In

covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...

.

^{2}, then, since division by ''n'' is a linear transformation, this formula immediately implies that the variance of their mean is
:$\backslash operatorname\backslash left(\backslash overline\backslash right)\; =\; \backslash operatorname\backslash left(\backslash frac\; \backslash sum\_^n\; X\_i\backslash right)\; =\; \backslash frac\backslash sum\_^n\; \backslash operatorname\backslash left(X\_i\backslash right)\; =\; \backslash fracn\backslash sigma^2\; =\; \backslash frac.$
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 the

covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...

s:
:$\backslash operatorname\backslash left(\backslash sum\_^n\; X\_i\backslash right)\; =\; \backslash sum\_^n\; \backslash sum\_^n\; \backslash operatorname\backslash left(X\_i,\; X\_j\backslash right)\; =\; \backslash sum\_^n\; \backslash operatorname\backslash left(X\_i\backslash right)\; +\; 2\backslash sum\_\backslash operatorname\backslash left(X\_i,\; X\_j\backslash right).$
(Note: The second equality comes from the fact that .)
Here, $\backslash operatorname(\backslash cdot,\backslash cdot)$ is the covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...

, 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. The next expression states equivalently that the variance of the sum is the sum of the diagonal of covariance matrix plus two times the sum of its upper triangular elements (or its lower triangular elements); this emphasizes that the covariance matrix is symmetric. This formula is used in the theory of Cronbach's alpha in classical test theory.
So if the variables have equal variance ''σ''^{2} and the average

covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...

jointly imply that
:$\backslash operatorname(aX\; \backslash pm\; bY)\; =a^2\; \backslash operatorname(X)\; +\; b^2\; \backslash operatorname(Y)\; \backslash pm\; 2ab\backslash ,\; \backslash operatorname(X,\; Y).$
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''.
The expression above can be extended to a weighted sum of multiple variables:
:$\backslash operatorname\backslash left(\backslash sum\_^n\; a\_iX\_i\backslash right)\; =\; \backslash sum\_^na\_i^2\; \backslash operatorname(X\_i)\; +\; 2\backslash sum\_\backslash sum\_a\_ia\_j\backslash operatorname(X\_i,X\_j)$

_{''i''} is given by
:$$\backslash begin\; \backslash sigma^2\; \&=\; \backslash frac\; \backslash sum\_^N\; \backslash left(x\_i\; -\; \backslash mu\backslash right)^2\; =\; \backslash frac\; \backslash sum\_^N\; \backslash left(x\_i^2\; -\; 2\backslash mu\; x\_i\; +\; \backslash mu^2\; \backslash right)\; \backslash \backslash ;\; href="/html/ALL/s/pt.html"\; ;"title="pt">pt$$
where the population mean is
: $\backslash mu\; =\; \backslash frac\; 1N\; \backslash sum\_^N\; x\_i.$
The population variance can also be computed using
:$\backslash sigma^2\; =\; \backslash frac\; \backslash sum\_\backslash left(\; x\_i-x\_j\; \backslash right)^2\; =\; \backslash frac\; \backslash sum\_^N\backslash left(\; x\_i-x\_j\; \backslash right)^2.$
This is true because
:$$\backslash begin\; \&\backslash frac\; \backslash sum\_^N\backslash left(\; x\_i\; -\; x\_j\; \backslash right)^2\; \backslash \backslash ;\; href="/html/ALL/s/pt.html"\; ;"title="pt">pt$$
The population variance matches the variance of the generating probability distribution. In this sense, the concept of population can be extended to continuous random variables with infinite populations.

_{1}, ..., ''Y''_{''n''} from the population, where ''n'' < ''N'', and estimate the variance on the basis of this sample. Directly taking the variance of the sample data gives the average of the squared deviations:
:$\backslash tilde\_Y^2\; =\; \backslash frac\; \backslash sum\_^n\; \backslash left(Y\_i\; -\; \backslash overline\backslash right)^2\; =\; \backslash left(\backslash frac\; 1n\; \backslash sum\_^n\; Y\_i^2\backslash right)\; -\; \backslash overline^2\; =\; \backslash frac\; \backslash sum\_\backslash left(Y\_i\; -\; Y\_j\backslash right)^2.$
Here, $\backslash overline$ denotes the _{''i''} are selected randomly, both $\backslash overline$ and $\backslash tilde\_Y^2$ are random variables. Their expected values can be evaluated by averaging over the ensemble of all possible samples of size ''n'' from the population. For $\backslash tilde\_Y^2$ this gives:
:$\backslash begin\; \backslash operatorname;\; href="/html/ALL/s/tilde\_Y^2.html"\; ;"title="tilde\_Y^2">tilde\_Y^2$
Hence $\backslash tilde\_Y^2$ gives an estimate of the population variance that is biased by a factor of $\backslash frac$. For this reason, $\backslash tilde\_Y^2$ is referred to as the ''biased sample variance''.

_{1}, ''y''_{2}) = (''y''_{1} − ''y''_{2})^{2}/2, meaning that it is obtained by averaging a 2-sample statistic over 2-element subsets of the population.

_{''i''} are independent observations from a ^{2} follows a scaled _{''i''} are independent and identically distributed, but not necessarily normally distributed, then
:$\backslash operatorname\backslash left;\; href="/html/ALL/s/^2\backslash right.html"\; ;"title="^2\backslash right">^2\backslash right$
where ''κ'' is the _{4} is the fourth ^{2} is a ^{2}. One can see indeed that the variance of the estimator tends asymptotically to zero. An asymptotically equivalent formula was given in Kenney and Keeping (1951:164), Rose and Smith (2002:264), and Weisstein (n.d.).

_{max} is the maximum of the sample, ''A'' is the arithmetic mean, ''H'' is the _{min} is the minimum of the sample.

probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

and statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, variance is the expectation of the squared deviation of a random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...

from its population mean
In statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and ...

or sample mean
The sample mean (or "empirical mean") and the sample covariance are statistics computed from a Sample (statistics), sample of data on one or more random variables.
The sample mean is the average value (or mean, mean value) of a sample (statistic ...

. 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
A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features from a collection of information, while descriptive statistics (in the mass noun sense) is the process of using and ana ...

, statistical inference
Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical ...

, hypothesis testing, goodness of fit
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measure ...

, 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
In statistics, the standard Deviation (statistics), deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (al ...

, the second central moment
In probability theory and statistics, a central moment is a moment (mathematics), moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the de ...

of a distribution, and the covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...

of the random variable with itself, and it is often represented by $\backslash sigma^2$, $s^2$, $\backslash operatorname(X)$, $V(X)$, or $\backslash mathbb(X)$.
An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard deviation is more commonly reported as a measure of dispersion once the calculation is finished.
There are two distinct concepts that are both called "variance". One, as discussed above, is part of a theoretical probability distribution
In probability theory and statistics, a probability distribution is the mathematical Function (mathematics), function that gives the probabilities of occurrence of different possible outcomes for an Experiment (probability theory), experiment. ...

and is defined by an equation. The other variance is a characteristic of a set of observations. When variance is calculated from observations, those observations are typically measured from a real world system. If all possible observations of the system are present then the calculated variance is called the population variance. Normally, however, only a subset is available, and the variance calculated from this is called the sample variance. The variance calculated from a sample is considered an estimate of the full population variance. There are multiple ways to calculate an estimate of the population variance, as discussed in the section below.
The two kinds of variance are closely related. To see how, consider that a theoretical probability distribution can be used as a generator of hypothetical observations. If an infinite number of observations are generated using a distribution, then the sample variance calculated from that infinite set will match the value calculated using the distribution's equation for variance.
Etymology

The term ''variance'' was first introduced byRonald Fisher
Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who a ...

in his 1918 paper '' The Correlation Between Relatives on the Supposition of Mendelian Inheritance'':
The great body of available statistics show us that the deviations of a human measurement from its mean follow very closely the Normal Law of Errors, and, therefore, that the variability may be uniformly measured by thestandard deviation In statistics, the standard Deviation (statistics), deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (al ...corresponding to thesquare root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square (algebra), square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots o ...of themean square error In statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and .... When there are two independent causes of variability capable of producing in an otherwise uniform population distributions with standard deviations $\backslash sigma\_1$ and $\backslash sigma\_2$, it is found that the distribution, when both causes act together, has a standard deviation $\backslash sqrt$. 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...

Definition

The variance of a random variable $X$ is theexpected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...

of the squared deviation from the mean of $X$, $\backslash mu\; =\; \backslash operatorname;\; href="/html/ALL/s/.html"\; ;"title="">$covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...

of a random variable with itself:
: $\backslash operatorname(X)\; =\; \backslash operatorname(X,\; X).$
The variance is also equivalent to the second cumulant
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expres ...

of a probability distribution that generates $X$. The variance is typically designated as $\backslash operatorname(X)$, or sometimes as $V(X)$ or $\backslash mathbb(X)$, or symbolically as $\backslash sigma^2\_X$ or simply $\backslash sigma^2$ (pronounced "sigma
Sigma (; uppercase Σ, lowercase
Letter case is the distinction between the Letter (alphabet), letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (or more formally ''minuscule'') in the wr ...

squared"). The expression for the variance can be expanded as follows:
:$\backslash begin\; \backslash operatorname(X)\; \&=\; \backslash operatorname\backslash left;\; href="/html/ALL/s/X\_-\_\backslash operatorname;\; \_;"title="X\_-\_\backslash operatorname$
In other words, the variance of is equal to the mean of the square of minus the square of the mean of . This equation should not be used for computations using floating point arithmetic
In computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computer, computing machinery. It includes the study and experimentation of algorithmic processes, and development of both computer hardware , hard ...

, because it suffers from catastrophic cancellation if the two components of the equation are similar in magnitude. For other numerically stable alternatives, see Algorithms for calculating variance.
Discrete random variable

If the generator of random variable $X$ is discrete withprobability mass function
In probability theory, probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. T ...

$x\_1\; \backslash mapsto\; p\_1,\; x\_2\; \backslash mapsto\; p\_2,\; \backslash ldots,\; x\_n\; \backslash mapsto\; p\_n$, then
:$\backslash operatorname(X)\; =\; \backslash sum\_^n\; p\_i\backslash cdot(x\_i\; -\; \backslash mu)^2,$
where $\backslash mu$ is the expected value. That is,
:$\backslash mu\; =\; \backslash sum\_^n\; p\_i\; x\_i\; .$
(When such a discrete weighted variance is specified by weights whose sum is not 1, then one divides by the sum of the weights.)
The variance of a collection of $n$ equally likely values can be written as
:$\backslash operatorname(X)\; =\; \backslash frac\; \backslash sum\_^n\; (x\_i\; -\; \backslash mu)^2$
where $\backslash mu$ is the average value. That is,
:$\backslash mu\; =\; \backslash frac\backslash sum\_^n\; x\_i\; .$
The variance of a set of $n$ equally likely values can be equivalently expressed, without directly referring to the mean, in terms of squared deviations of all pairwise squared distances of points from each other:
:$\backslash operatorname(X)\; =\; \backslash frac\; \backslash sum\_^n\; \backslash sum\_^n\; \backslash frac(x\_i\; -\; x\_j)^2\; =\; \backslash frac\backslash sum\_i\; \backslash sum\_\; (x\_i-x\_j)^2.$
Absolutely continuous random variable

If the random variable $X$ has aprobability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ...

$f(x)$, and $F(x)$ is the corresponding cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ever ...

, then
:$\backslash begin\; \backslash operatorname(X)\; =\; \backslash sigma^2\; \&=\; \backslash int\_\; (x-\backslash mu)^2\; f(x)\; \backslash ,\; dx\; \backslash \backslash ;\; href="/html/ALL/s/pt.html"\; ;"title="pt">pt$
or equivalently,
:$\backslash operatorname(X)\; =\; \backslash int\_\; x^2\; f(x)\; \backslash ,dx\; -\; \backslash mu^2\; ,$
where $\backslash mu$ is the expected value of $X$ given by
:$\backslash mu\; =\; \backslash int\_\; x\; f(x)\; \backslash ,\; dx\; =\; \backslash int\_\; x\; \backslash ,\; d\; F(x).$
In these formulas, the integrals with respect to $dx$ and $dF(x)$
are Lebesgue and Lebesgue–Stieltjes integrals, respectively.
If the function $x^2f(x)$ is Riemann-integrable on every finite interval $;\; href="/html/ALL/s/,b.html"\; ;"title=",b">,b$ then
:$\backslash operatorname(X)\; =\; \backslash int^\_\; x^2\; f(x)\; \backslash ,\; dx\; -\; \backslash mu^2,$
where the integral is an improper Riemann integral.
Examples

Exponential distribution

Theexponential distribution
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...

with parameter is a continuous distribution whose probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ...

is given by
:$f(x)\; =\; \backslash lambda\; e^$
on the interval . Its mean can be shown to be
:$\backslash operatorname;\; href="/html/ALL/s/.html"\; ;"title="">$
Using integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral (mathematics), integral of a product (mathematics), product of Function (mathematics), functions in terms o ...

and making use of the expected value already calculated, we have:
:$\backslash begin\; \backslash operatorname\backslash left;\; href="/html/ALL/s/^2\backslash right.html"\; ;"title="^2\backslash right">^2\backslash right$
Thus, the variance of is given by
:$\backslash operatorname(X)\; =\; \backslash operatorname\backslash left;\; href="/html/ALL/s/^2\backslash right.html"\; ;"title="^2\backslash right">^2\backslash right$
Fair die

A fair six-sided die can be modeled as a discrete random variable, , with outcomes 1 through 6, each with equal probability 1/6. The expected value of is $(1\; +\; 2\; +\; 3\; +\; 4\; +\; 5\; +\; 6)/6\; =\; 7/2.$ Therefore, the variance of is :$\backslash begin\; \backslash operatorname(X)\; \&=\; \backslash sum\_^6\; \backslash frac\backslash left(i\; -\; \backslash frac\backslash right)^2\; \backslash \backslash ;\; href="/html/ALL/s/pt.html"\; ;"title="pt">pt$ The general formula for the variance of the outcome, , of an die is :$\backslash begin\; \backslash operatorname(X)\; \&=\; \backslash operatorname\backslash left(X^2\backslash right)\; -\; (\backslash operatorname(X))^2\; \backslash \backslash ;\; href="/html/ALL/s/pt.html"\; ;"title="pt">pt$Commonly used probability distributions

The following table lists the variance for some commonly used probability distributions.Properties

Basic properties

Variance is non-negative because the squares are positive or zero: :$\backslash operatorname(X)\backslash ge\; 0.$ The variance of a constant is zero. :$\backslash operatorname(a)\; =\; 0.$ Conversely, if the variance of a random variable is 0, then it isalmost surely
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by exp ...

a constant. That is, it always has the same value:
:$\backslash operatorname(X)=\; 0\; \backslash iff\; \backslash exists\; a\; :\; P(X=a)\; =\; 1.$
Issues of finiteness

If a distribution does not have a finite expected value, as is the case for theCauchy distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) f ...

, then the variance cannot be finite either. However, some distributions may not have a finite variance, despite their expected value being finite. An example is a Pareto distribution
The Pareto distribution, named after the Italian civil engineer, economist
An economist is a professional and practitioner in the social sciences, social science discipline of economics.
The individual may also study, develop, and apply th ...

whose index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...

$k$ satisfies $1\; <\; k\; \backslash leq\; 2.$
Decomposition

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 :$\backslash operatorname;\; href="/html/ALL/s/.html"\; ;"title="">$ Theconditional expectation
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an Law of large numbers, arbitrarily large number of ...

$\backslash operatorname\; E(X\backslash mid\; Y)$ of $X$ given $Y$, and the conditional variance In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by express ...

$\backslash operatorname(X\backslash mid\; Y)$ may be understood as follows. Given any particular value ''y'' of the random variable ''Y'', there is a conditional expectation $\backslash operatorname\; E(X\backslash mid\; Y=y)$ given the event ''Y'' = ''y''. This quantity depends on the particular value ''y''; it is a function $g(y)\; =\; \backslash operatorname\; E(X\backslash mid\; Y=y)$. That same function evaluated at the random variable ''Y'' is the conditional expectation $\backslash operatorname\; E(X\backslash mid\; Y)\; =\; g(Y).$
In particular, if $Y$ is a discrete random variable assuming possible values $y\_1,\; y\_2,\; y\_3\; \backslash ldots$ with corresponding probabilities $p\_1,\; p\_2,\; p\_3\; \backslash ldots,$, then in the formula for total variance, the first term on the right-hand side becomes
:$\backslash operatorname(\backslash operatorname;\; href="/html/ALL/s/\_\backslash mid\_Y.html"\; ;"title="\; \backslash mid\; Y">\; \backslash mid\; Y$
where $\backslash sigma^2\_i\; =\; \backslash operatorname;\; href="/html/ALL/s/\_\backslash mid\_Y\_=\_y\_i.html"\; ;"title="\; \backslash mid\; Y\; =\; y\_i">\; \backslash mid\; Y\; =\; y\_i$analysis of variance
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician ...

, where the corresponding formula is
:$\backslash mathit\_\backslash text\; =\; \backslash mathit\_\backslash text\; +\; \backslash mathit\_\backslash text;$
here $\backslash mathit$ refers to the Mean of the Squares. In linear regression
In statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, an ...

analysis the corresponding formula is
:$\backslash mathit\_\backslash text\; =\; \backslash mathit\_\backslash text\; +\; \backslash mathit\_\backslash text.$
This can also be derived from the additivity of variances, since the total (observed) score is the sum of the predicted score and the error score, where the latter two are uncorrelated.
Similar decompositions are possible for the sum of squared deviations (sum of squares, $\backslash mathit$):
:$\backslash mathit\_\backslash text\; =\; \backslash mathit\_\backslash text\; +\; \backslash mathit\_\backslash text,$
:$\backslash mathit\_\backslash text\; =\; \backslash mathit\_\backslash text\; +\; \backslash mathit\_\backslash text.$
Calculation from the CDF

The population variance for a non-negative random variable can be expressed in terms of thecumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ever ...

''F'' using
:$2\backslash int\_0^\backslash infty\; u(1\; -\; F(u))\backslash ,du\; -\; \backslash left(\backslash int\_0^\backslash infty\; (1\; -\; F(u))\backslash ,du\backslash right)^2.$
This expression can be used to calculate the variance in situations where the CDF, but not the density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek language, Greek letter Rho (letter), rho), although the Latin letter ''D'' ca ...

, 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 variable, i.e. $\backslash mathrm\_m\backslash ,\backslash mathrm\backslash left(\backslash left(X\; -\; m\backslash right)^2\backslash right)\; =\; \backslash mathrm(X)$. Conversely, if a continuous function $\backslash varphi$ satisfies $\backslash mathrm\_m\backslash ,\backslash mathrm(\backslash varphi(X\; -\; m))\; =\; \backslash mathrm(X)$ for all random variables ''X'', then it is necessarily of the form $\backslash varphi(x)\; =\; a\; x^2\; +\; b$, where . This also holds in the multidimensional case.Units of measurement

Unlike the expected absolute deviation, the variance of a variable has units that are the square of the units of the variable itself. For example, a variable measured in meters will have a variance measured in meters squared. For this reason, describing data sets via theirstandard deviation
In statistics, the standard Deviation (statistics), deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (al ...

or root mean square deviation
The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) is a frequently used measure of the differences between values (sample or population values) predicted by a model or an estimator and the values observed. The RMSD represents ...

is often preferred over using the variance. In the dice example the standard deviation is , slightly larger 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 standard deviation is more amenable to algebraic manipulation than the expected absolute deviation, and, together with variance and its generalization covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...

, is used frequently in theoretical statistics; however the expected absolute deviation tends to be more robust as it is less sensitive to outlier
In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are ...

s arising from measurement anomalies or an unduly heavy-tailed distribution
In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution. In many applications it is the right tail of the distrib ...

.
Propagation

Addition and multiplication by a constant

Variance is invariant with respect to changes in alocation parameter
In geography
Geography (from Ancient Greek, Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the la ...

. That is, if a constant is added to all values of the variable, the variance is unchanged:
:$\backslash operatorname(X+a)=\backslash operatorname(X).$
If all values are scaled by a constant, the variance is scaled by the square of that constant:
:$\backslash operatorname(aX)=a^2\backslash operatorname(X).$
The variance of a sum of two random variables is given by
:$\backslash operatorname(aX+bY)=a^2\backslash operatorname(X)+b^2\backslash operatorname(Y)+2ab\backslash ,\; \backslash operatorname(X,Y),$
:$\backslash operatorname(aX-bY)=a^2\backslash operatorname(X)+b^2\backslash operatorname(Y)-2ab\backslash ,\; \backslash operatorname(X,Y),$
where $\backslash operatorname(X,Y)$ is the Linear combinations

In general, for the sum of $N$ random variables $\backslash $, the variance becomes: :$\backslash operatorname\backslash left(\backslash sum\_^N\; X\_i\backslash right)=\backslash sum\_^N\backslash operatorname(X\_i,X\_j)=\backslash sum\_^N\backslash operatorname(X\_i)+\backslash sum\_\backslash operatorname(X\_i,X\_j),$ see also general Bienaymé's identity. These results lead to the variance of a linear combination as: :$\backslash begin\; \backslash operatorname\backslash left(\; \backslash sum\_^N\; a\_iX\_i\backslash right)\; \&=\backslash sum\_^\; a\_ia\_j\backslash operatorname(X\_i,X\_j)\; \backslash \backslash \; \&=\backslash sum\_^N\; a\_i^2\backslash operatorname(X\_i)+\backslash sum\_a\_ia\_j\backslash operatorname(X\_i,X\_j)\backslash \backslash \; \&\; =\backslash sum\_^N\; a\_i^2\backslash operatorname(X\_i)+2\backslash sum\_a\_ia\_j\backslash operatorname(X\_i,X\_j).\; \backslash end$ If the random variables $X\_1,\backslash dots,X\_N$ are such that :$\backslash operatorname(X\_i,X\_j)=0\backslash \; ,\backslash \; \backslash forall\backslash \; (i\backslash ne\; j)\; ,$ then they are said to beuncorrelated
In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname Y- \operatorname \operatorname is zero. If two variables are uncorrelated, t ...

. It follows immediately from the expression given earlier that if the random variables $X\_1,\backslash dots,X\_N$ are uncorrelated, then the variance of their sum is equal to the sum of their variances, or, expressed symbolically:
:$\backslash operatorname\backslash left(\backslash sum\_^N\; X\_i\backslash right)=\backslash sum\_^N\backslash operatorname(X\_i).$
Since independent random variables are always uncorrelated (see ), the equation above holds in particular when the random variables $X\_1,\backslash dots,X\_n$ are independent. Thus, independence is sufficient but not necessary for the variance of the sum to equal the sum of the variances.
Matrix notation for the variance of a linear combination

Define $X$ as a column vector of $n$ random variables $X\_1,\; \backslash ldots,X\_n$, and $c$ as a column vector of $n$ scalars $c\_1,\; \backslash ldots,c\_n$. Therefore, $c^\backslash mathsf\; X$ is a linear combination of these random variables, where $c^\backslash mathsf$ denotes thetranspose
In linear algebra, the transpose of a Matrix (mathematics), matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among othe ...

of $c$. Also let $\backslash Sigma$ be the covariance matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square Matrix (mathematics), matrix giving the covariance between ea ...

of $X$. The variance of $c^\backslash mathsfX$ is then given by:
:$\backslash operatorname\backslash left(c^\backslash mathsf\; X\backslash right)\; =\; c^\backslash mathsf\; \backslash Sigma\; c\; .$
This implies that the variance of the mean can be written as (with a column vector of ones)
:$\backslash operatorname\backslash left(\backslash bar\backslash right)\; =\; \backslash operatorname\backslash left(\backslash frac\; 1\text{'}X\backslash right)\; =\; \backslash frac\; 1\text{'}\backslash Sigma\; 1.$
Sum of variables

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) ofuncorrelated
In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname Y- \operatorname \operatorname is zero. If two variables are uncorrelated, t ...

random variables is the sum of their variances:
:$\backslash operatorname\backslash left(\backslash sum\_^n\; X\_i\backslash right)\; =\; \backslash sum\_^n\; \backslash operatorname(X\_i).$
This statement is called the Bienaymé formula and was discovered in 1853. It is often made with the stronger condition that the variables are independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independen ...

, but being uncorrelated suffices. So if all the variables have the same variance σstandard error
The standard error (SE) of a statistic (usually an estimate of a Statistical parameter, parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the statistic is the sample mean, it is calle ...

of the sample mean, which is used in the central limit theorem
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

.
To prove the initial statement, it suffices to show that
:$\backslash operatorname(X\; +\; Y)\; =\; \backslash operatorname(X)\; +\; \backslash operatorname(Y).$
The general result then follows by induction. Starting with the definition,
:$\backslash begin\; \backslash operatorname(X\; +\; Y)\; \&=\; \backslash operatorname\backslash left;\; href="/html/ALL/s/X\_+\_Y)^2\backslash right.html"\; ;"title="X\; +\; Y)^2\backslash right">X\; +\; Y)^2\backslash right$
Using the linearity of the expectation operator and the assumption of independence (or uncorrelatedness) of ''X'' and ''Y'', this further simplifies as follows:
:$\backslash begin\; \backslash operatorname(X\; +\; Y)\; \&=\; \backslash operatorname\backslash left;\; href="/html/ALL/s/^2\backslash right.html"\; ;"title="^2\backslash right">^2\backslash right$
Sum of correlated variables

=Sum of correlated variables with fixed sample size

= In general, the variance of the sum of variables is the sum of theircorrelation
In statistics, correlation or dependence is any statistical relationship, whether causality, causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in ...

of distinct variables is ''ρ'', then the variance of their mean is
:$\backslash operatorname\backslash left(\backslash overline\backslash right)\; =\; \backslash frac\; +\; \backslash frac\backslash rho\backslash sigma^2.$
This implies that the variance of the mean increases with the average of the correlations. In other words, additional correlated observations are not as effective as additional independent observations at reducing the uncertainty of the mean. Moreover, if the variables have unit variance, for example if they are standardized, then this simplifies to
:$\backslash operatorname\backslash left(\backslash overline\backslash right)\; =\; \backslash frac\; +\; \backslash frac\backslash rho.$
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
:$\backslash lim\_\; \backslash operatorname\backslash left(\backslash overline\backslash right)\; =\; \backslash rho.$
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 not generally converge to the population mean, even though the law of large numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...

states that the sample mean will converge for independent variables.
=Sum of uncorrelated variables with random sample size

= There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. In such cases, the sample size is a random variable whose variation adds to the variation of , such that, : $\backslash operatorname\backslash left(\backslash sum\_^X\_i\backslash right)=\backslash operatorname\backslash left;\; href="/html/ALL/s/\backslash right.html"\; ;"title="\backslash right">\backslash right$ which follows from the law of total variance. If has aPoisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...

, then $\backslash operatorname;\; href="/html/ALL/s/.html"\; ;"title="">$ with estimator = . So, the estimator of $\backslash operatorname\backslash left(\backslash sum\_^X\_i\backslash right)$ becomes $n^2+n\backslash bar^2$ giving $\backslash operatorname(\backslash bar)=\backslash sqrt$
Weighted sum of variables

The scaling property and the Bienaymé formula, along with the property of theProduct of variables

Product of independent variables

If two variables X and Y areindependent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independen ...

, the variance of their product is given by
:$\backslash operatorname(XY)\; =;\; href="/html/ALL/s/operatorname(X).html"\; ;"title="operatorname(X)">operatorname(X)$
Equivalently, using the basic properties of expectation, it is given by
:$\backslash operatorname(XY)\; =\; \backslash operatorname\backslash left(X^2\backslash right)\; \backslash operatorname\backslash left(Y^2\backslash right)\; -;\; href="/html/ALL/s/operatorname(X).html"\; ;"title="operatorname(X)">operatorname(X)$
Product of statistically dependent variables

In general, if two variables are statistically dependent, then the variance of their product is given by: :$\backslash begin\; \backslash operatorname(XY)\; =\; \&\backslash operatorname\backslash left;\; href="/html/ALL/s/^2\_Y^2\backslash right.html"\; ;"title="^2\; Y^2\backslash right">^2\; Y^2\backslash right$Arbitrary functions

The delta method uses second-orderTaylor expansion
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

s to approximate the variance of a function of one or more random variables: see Taylor expansions for the moments of functions of random variables. For example, the approximate variance of a function of one variable is given by
:$\backslash operatorname\backslash left;\; href="/html/ALL/s/(X)\backslash right.html"\; ;"title="(X)\backslash right">(X)\backslash right$Population variance and sample variance

Real-world observations such as the measurements of yesterday's rain throughout the day typically cannot be complete sets of all possible observations that could be made. As such, the variance calculated from the finite set will in general not match the variance that would have been calculated from the full population of possible observations. This means that one estimates the mean and variance from a limited set of observations by using anestimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, th ...

equation. The estimator is a function of the sample of ''n'' observations drawn without observational bias from the whole population
Population typically refers to the number of people in a single area, whether it be a city
A city is a human settlement of notable size.Goodall, B. (1987) ''The Penguin Dictionary of Human Geography''. London: Penguin.Kuper, A. and Kuper, ...

of potential observations. In this example that sample would be the set of actual measurements of yesterday's rainfall from available rain gauges within the geography of interest.
The simplest estimators for population mean and population variance are simply the mean and variance of the sample, the sample mean and (uncorrected) sample variance – these are consistent estimator
In statistics, a consistent estimator or asymptotically consistent estimator is an estimator—a rule for computing estimates of a parameter ''θ''0—having the property that as the number of data points used increases indefinitely, the result ...

s (they converge to the correct value as the number of samples increases), but can be improved. Estimating the population variance by taking the sample's variance is close to optimal in general, but can be improved in two ways. Most simply, the sample variance is computed as an average of squared deviations about the (sample) mean, by dividing by ''n.'' However, using values other than ''n'' improves the estimator in various ways. Four common values for the denominator are ''n,'' ''n'' − 1, ''n'' + 1, and ''n'' − 1.5: ''n'' is the simplest (population variance of the sample), ''n'' − 1 eliminates bias, ''n'' + 1 minimizes mean squared error
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the expected value, average of the squares of the Error (statistics), errors—that is, the ...

for the normal distribution, and ''n'' − 1.5 mostly eliminates bias in unbiased estimation of standard deviation for the normal distribution.
Firstly, if the true population mean is unknown, then the sample variance (which uses the sample mean in place of the true mean) is a biased estimator: it underestimates the variance by a factor of (''n'' − 1) / ''n''; correcting by this factor (dividing by ''n'' − 1 instead of ''n'') is called ''Bessel's correction
In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a Sample (statistics), sample. This method c ...

''. The resulting estimator is unbiased, and is called the (corrected) sample variance or unbiased sample variance. For example, when ''n'' = 1 the variance of a single observation about the sample mean (itself) is obviously zero regardless of the population variance. If the mean is determined in some other way than from the same samples used to estimate the variance then this bias does not arise and the variance can safely be estimated as that of the samples about the (independently known) mean.
Secondly, the sample variance does not generally minimize mean squared error
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the expected value, average of the squares of the Error (statistics), errors—that is, the ...

between sample variance and population variance. Correcting for bias often makes this worse: one can always choose a scale factor that performs better than the corrected sample variance, though the optimal scale factor depends on the excess kurtosis of the population (see mean squared error: variance), and introduces bias. This always consists of scaling down the unbiased estimator (dividing by a number larger than ''n'' − 1), and is a simple example of a shrinkage estimator: one "shrinks" the unbiased estimator towards zero. For the normal distribution, dividing by ''n'' + 1 (instead of ''n'' − 1 or ''n'') minimizes mean squared error. The resulting estimator is biased, however, and is known as the biased sample variation.
Population variance

In general, the ''population variance'' of a ''finite''population
Population typically refers to the number of people in a single area, whether it be a city
A city is a human settlement of notable size.Goodall, B. (1987) ''The Penguin Dictionary of Human Geography''. London: Penguin.Kuper, A. and Kuper, ...

of size ''N'' with values ''x''Sample variance

Biased sample variance

In many practical situations, the true variance of a population is not known ''a priori'' and must be computed somehow. When dealing with extremely large populations, it is not possible to count every object in the population, so the computation must be performed on a sample of the population. Sample variance can also be applied to the estimation of the variance of a continuous distribution from a sample of that distribution. We take a sample with replacement of ''n'' values ''Y''sample mean
The sample mean (or "empirical mean") and the sample covariance are statistics computed from a Sample (statistics), sample of data on one or more random variables.
The sample mean is the average value (or mean, mean value) of a sample (statistic ...

:
:$\backslash overline\; =\; \backslash frac\; \backslash sum\_^n\; Y\_i\; .$
Since the ''Y''Unbiased sample variance

Correcting for this bias yields the ''unbiased sample variance'', denoted $S^2$: :$S^2\; =\; \backslash frac\; \backslash tilde\_Y^2\; =\; \backslash frac\; \backslash left;\; href="/html/ALL/s/\backslash frac\_\backslash sum\_^n\_\backslash left(Y\_i\_-\_\backslash overline\backslash right)^2\_\backslash right.html"\; ;"title="\backslash frac\; \backslash sum\_^n\; \backslash left(Y\_i\; -\; \backslash overline\backslash right)^2\; \backslash right">\backslash frac\; \backslash sum\_^n\; \backslash left(Y\_i\; -\; \backslash overline\backslash right)^2\; \backslash right$ Either estimator may be simply referred to as the ''sample variance'' when the version can be determined by context. The same proof is also applicable for samples taken from a continuous probability distribution. The use of the term ''n'' − 1 is calledBessel's correction
In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a Sample (statistics), sample. This method c ...

, and it is also used in sample covariance and the sample standard deviation (the square root of variance). The square root is a concave function
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...

and thus introduces negative bias (by Jensen's inequality
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen (mathematician), Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was mathematical proof, pro ...

), which depends on the distribution, and thus the corrected sample standard deviation (using Bessel's correction) is biased. The unbiased estimation of standard deviation 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 ''ƒ''(''y''Distribution of the sample variance

Being a function ofrandom variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...

s, the sample variance is itself a random variable, and it is natural to study its distribution. In the case that ''Y''normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The param ...

, Cochran's theorem shows that ''S''chi-squared distribution
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by ex ...

(see also: asymptotic properties):
:$(n\; -\; 1)\backslash frac\backslash sim\backslash chi^2\_.$
As a direct consequence, it follows that
:$\backslash operatorname\backslash left(S^2\backslash right)\; =\; \backslash operatorname\backslash left(\backslash frac\; \backslash chi^2\_\backslash right)\; =\; \backslash sigma^2\; ,$
and
:$\backslash operatorname\backslash left;\; href="/html/ALL/s/^2\backslash right.html"\; ;"title="^2\backslash right">^2\backslash right$
If the ''Y''kurtosis
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real number, real-valued random variable. Like skew ...

of the distribution and ''μ''central moment
In probability theory and statistics, a central moment is a moment (mathematics), moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the de ...

.
If the conditions of the law of large numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...

hold for the squared observations, ''S''consistent estimator
In statistics, a consistent estimator or asymptotically consistent estimator is an estimator—a rule for computing estimates of a parameter ''θ''0—having the property that as the number of data points used increases indefinitely, the result ...

of ''σ''Samuelson's inequality

Samuelson's inequality is a result that states bounds on the values that individual observations in a sample can take, given that the sample mean and (biased) variance have been calculated. Values must lie within the limits $\backslash bar\; y\; \backslash pm\; \backslash sigma\_Y\; (n-1)^.$Relations with the harmonic and arithmetic means

It has been shown that for a sample of positive real numbers, : $\backslash sigma\_y^2\; \backslash le\; 2y\_\; (A\; -\; H),$ where ''y''harmonic mean
In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate (mathematics), rate is desired.
The harmonic mean can be expres ...

of the sample and $\backslash sigma\_y^2$ is the (biased) variance of the sample.
This bound has been improved, and it is known that variance is bounded by
: $\backslash sigma\_y^2\; \backslash le\; \backslash frac,$
: $\backslash sigma\_y^2\; \backslash ge\; \backslash frac,$
where ''y''Tests of equality of variances

The F-test of equality of variances and the chi square tests are adequate when the sample is normally distributed. Non-normality makes testing for the equality of two or more variances more difficult. Several non parametric tests have been proposed: these include the Barton–David–Ansari–Freund–Siegel–Tukey test, the Capon test, Mood test, the Klotz test and the Sukhatme test. The Sukhatme test applies to two variances and requires that bothmedian
In statistics and probability theory, the median is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be th ...

s be known and equal to zero. The Mood, Klotz, Capon and Barton–David–Ansari–Freund–Siegel–Tukey tests also apply to two variances. They allow the median to be unknown but do require that the two medians are equal.
The Lehmann test is a parametric test of two variances. Of this test there are several variants known. Other tests of the equality of variances include 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.
Moment of inertia

The variance of a probability distribution is analogous to themoment of inertia
The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accelera ...

in classical mechanics
Classical mechanics is a Theoretical physics, physical theory describing the motion of macroscopic objects, from projectiles to parts of Machine (mechanical), machinery, and astronomical objects, such as spacecraft, planets, stars, and galax ...

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 '' moments'' of probability distribution
In probability theory and statistics, a probability distribution is the mathematical Function (mathematics), function that gives the probabilities of occurrence of different possible outcomes for an Experiment (probability theory), experiment. ...

s. The covariance matrix is related to the moment of inertia tensor for multivariate distributions. The moment of inertia of a cloud of ''n'' points with a covariance matrix of $\backslash Sigma$ is given by
:$I\; =\; n\backslash left(\backslash mathbf\_\; \backslash operatorname(\backslash Sigma)\; -\; \backslash Sigma\backslash right).$
This difference between moment of inertia in physics and in statistics is clear for points that are gathered along a line. Suppose many points are close to the ''x'' axis and distributed along it. The covariance matrix might look like
:$\backslash Sigma\; =\; \backslash begin10\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0.1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0.1\backslash end.$
That is, there is the most variance in the ''x'' direction. Physicists would consider this to have a low moment ''about'' the ''x'' axis so the moment-of-inertia tensor is
:$I\; =\; n\backslash begin0.2\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 10.1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 10.1\backslash end.$
Semivariance

The ''semivariance'' is calculated in the same manner as the variance but only those observations that fall below the mean are included in the calculation:$$\backslash text\; =\; \backslash sum\_(x\_-\backslash mu)^$$It is also described as a specific measure in different fields of application. For skewed distributions, the semivariance can provide additional information that a variance does not. For inequalities associated with the semivariance, see .Generalizations

For complex variables

If $x$ is a scalar complex-valued random variable, with values in $\backslash mathbb,$ then its variance is $\backslash operatorname\backslash left;\; href="/html/ALL/s/x\_-\_\backslash mu)(x\_-\_\backslash mu)^*\backslash right.html"\; ;"title="x\; -\; \backslash mu)(x\; -\; \backslash mu)^*\backslash right">x\; -\; \backslash mu)(x\; -\; \backslash mu)^*\backslash right$ where $x^*$ is thecomplex conjugate
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...

of $x.$ This variance is a real scalar.
For vector-valued random variables

As a matrix

If $X$ is a vector-valued random variable, with values in $\backslash mathbb^n,$ and thought of as a column vector, then a natural generalization of variance is $\backslash operatorname\backslash left;\; href="/html/ALL/s/X\_-\_\backslash mu)(X\_-\_\backslash mu)^\backslash right.html"\; ;"title="X\; -\; \backslash mu)(X\; -\; \backslash mu)^\backslash right">X\; -\; \backslash mu)(X\; -\; \backslash mu)^\backslash right$ where $\backslash mu\; =\; \backslash operatorname(X)$ and $X^$ is the transpose of $X,$ and so is a row vector. The result is a positive semi-definite square matrix, commonly referred to as the variance-covariance matrix (or simply as the ''covariance matrix''). If $X$ is a vector- and complex-valued random variable, with values in $\backslash mathbb^n,$ then the covariance matrix is $\backslash operatorname\backslash left;\; href="/html/ALL/s/X\_-\_\backslash mu)(X\_-\_\backslash mu)^\backslash dagger\backslash right.html"\; ;"title="X\; -\; \backslash mu)(X\; -\; \backslash mu)^\backslash dagger\backslash right">X\; -\; \backslash mu)(X\; -\; \backslash mu)^\backslash dagger\backslash right$ where $X^\backslash dagger$ is theconjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n Complex number, complex matrix (mathematics), matrix \boldsymbol is an n \times m matrix obtained by transpose, transposing \boldsymbol and applying ...

of $X.$ This matrix is also positive semi-definite and square.
As a scalar

Another generalization of variance for vector-valued random variables $X$, which results in a scalar value rather than in a matrix, is the generalized variance $\backslash det(C)$, thedeterminant
In mathematics, the determinant is a Scalar (mathematics), scalar value that is a function (mathematics), function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In p ...

of the covariance matrix. The generalized variance can be shown to be related to the multidimensional scatter of points around their mean.
A different generalization is obtained by considering the Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two Point (geometry), points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theo ...

between the random variable and its mean. This results in $\backslash operatorname\backslash left;\; href="/html/ALL/s/X\_-\_\backslash mu)^(X\_-\_\backslash mu)\backslash right.html"\; ;"title="X\; -\; \backslash mu)^(X\; -\; \backslash mu)\backslash right">X\; -\; \backslash mu)^(X\; -\; \backslash mu)\backslash right$ which is the trace
Trace may refer to:
Arts and entertainment Music
* Trace (Son Volt album), ''Trace'' (Son Volt album), 1995
* Trace (Died Pretty album), ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* The Trace (album), ''The ...

of the covariance matrix.
See also

* Bhatia–Davis inequality *Coefficient of variation
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by ex ...

* Homoscedasticity
In statistics, a sequence (or a vector) of random variables is homoscedastic () if all its random variables have the same finite variance. This is also known as homogeneity of variance. The complementary notion is called heteroscedasticity. The s ...

* Least-squares spectral analysis
Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of Sine wave, sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the most used spectral method in science ...

for computing a frequency spectrum
The power spectrum S_(f) of a time series
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and ...

with spectral magnitudes in % of variance or in dB
* Popoviciu's inequality on variances
* Measures for statistical dispersion
* Variance-stabilizing transformation
Types of variance

*Correlation
In statistics, correlation or dependence is any statistical relationship, whether causality, causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in ...

* Distance variance
* Explained variance In statistics, explained variation measures the proportion to which a mathematical model accounts for the variation (dispersion (statistics), dispersion) of a given data set. Often, variation is quantified as variance; then, the more specific term e ...

* Pooled variance
* Pseudo-variance
References

{{Authority control Moment (mathematics) Statistical deviation and dispersion Articles containing proofs