5 × 5) = 0.4, or 40% of the values. # The variance of a finite sum of uncorrelated random variables is equal to the sum of their variances. For example, suppose that in a population of married couples the hourly income of the women is independent of the income of the men. Suppose that income of the women has variance 100 and the income of the men has variance 200. Then the variance of their joint income is 100 + 200 = 300. Another example is this: We have seen that the variance of 1, 2, 3, 4 is 1.25. So if 40 persons draw independently a random number from this list, and we add their choices, then the variance of the sum will be 40 × 1.25 = 50. # Suppose that the observations can be partitioned into subgroups according to some second variable. Then the variance of the total group is equal to the mean of the variances of the subgroups plus the variance of the means of the subgroups. This property is known as Variance Decomposition or the Law Of Total Variance and plays an important role in the Analysis Of Variance . For example, suppose that a group consists of a subgroup of men and an equally large subgroup of women. Suppose that the men have a mean body length of 180 and that the variance of their lengths is 100. Suppose that the women have a mean length of 160 and that the variance of their lengths is 50. Then the mean of the variances is (100 + 50) / 2 = 75; the variance of the means is the variance of 180, 160 which is 100. Then, for the total group of men and women combined, the variance of the body lengths will be 75 + 100 = 175. In a more general case, if the subgroups have unequal sizes, then they must be weighted proportionally to their size in the computations of the means and variances. The formula is also valid with more than two groups, and even if the grouping variable is continuous. This formula implies that the variance of the total group cannot be smaller than the mean of the variances of the subgroups. In general, if you combine subgroups with different means, then the variance will become larger. In the above example, when the subgroups are analyzed separately, the variance is influenced only by the man-man differences and the woman-woman differences. If the two groups are combined, however, then the men-women differences enter into the variance also. # Many computational formulas for the variance are based on this equality: The variance is equal to the mean of the squares minus the square of the mean. For example, if we consider the numbers 1, 2, 3, 4 then the mean of the squares is (1 × 1 + 2 × 2 + 3 × 3 + 4 × 4) / 4 = 7.5. The mean is 2.5, so the square of the mean is 6.25. Therefore the variance is 7.5 − 6.25 = 1.25, which is indeed the same result obtained earlier with the definition formulas. Many pocket calculators use an algorithm that is based on this formula and that allows them to compute the variance while the data are entered, without storing all values in memory. The algorithm is to adjust only three variables when a new data value is entered: The number of data up to that moment (''n''), the sum of the values up to that moment (''S''), and the sum of the squared values up to that moment (''SS''). For example, if the data are 1, 2, 3, 4, then after entering the first value, the algorithm would have ''n'' = 1, ''S'' = 1 and ''SS'' = 1. After entering the second value (2), it would have ''n'' = 2, ''S'' = 3 and ''SS'' = 5. When all data are entered, it would have ''n'' = 4, ''S'' = 10 and ''SS'' = 30. Next, the mean is computed as ''M'' = ''S'' / ''n'', and finally the variance is computed as ''SS'' / ''n'' − ''M'' × ''M''. In this example the outcome would be 30 / 4 - 2.5 × 2.5 = 7.5 − 6.25 = 1.25. If the unbiased sample estimate is to be computed, the outcome will be multiplied by ''n'' / (''n'' − 1), which yields 1.667 in this example. PROPERTIES, FORMAL Some of the properties listed in the previous section deserve a more formal treatment, which is done in this section. The numbering is the same as in the previous section. 6. ''Effect of a linear transformation'' It can be shown from the definition that the variance does not depend on the mean value μ. That is, if the variable is "displaced" an amount ''b'' by taking ''X'' + ''b'', the variance of the resulting random variable is left untouched. By contrast, if the variable is multiplied by a scaling factor ''a'', the variance is multiplied by ''a''2. More formally, if ''a'' and ''b'' are real constants and ''X'' is a Random Variable whose variance is defined, then :$\backslash operatorname\{Var\}(aX+b)=a^2\backslash operatorname\{Var\}(X).$ 8.a. ''Variance of the sum of uncorrelated 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) of Uncorrelated random variables is the sum of their variances: :$\backslash operatorname\{Var\}\backslash Big(\backslash sum\_\{i=1\}^n\; X\_i\backslash Big)\; =\; \backslash sum\_\{i=1\}^n\; \backslash operatorname\{Var\}(X\_i).$ This statement is often made with the stronger condition that the variables are Independent , but Uncorrelated ness suffices. So if the variables have the same variance σ2, then, since division by ''n'' is a linear transformation, this formula immediately implies that the variance of their mean is : That is, the variance of the mean decreases with ''n''. This fact is used in the definition of the Standard Error of the sample mean, which is used in the Central Limit Theorem . 8.b. ''Variance of the sum of correlated variables'' In general, if the variables are correlated, then the variance of their sum is the sum of their Covariance s: :$\backslash operatorname\{Var\}\backslash left(\backslash sum\_\{i=1\}^n\; X\_i\; ight)\; =\; \backslash sum\_\{i=1\}^n\; \backslash sum\_\{j=1\}^n\; \backslash operatorname\{Cov\}(X\_i,\; X\_j).$ Here Cov is the Covariance , which is zero for independent random variables (if it exists). The formula states that the variance of a sum is equal 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 Classical Test Theory . So if the variables have equal variance σ2 and the average correlation of distinct variables is ρ, 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, for example if they are standardized, then this simplifies to : This formula is used in the Spearman-Brown Prediction Formula of Classical Test Theory . This converges to ρ if ''n'' goes to infinity, provided that the average correlation remains constant or converges too. So for the variance of the mean of standardized variables with equal correlations or converging average correlation we have :$\backslash lim\_\{n\; o\; \backslash infty\}\; \backslash operatorname\{Var\}(\backslash overline\{X\})\; =\; ho.$ Therefore, the variance of the mean of a large number of standardized variables is approximately equal to their average correlation. This makes clear that the sample mean of correlated variables does generally not converge to the population mean, even though the Law Of Large Numbers states that the sample mean will converge for independent variables. 8.c. ''Variance of a weighted sum of variables'' Properties 6 and 8, along with this property from the Covariance page: Cov(''aX'', ''bY'') = ''ab'' Cov(''X'', ''Y'') jointly imply that :$\backslash operatorname\{Var\}(aX+bY)\; =a^2\; \backslash operatorname\{Var\}(X)\; +\; b^2\; \backslash operatorname\{Var\}(Y)\; +\; 2ab\backslash ,\; \backslash operatorname\{Cov\}(X,\; Y).$ 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, if ''X'' and ''Y'' are uncorrelated and the weight of ''X'' is two times the weight of ''Y'', then the weight of the variance of ''X'' will be four times the weight of the variance of ''Y''. 9. ''Decomposition of variance'' The general formula for variance decomposition or the Law Of Total Variance is: If ''X'' and ''Y'' are two random variables and the variance of ''X'' exists, then
	Here, E(''X''''Y'') Is The	"http://wwwinformationdelightinfo/information/entry/conditional_expectation" class="copylinks">Conditional Expectation of ''X'' given ''Y'', and Var(''X''''Y'') is the conditional variance of ''X'' given ''Y'' (A more intuitive explanation is that given a particular value of ''Y'', then ''X'' follows a distribution with mean E(''X''''Y'') and variance Var(''X''''Y'') The above formula tells how to find Var(''X'') based on the distributions of these two quantities when ''Y'' is allowed to vary) This formula is often applied in Analysis Of Variance , where the corresponding formula is

It is also used in Linear Regression analysis, where the corresponding formula is

:$SS\_\{\backslash mbox\{Total\}\}\; =\; SS\_\{\backslash mbox\{Regression\}\}\; +\; SS\_\{\backslash mbox\{Residual\}\}.$

This can also be derived from the additivity of variances (property 8), since the total (observed) score is the sum of the predicted score and the error score, where the latter two are uncorrelated.

10. ''Computational formula for variance''

The Computational Formula For The Variance follows in a straightforward manner from the linearity of expected values and the above definition:

:$\{\}\backslash operatorname\{Var\}(X)=\; \backslash operatorname\{E\}(X^2\; -\; 2\backslash ,X\backslash ,\backslash operatorname\{E\}(X)\; +\; (\backslash operatorname\{E\}(X))^2),$

:$\{\}=\backslash operatorname\{E\}(X^2)\; -\; 2(\backslash operatorname\{E\}(X))^2\; +\; (\backslash operatorname\{E\}(X))^2,$

:$\{\}=\backslash operatorname\{E\}(X^2)\; -\; (\backslash operatorname\{E\}(X))^2.$

This is often used to calculate the variance in practice, although it suffers from Numerical Approximation Error if the two components of the equation are similar in magnitude.


APPROXIMATING THE VARIANCE OF A FUNCTION

The Delta Method uses second-order Taylor Expansion s to approximate the variance of a function of one or more random variables. For example, the approximate variance of a function of one variable is given by

::$\backslash operatorname\{Var\}\backslash left[f(X)\; ight]\backslash approx\; \backslash left(f\text{'}(\backslash operatorname\{E\}\backslash left[X\; ight])\; ight)^2\backslash operatorname\{Var\}\backslash left[X\; ight]$

provided that ''f'' is twice differentiable and that the mean and variance of ''X'' are finite.


POPULATION VARIANCE AND SAMPLE VARIANCE


In general, the population variance of a ''finite'' Population of size ''N'' is given by

:
\left(x_i - \overline{x} ight)^ 2 \,

or if the population is an Abstract Population with probability distribution Pr:

:$\{\}\backslash sigma^2\; =\; \backslash sum\_\{i=1\}^N$
\left(x_i - \overline{x} ight)^ 2 \, \Pr(x_i),

where $\backslash overline\{x\}$ is the population mean. This is merely a special case of the general definition of variance introduced above, but restricted to finite populations.

In many practical situations, the true variance of a population is not known ''a priori'' and must be computed somehow. When dealing with infinite populations, this is generally impossible.

A common method is estimating the variance of large (finite or infinite) populations from a Sample . We take a Sample $(y\_1,\backslash dots,y\_n)$ of ''n'' values from the population, and estimate the variance on the basis of this sample. There are several good estimators. Two of them are well known:

:
and
:

Both are referred to as ''sample variance''. Most advanced electronic calculators can calculate both $s\_n^2$ and $s^2$at the press of a button, in which case that button is usually labeled $\backslash sigma^2$ or $\backslash sigma\_n^2$ for $s\_n^2$ and $\backslash sigma\_\{n-1\}^2$ for $s^2$.

The two estimators only differ slightly as we see, and for larger values of the Sample Size ''n'' the difference is negligible. The second one is an Unbiased Estimator of the population variance, meaning that its expected value $E${Link without Title} is equal to the true variance of the sampled random variable. The first one may be seen as the variance of the sample considered as a population.

Common sense would suggest to apply the population formula to the sample as well. The reason that it is biased is that the sample mean is generally somewhat closer to the observations in the sample than the population mean is to these observations. This is so because the sample mean is by definition in the middle of the sample, while the population mean may even lie outside the sample. So the deviations to the sample mean will often be smaller than the deviations to the population mean, and so, if the same formula is applied to both, then this variance estimate will on average be somewhat smaller in the sample than in the population.

One common source of confusion is that the term ''sample variance'' may refer to either the unbiased estimator $s^2$ of the population variance, or to the variance $\backslash sigma^2$ of the sample viewed as a finite population. Both can be used to estimate the true population variance. Apart from theoretical considerations, it doesn't really matter which one is used, as for small Sample Size s both are inaccurate and for large values of ''n'' they are practically the same. Naively computing the variance by dividing by ''n'' instead of ''n''-1 systematically underestimates the population variance. Moreover, in practical applications most people report the standard deviation rather than the sample variance, and the standard deviation that is obtained from the unbiased ''n''-1 version of the sample variance has a slight negative bias (though for normally distributed samples a theoretically interesting but rarely used Slight Correction exists to eliminate this bias). Nevertheless, in applied statistics it is a convention to use the ''n''-1 version if the variance or the standard deviation is computed from a sample.

In practice, for large $n$, the distinction is often a minor one. In the course of statistical measurements, Sample Size s so small as to warrant the use of the unbiased variance
virtually never occur. In this context Press et al.Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. (1986) ''''. Cambridge: Cambridge University Press. ( online ) commented that ''if the difference between ''n'' and ''n''−1 ever matters to you, then you are probably up to no good anyway - e.g., trying to substantiate a questionable hypothesis with marginal data.''


Distribution of the sample variance

Being a function of Random Variable s, the sample variance is itself a random variable, and it is natural to study its distribution. In the case that $y\_i$ are independent Gaussian realizations, Cochran's Theorem shows that $s^2$ follows a scaled Chi-square Distribution :

:$$
(n-1)rac{s^2}{\sigma^2}\sim\chi^2_{n-1}


As a direct consequence, it follows that $\backslash operatorname\{E\}(s^2)=\backslash sigma^2$.

However, even in the absence of the Gaussian assumption, it is still possible to prove that $s^2$ is unbiased for $\backslash sigma^2$:


An unbiased estimator


Abstract proof

Suppose the variables $X\_1,\; \backslash dots,\; X\_n$ are independently identically distributed, representing a sample of size ''n''. Let ''X'' be the variable that is obtained by drawing a random integer number ''i'' between 1 and ''n'' and then observing the outcome of $X\_i$. That is, ''i'' is considered as a random variable. Then the variance of ''X'' is the same as the variance of each of the $X\_i$. Let $\backslash overline\{X\}$ be the sample mean and $V$ the (not bias corrected) sample variance, both computed from $X\_1,\; \backslash dots,\; X\_n$. Using the Variance Decomposition with $Y\; =\; (X\_1,\; \backslash dots,\; X\_n)$, we obtain

:$\backslash operatorname\{Var\}(X)\; =\; \backslash operatorname\{Var\}(\backslash overline\{X\})\; +\; \backslash operatorname\{E\}(V).$

Substitution of the formula for the variance of a mean (see the section about the variance of sums) yields

:$\backslash operatorname\{Var\}(X)\; =\; \backslash operatorname\{Var\}(X)/n\; +\; \backslash operatorname\{E\}(V).$

Solving this yields $\backslash operatorname\{E\}(V)\; =\; \backslash operatorname\{Var\}(X)(n\; -1)/n$. This implies that $V$ is biased and that $V\; n/(n-1)$ is unbiased. The latter statistic is equal to $s^2$.


Specific proof

We will demonstrate why $s^2$ is an Unbiased Estimator of the population variance. An estimator $\backslash hat\{\; heta\}$ for a parameter $heta$ is unbiased if $\backslash operatorname\{E\}(\; \backslash hat\{\; heta\})\; =\; heta$. Therefore, to prove that $s^2$ is unbiased, we will show that $\backslash operatorname\{E\}(\; s^2)\; =\; \backslash sigma^2$. As an assumption, the population which the $x\_i$ are drawn from has mean $\backslash mu$ and variance $\backslash sigma^2$.

:$\backslash operatorname\{E\}\; (\; s^2\; )$

= \operatorname{E} \left( rac{1}{n-1} \sum_{i=1}^n \left( x_i - \overline{x} ight) ^ 2 ight)



::$$

= rac{1}{n-1} \sum_{i=1}^n \operatorname{E} \left( \left( x_i - \overline{x} ight) ^ 2 ight)



::$$

= rac{1}{n-1} \sum_{i=1}^n \operatorname{E} \left( \left( (x_i - \mu) - (\overline{x} - \mu) ight) ^ 2 ight)



::$$

= rac{1}{n-1} \sum_{i=1}^n \left\{ \operatorname{E} \left( (x_i - \mu)^2 ight)

- 2 \operatorname{E} \left( (x_i - \mu) (\overline{x} - \mu) ight)

+ \operatorname{E} \left( (\overline{x} - \mu) ^ 2 ight) ight\}



::$$

= rac{1}{n-1} \sum_{i=1}^n \left[ \sigma^2

- 2 \left( rac{1}{n} \sum_{j=1}^n \operatorname{E} \left( (x_i - \mu) (x_j - \mu) ight) ight)

+ rac{1}{n^2} \sum_{j=1}^n \sum_{k=1}^n \operatorname{E} \left( (x_j - \mu) (x_k - \mu) ight) ight]



::$$

= rac{1}{n-1} \sum_{i=1}^n \left( \sigma^2

- rac{2 \sigma^2}{n}

+ rac{\sigma^2}{n} ight)



::$\{\}$
= rac{1}{n-1} \sum_{i=1}^n rac{(n-1)\sigma^2}{n}


::$\{\}$

= rac{(n-1)\sigma^2}{n-1} = \sigma^2

.


Alternative proof


:$\backslash operatorname\{E\}\backslash left(\; \backslash sum\_\{i=1\}^n\; \{(X\_i-\backslash overline\{X\})^2\}\; ight)$
=\operatorname{E}\left( \sum_{i=1}^n {X_i^2} ight) - n\operatorname{E}\left( \overline{X}^2 ight)


::$$
=n\operatorname{E}\left(X_j^2 ight) - rac{1}{n} \operatorname{E}\left(\left(\sum_{i=1}^n X_i ight)^2 ight)
ext{ for any } j=1\ldots n

::$$
=n(\operatorname{Var}\left(X_j ight) + (\operatorname{E}\left(X_j ight))^2) - rac{1}{n} \operatorname{E}\left(\left(\sum_{i=1}^n X_i ight)^2 ight)


::$$
=n\sigma^2 + rac{1}{n}\left(n\operatorname{E}\left(X_j ight) ight)^2 - rac{1}{n}\operatorname{E}\left(\left(\sum_{i=1}^n X_i ight)^2 ight)


::$$
=n\sigma^2 - rac{1}{n}\left \operatorname{E}\left(\left(\sum_{i=1}^n X_i ight)^2 ight) - \left(\operatorname{E}\left(\sum_{i=1}^n X_i ight) ight)^2 ight


::$$
=n\sigma^2 - rac{1}{n}\operatorname{Var}\left(\sum_{i=1}^n X_i ight)
=n\sigma^2 - rac{1}{n}(n\sigma^2)
=(n-1)\sigma^2.



GENERALIZATIONS


If $X$ is a Vector -valued random variable, with values in $\backslash mathbb\{R\}^n$, and thought of as a column vector, then the natural generalization of variance is $\backslash operatorname\{E\}((X\; -\; \backslash mu)(X\; -\; \backslash mu)^\backslash operatorname\{T\})$, where $\backslash mu\; =\; \backslash operatorname\{E\}(X)$ and $X^\backslash operatorname\{T\}$ is the transpose of $X$, and so is a row vector. This variance is a Positive Semi-definite Square Matrix , commonly referred to as the Covariance Matrix .

• ), where $X^---$ is the Complex Conjugate of $X$. This variance is also a positive semi-definite square matrix.

CHARACTERISTIC PROPERTY

The second moment of a random variable attains the minimum value when taken around the mean of the random variable, i.e. $\backslash mathrm\{E\}\; X\; =\; \backslash mathrm\{argmin\}\_a\; \backslash mathrm\{E\}\; (X\; -\; a)^2$. This property could be reversed, i.e. if the function $\backslash phi$ satisfies $\backslash mathrm\{E\}\; X\; =\; \backslash mathrm\{argmin\}\_a\; \backslash mathrm\{E\}\; \backslash phi(X\; -\; a)$ then it is necessary of the form $\backslash phi\; =\; a\; x^2\; +\; b$. This is also true in multidimensional case A. Kagan and L. A. Shepp, ''Why the variance?''.


HISTORY


The term ''variance'' was first introduced by Ronald Fisher in his 1918 paper '' The Correlation Between Relatives On The Supposition Of Mendelian Inheritance ''.


MOMENT OF INERTIA


The variance of a probability distribution is analogous to the Moment Of Inertia in Classical Mechanics of a corresponding mass distribution along a line, with respect to rotation about its center of mass. It is because of this analogy that such things as the variance are called '' Moment s'' of Probability Distribution s. (The Covariance Matrix is analogous to the moment of inertia tensor for multivariate distributions.)


SEE ALSO





REFERENCES



EXTERNAL LINKS

• Fisher's original paper (pdf format)