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In [[Probability theory|probability theory]] and [[Statistics|statistics]], '''variance''' is the [[Expected value|expected value]] of the [[Squared deviations from the mean|squared deviation from the mean]] of a [[Random variable|random variable]]. The [[Standard deviation|standard deviation]] is obtained as the square root of the variance. Variance is a measure of [[Statistical dispersion|dispersion]], meaning it is a measure of how far a set of numbers are spread out from their average value. It is the second [[Central moment|central moment]] of a [[Probability distribution|distribution]], and the [[Covariance|covariance]] of the random variable with itself, and it is often represented by {{tmath| \sigma^2 }}, {{tmath| s^2 }}, {{tmath| \operatorname{Var}(X) }}, {{tmath| V(X) }}, or {{tmath| \mathbb{V}(X) }}.<ref>{{cite book |last1=Wasserman |first1=Larry |title=All of Statistics: a concise course in statistical inference |date=2005 |publisher=Springer texts in statistics |isbn=978-1-4419-2322-6 |page=51 }}</ref>

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 [[Average absolute deviation|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. Another disadvantage is that the variance is not finite for many distributions.

There are two distinct concepts that are both called "variance". One, as discussed above, is part of a theoretical [[Probability distribution|probability distribution]] 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 estimate the population variance on the basis of the sample 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. Variance has a central role in statistics, where some ideas that use it include [[Descriptive statistics|descriptive statistics]], [[Statistical inference|statistical inference]], [[Hypothesis testing|hypothesis testing]], [[Goodness of fit|goodness of fit]], and [[Monte Carlo method|Monte Carlo sampling]].


[[File:variance_visualisation.svg|thumb|Geometric visualisation of the variance of an arbitrary distribution (2, 4, 4, 4, 5, 5, 7, 9): {{ordered list
    |A frequency distribution is constructed.
    |The centroid of the distribution gives its mean.
    |A square with sides equal to the difference of each value from the mean is formed for each value.
    |Arranging the squares into a rectangle with one side equal to the number of values, ''n'', results in the other side being the distribution's variance, ''σ''<sup>2</sup>.
}}]]

== Definition ==
The variance of a random variable <math>X</math> is the [[Expected value|expected value]] of the [[Squared deviations from the mean|squared deviation from the mean]] of {{tmath| X }}, {{tmath|1= \mu = \operatorname{E}[X] }}:
<math display="block"> \operatorname{Var}(X) = \operatorname{E}\left[(X - \mu)^2 \right]. </math>
This definition encompasses random variables that are generated by processes that are discrete, continuous, [[Cantor distribution|neither]], or mixed. The variance can also be thought of as the [[Covariance|covariance]] of a random variable with itself:
<math display="block">\operatorname{Var}(X) = \operatorname{Cov}(X, X).</math>

The variance is also equivalent to the second [[Cumulant|cumulant]] of a probability distribution that generates {{tmath| X }}. The variance is typically designated as {{tmath| \operatorname{Var}(X) }}, or sometimes as <math>V(X)</math> or {{tmath| \mathbb{V}(X) }}, or symbolically as {{tmath| \sigma^2_X }} or simply <math>\sigma^2</math> (pronounced "sigma squared"). The expression for the variance can be expanded as follows:
<math display="block">\begin{align}
\operatorname{Var}(X) &= \operatorname{E}\left[{\left(X - \operatorname{E}[X]\right)}^2\right] \\[4pt]
&= \operatorname{E}\left[X^2 - 2 X \operatorname{E}[X] + \operatorname{E}[X]^2\right] \\[4pt]
&= \operatorname{E}\left[X^2\right] - 2 \operatorname{E}[X] \operatorname{E}[X] + \operatorname{E}[X]^2 \\[4pt]
&= \operatorname{E}\left[X^2\right] - 2\operatorname{E}[X]^2 + \operatorname{E}[X]^2 \\[4pt]
&= \operatorname{E}\left[X^2\right] -  \operatorname{E}[X]^2
\end{align}</math>

In other words, the variance of {{mvar|X}} is equal to the mean of the square of {{mvar|X}} minus the square of the mean of {{mvar|X}}.  This equation should not be used for computations using [[Floating-point arithmetic|floating-point arithmetic]], because it suffers from [[Catastrophic cancellation|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 <math>X</math> is discrete with [[Probability mass function|probability mass function]] <math>x_1 \mapsto p_1, x_2 \mapsto p_2, \ldots, x_n \mapsto p_n</math>, then
<math display="block">\operatorname{Var}(X) = \sum_{i=1}^n p_i \cdot {\left(x_i - \mu\right)}^2,</math>
where <math>\mu</math> is the expected value. That is,
<math display="block">\mu = \sum_{i=1}^n p_i x_i .</math>
(When such a discrete weighted variance is specified by weights whose sum is not&nbsp;1, then one divides by the sum of the weights.)

The variance of a collection of <math>n</math> equally likely values can be written as
<math display="block"> \operatorname{Var}(X) = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2 </math>
where <math>\mu</math> is the average value. That is,
<math display="block">\mu = \frac{1}{n}\sum_{i=1}^n x_i .</math>

The variance of a set of <math>n</math> 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:<ref>{{cite conference|author=Yuli Zhang |author2=Huaiyu Wu |author3=Lei Cheng |date=June 2012|title=Some new deformation formulas about variance and covariance|conference=Proceedings of 4th International Conference on Modelling, Identification and Control(ICMIC2012)|pages=987–992}}</ref>
<math display="block"> \operatorname{Var}(X) = \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n \frac{1}{2} {\left(x_i - x_j\right)}^2 = \frac{1}{n^2} \sum_i \sum_{j>i} {\left(x_i - x_j\right)}^2. </math>

=== Absolutely continuous random variable ===

If the random variable <math>X</math> has a [[Physics:Probability density function|probability density function]] {{tmath| f(x) }}, and <math>F(x)</math> is the corresponding [[Cumulative distribution function|cumulative distribution function]], then
<math display="block">\begin{align}
   \operatorname{Var}(X) = \sigma^2 &= \int_{\R} {\left(x - \mu\right)}^2 f(x) \, dx \\[4pt]
     &= \int_{\R} x^2 f(x)\,dx -2\mu\int_{\R} xf(x)\,dx + \mu^2\int_{\R} f(x)\,dx \\[4pt]
     &= \int_{\R} x^2 \,dF(x) - 2 \mu \int_{\R} x \,dF(x) + \mu^2 \int_{\R} \,dF(x) \\[4pt]
     &= \int_{\R} x^2 \,dF(x) - 2 \mu \cdot \mu + \mu^2 \cdot 1 \\[4pt]
     &= \int_{\R} x^2 \,dF(x) - \mu^2,
 \end{align}</math>
or equivalently,
<math display="block">\operatorname{Var}(X) = \int_{\R} x^2 f(x) \,dx - \mu^2 ,</math>
where <math>\mu</math> is the expected value of <math>X</math> given by
<math display="block">\mu = \int_{\R} x f(x) \, dx = \int_{\R} x \, dF(x). </math>

In these formulas, the integrals with respect to <math>dx</math> and <math>dF(x)</math> are [[Lebesgue integral|Lebesgue]] and [[Lebesgue–Stieltjes integration|Lebesgue–Stieltjes]] integrals, respectively.

If the function <math>x^2f(x)</math> is Riemann-integrable on every finite interval <math>[a,b]\subset\R,</math> then
<math display="block">\operatorname{Var}(X) = \int^{+\infty}_{-\infty} x^2 f(x) \, dx - \mu^2, </math>
where the integral is an improper Riemann integral.

== Examples ==

=== Exponential distribution ===
The [[Exponential distribution|exponential distribution]] with parameter {{mvar|&lambda;}} > 0 is a continuous distribution whose [[Physics:Probability density function|probability density function]] is given by
<math display="block">f(x) = \lambda e^{-\lambda x}</math>
on the interval {{closed-open|0, ∞}}.  Its mean can be shown to be
<math display="block">\operatorname{E}[X] = \int_0^\infty x \lambda e^{-\lambda x} \, dx = \frac{1}{\lambda}.</math>

Using [[Integration by parts|integration by parts]] and making use of the expected value already calculated, we have:
<math display="block">\begin{align}
  \operatorname{E}\left[X^2\right] &= \int_0^\infty x^2 \lambda e^{-\lambda x} \, dx \\
    &= {\left[ -x^2 e^{-\lambda x} \right]}_0^\infty + \int_0^\infty 2 x e^{-\lambda x} \,dx \\
    &= 0 + \frac{2}{\lambda}\operatorname{E}[X] \\
    &= \frac{2}{\lambda^2}.
\end{align}</math>

Thus, the variance of {{mvar|X}} is given by
<math display="block">\operatorname{Var}(X) = \operatorname{E}\left[X^2\right] - \operatorname{E}[X]^2 = \frac{2}{\lambda^2} - \left(\frac{1}{\lambda}\right)^2 = \frac{1}{\lambda^2}.</math>

=== Fair die <!--Singular: die; plural: dice. Don't change--> ===
A fair six-sided die can be modeled as a discrete random variable, {{mvar|X}}, with outcomes 1 through 6, each with equal probability 1/6.  The expected value of {{mvar|X}} is <math>(1 + 2 + 3 + 4 + 5 + 6)/6 = 7/2.</math> Therefore, the variance of {{mvar|X}} is
<math display="block">\begin{align}
  \operatorname{Var}(X) &= \sum_{i=1}^6 \frac{1}{6}\left(i - \frac{7}{2}\right)^2 \\[5pt]
    &= \frac{1}{6}\left((-5/2)^2 + (-3/2)^2 + (-1/2)^2 + (1/2)^2 + (3/2)^2 + (5/2)^2\right) \\[5pt]
    &= \frac{35}{12} \approx 2.92.
\end{align}</math>

The general formula for the variance of the outcome, {{mvar|X}}, of an {{nowrap|{{mvar|n}}-sided}} die is
<math display="block">\begin{align}
  \operatorname{Var}(X) &= \operatorname{E}\left(X^2\right) - (\operatorname{E}(X))^2 \\[5pt]
    &= \frac{1}{n}\sum_{i=1}^n i^2 - \left(\frac{1}{n}\sum_{i=1}^n i\right)^2 \\[5pt]
    &= \frac{(n + 1)(2n + 1)}{6} - \left(\frac{n + 1}{2}\right)^2 \\[4pt]
    &= \frac{n^2 - 1}{12}.
\end{align}</math>

=== Commonly used probability distributions ===
The following table lists the variance for some commonly used probability distributions.
{| class="wikitable"
|-
! Name of the probability distribution
! Probability distribution function
! Mean
! Variance
|-
| [[Binomial distribution]]
| <math>\Pr\,(X=k) = \binom{n}{k}p^k(1 - p)^{n-k}</math>
| <math>np</math>
! <math>np(1 - p)</math>
|-
| [[Geometric distribution]]
| <math>\Pr\,(X=k) = (1 - p)^{k-1}p</math>
| <math>\frac{1}{p}</math>
! <math>\frac{(1 - p)}{p^2}</math>
|-
| [[Normal distribution]]
| <math>f\left(x \mid \mu, \sigma^2\right) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{1}{2} {\left(\frac{x - \mu}{\sigma}\right)}^2}</math>
| <math>\mu</math>
! <math>\sigma^2</math>
|-
| [[Uniform distribution (continuous)]]
| <math>f(x \mid a, b) = \begin{cases}
  \frac{1}{b - a} & \text{for } a \le x \le b, \\[3pt]
                0 & \text{for } x < a \text{ or } x > b
  \end{cases}</math>
| <math>\frac{a + b}{2}</math>
! <math>\frac{(b - a)^2}{12}</math>
|-
| [[Exponential distribution]]
| <math>f(x \mid \lambda) = \lambda e^{-\lambda x}</math>
| <math>\frac{1}{\lambda}</math>
! <math>\frac{1}{\lambda^2}</math>
|-
| [[Poisson distribution]]
| <math>f(k \mid \lambda) = \frac{e^{-\lambda}\lambda^{k}}{k!}</math>
| <math>\lambda </math>
! <math>\lambda </math>
|}

== Properties ==

=== Basic properties ===
Variance is non-negative because the squares are positive or zero:
<math display="block">\operatorname{Var}(X)\ge 0.</math>

The variance of a constant is zero.
<math display="block">\operatorname{Var}(a) = 0.</math>

Conversely, if the variance of a random variable is 0, then it is [[Almost surely|almost surely]] a constant. That is, it always has the same value:
<math display="block">\operatorname{Var}(X)= 0 \iff \exists a : P(X=a) = 1.</math>

=== Issues of finiteness ===
If a distribution does not have a finite expected value, as is the case for the [[Cauchy distribution]], 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]] whose index <math>k</math> satisfies <math>1 < k \leq 2.</math>

=== Decomposition ===

The general formula for variance decomposition or the [[Law of total variance|law of total variance]] is: If <math>X</math> and <math>Y</math> are two random variables, and the variance of <math>X</math> exists, then
<math display="block">\operatorname{Var}[X] = \operatorname{E}(\operatorname{Var}[X\mid Y]) + \operatorname{Var}(\operatorname{E}[X\mid Y]).</math>

The [[Conditional expectation|conditional expectation]] <math>\operatorname E(X\mid Y)</math> of <math>X</math> given <math>Y</math>, and the [[Conditional variance|conditional variance]] <math>\operatorname{Var}(X\mid Y)</math> may be understood as follows. Given any particular value ''y'' of&nbsp;the random variable&nbsp;''Y'', there is a conditional expectation <math>\operatorname E(X\mid Y=y)</math>  given the event&nbsp;''Y''&nbsp;=&nbsp;''y''. This quantity depends on the particular value&nbsp;''y''; it is a function <math> g(y) = \operatorname E(X\mid Y=y)</math>. That same function evaluated at the random variable ''Y'' is the conditional expectation {{tmath|1= \operatorname E(X\mid Y) = g(Y) }}.

In particular, if <math>Y</math> is a discrete random variable assuming possible values <math>y_1, y_2, y_3 \ldots</math> with corresponding probabilities <math>p_1, p_2, p_3 \ldots, </math>, then in the formula for total variance, the first term on the right-hand side becomes
<math display="block">\operatorname{E}(\operatorname{Var}[X \mid Y]) = \sum_i p_i \sigma^2_i,</math>
where <math>\sigma^2_i = \operatorname{Var}[X \mid Y = y_i]</math>. Similarly, the second term on the right-hand side becomes
<math display="block">\operatorname{Var}(\operatorname{E}[X \mid Y]) = \sum_i p_i \mu_i^2 - \left(\sum_i p_i \mu_i\right)^2 = \sum_i p_i \mu_i^2 - \mu^2,</math>
where {{tmath|= \mu_i = \operatorname{E}[X \mid Y = y_i] }} and {{tmath|1= \textstyle \mu = \sum_i p_i \mu_i }}. Thus the total variance is given by
<math display="block">\operatorname{Var}[X] = \sum_i p_i \sigma^2_i + \left( \sum_i p_i \mu_i^2 - \mu^2 \right).</math>

A similar formula is applied in [[Analysis of variance|analysis of variance]], where the corresponding formula is\
<math display="block">\mathit{MS}_\text{total} = \mathit{MS}_\text{between} + \mathit{MS}_\text{within};</math>
here <math>\mathit{MS}</math> refers to the Mean of the Squares. In [[Linear regression|linear regression]] analysis the corresponding formula is
<math display="block">\mathit{MS}_\text{total} = \mathit{MS}_\text{regression} + \mathit{MS}_\text{residual}.</math>

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, <math>\mathit{SS}</math>):
<math display="block">\mathit{SS}_\text{total} = \mathit{SS}_\text{between} + \mathit{SS}_\text{within},</math>
<math display="block">\mathit{SS}_\text{total} = \mathit{SS}_\text{regression} + \mathit{SS}_\text{residual}.</math>

=== Calculation from the CDF ===

The population variance for a non-negative random variable can be expressed in terms of the [[Cumulative distribution function|cumulative distribution function]] ''F'' using
<math display="block">2\int_0^\infty u(1 - F(u))\,du - {\left[\int_0^\infty (1 - F(u))\,du\right]}^2.</math>

This expression can be used to calculate the variance in situations where the CDF, but not the [[Physics:Probability density function|density]], can be conveniently expressed.

=== Characteristic property ===
The second [[Moment (mathematics)|moment]] of a random variable attains the minimum value when taken around the first moment (i.e., mean) of the random variable, i.e. {{tmath|1= \mathrm{argmin}_m \, \mathrm{E}\left(\left(X - m\right)^2\right) = \mathrm{E}(X) }}. Conversely, if a continuous function <math>\varphi</math> satisfies <math>\mathrm{argmin}_m\,\mathrm{E}(\varphi(X - m)) = \mathrm{E}(X)</math> for all random variables {{math|''X''}}, then it is necessarily of the form {{tmath|1= \varphi(x) = a x^2 + b }}, where {{math|''a'' > 0}}. This also holds in the multidimensional case.<ref>{{Cite journal | last1 = Kagan | first1 = A. | last2 = Shepp | first2 = L. A. | doi = 10.1016/S0167-7152(98)00041-8 | title = Why the variance? | journal = Statistics & Probability Letters | volume = 38 | issue = 4 | pages = 329–333 | year = 1998 }}</ref>

=== Units of measurement ===

Unlike the [[Average absolute deviation|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 their [[Standard deviation|standard deviation]] or [[Root mean square deviation|root mean square deviation]] is often preferred over using the variance.  In the dice example the standard deviation is {{math|{{sqrt|2.9}} ≈ 1.7}}, slightly larger than the expected absolute deviation of&nbsp;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|covariance]], is used frequently in theoretical statistics; however the expected absolute deviation tends to be more [[Robust statistics|robust]] as it is less sensitive to [[Outlier|outlier]]s arising from measurement anomalies or an unduly [[Heavy-tailed distribution|heavy-tailed distribution]].

== Propagation ==

=== Addition and multiplication by a constant ===
Variance is [[Invariant (mathematics)|invariant]] with respect to changes in a [[Location parameter|location parameter]].  That is, if a constant is added to all values of the variable, the variance is unchanged:
<math display="block">\operatorname{Var}(X+a)=\operatorname{Var}(X).</math>

If all values are scaled by a constant, the variance is [[Homogeneous function|scaled]] by the square of that constant:
<math display="block">\operatorname{Var}(aX)=a^2\operatorname{Var}(X).</math>

The variance of a sum of two random variables is given by
<math display="block">\begin{align}
\operatorname{Var}(aX + bY) &= a^2\operatorname{Var}(X) + b^2\operatorname{Var}(Y) + 2ab\, \operatorname{Cov}(X,Y) \\[1ex]
\operatorname{Var}(aX - bY) &= a^2\operatorname{Var}(X) + b^2\operatorname{Var}(Y) - 2ab\, \operatorname{Cov}(X,Y)
\end{align}</math>
where <math>\operatorname{Cov}(X,Y)</math> is the [[Covariance|covariance]].

=== Linear combinations ===
In general, for the sum of <math>N</math> random variables {{tmath| \{X_1,\dots,X_N\} }}, the variance becomes:
<math display="block">\operatorname{Var}\left(\sum_{i=1}^N X_i\right) = \sum_{i,j=1}^N\operatorname{Cov}(X_i,X_j) = \sum_{i=1}^N\operatorname{Var}(X_i) + \sum_{i,j=1,i\ne j}^N\operatorname{Cov}(X_i,X_j),</math> 
see also general [[Bienaymé's identity]].

These results lead to the variance of a [[Linear combination|linear combination]] as:
<math display="block">\begin{align}
\operatorname{Var}\left( \sum_{i=1}^N a_iX_i\right) &=\sum_{i,j=1}^{N} a_ia_j\operatorname{Cov}(X_i,X_j) \\
&= \sum_{i=1}^N a_i^2 \operatorname{Var}(X_i) +   \sum_{i \neq j}            a_i a_j \operatorname{Cov}(X_i,X_j)\\
&= \sum_{i=1}^N a_i^2 \operatorname{Var}(X_i) + 2 \sum_{1 \leq i < j \leq N} a_i a_j \operatorname{Cov}(X_i,X_j).
\end{align}</math>

If the random variables <math>X_1,\dots,X_N</math> are such that
<math display="block">\operatorname{Cov}(X_i,X_j)=0\ ,\ \forall\ (i\ne j) ,</math>
then they are said to be [[Covariance#Definition|uncorrelated]]. It follows immediately from the expression given earlier that if the random variables <math>X_1,\dots,X_N</math> are uncorrelated, then the variance of their sum is equal to the sum of their variances, or, expressed symbolically:
<math display="block">\operatorname{Var}\left(\sum_{i=1}^N X_i\right) = \sum_{i=1}^N\operatorname{Var}(X_i).</math>

Since independent random variables are always uncorrelated (see {{Section link|Covariance|Uncorrelatedness and independence}}), the equation above holds in particular when the random variables <math>X_1,\dots,X_n</math> 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 <math>X</math> as a column vector of <math>n</math> random variables {{tmath| X_1, \ldots,X_n }}, and <math>c</math> as a column vector of <math>n</math> scalars {{tmath| c_1, \ldots,c_n }}. Therefore, <math>c^\mathsf{T} X</math> is a [[Linear combination|linear combination]] of these random variables, where <math>c^\mathsf{T}</math> denotes the [[Transpose|transpose]] of {{tmath| c }}. Also let <math>\Sigma</math> be the [[Covariance matrix|covariance matrix]] of {{tmath| X }}. The variance of <math>c^\mathsf{T}X</math> is then given by:<ref>{{cite book | last1=Johnson | first1=Richard | last2=Wichern | first2=Dean | year=2001 | title=Applied Multivariate Statistical Analysis | url=https://archive.org/details/appliedmultivari00john_130 | url-access=limited | publisher=Prentice Hall | page=[https://archive.org/details/appliedmultivari00john_130/page/n96 76] | isbn=0-13-187715-1 }}</ref>
<math display="block">\operatorname{Var}\left(c^\mathsf{T} X\right) = c^\mathsf{T} \Sigma c .</math>

This implies that the variance of the mean can be written as (with a column vector of ones)
<math display="block">\operatorname{Var}\left(\bar{x}\right) = \operatorname{Var}\left(\frac{1}{n} 1'X\right) = \frac{1}{n^2} 1'\Sigma 1.</math>

=== Sum of variables ===

==== Sum of uncorrelated variables ====
{{main article|Bienaymé's identity}}
{{see also|Sum of normally distributed random variables}}
One reason for the use of the variance in preference to other measures of dispersion is that the variance of the sum (or the difference) of uncorrelated random variables is the sum of their variances:
<math display="block">\operatorname{Var}\left(\sum_{i=1}^n X_i\right) = \sum_{i=1}^n \operatorname{Var}(X_i).</math>

This statement is called the Bienaymé formula<ref>[[Biography:Michel Loève|Loève, M.]] (1977) "Probability Theory", ''Graduate Texts in Mathematics'', Volume 45, 4th edition, Springer-Verlag, p.&nbsp;12.</ref> and was discovered in 1853.<ref>Bienaymé, I.-J. (1853) "Considérations à l'appui de la découverte de Laplace sur la loi de probabilité dans la méthode des moindres carrés", ''Comptes rendus de l'Académie des sciences Paris'', 37, p.&nbsp;309–317; digital copy available [http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-2994&I=313] {{webarchive |url=https://web.archive.org/web/20180623145935/http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-2994&I=313 |date=2018-06-23 }}</ref><ref>Bienaymé, I.-J. (1867) "Considérations à l'appui de la découverte de Laplace sur la loi de probabilité dans la méthode des moindres carrés", ''Journal de Mathématiques Pures et Appliquées, Série 2'', Tome 12, p.&nbsp;158–167; digital copy available [http://gallica.bnf.fr/ark:/12148/bpt6k16411c/f166.image.n19][http://sites.mathdoc.fr/JMPA/PDF/JMPA_1867_2_12_A10_0.pdf]</ref> It is often made with the stronger condition that the variables are independent, but being uncorrelated suffices. So if all the variables have the same variance {{math|''σ''<sup>2</sup>}}, then, since division by {{math|''n''}} is a linear transformation, this formula immediately implies that the variance of their mean is
<math display="block">
  \operatorname{Var}\left(\overline{X}\right) = \operatorname{Var}\left(\frac{1}{n} \sum_{i=1}^n X_i\right) =
  \frac{1}{n^2}\sum_{i=1}^n \operatorname{Var}\left(X_i\right) =
  \frac{1}{n^2}n\sigma^2 =
  \frac{\sigma^2}{n}.
</math>

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 standard error of the sample mean, which is used in the [[Central limit theorem|central limit theorem]].

To prove the initial statement, it suffices to show that
<math display="block">\operatorname{Var}(X + Y) = \operatorname{Var}(X) + \operatorname{Var}(Y).</math>

The general result then follows by induction.  Starting with the definition,
<math display="block">\begin{align}
  \operatorname{Var}(X + Y) &= \operatorname{E}\left[(X + Y)^2\right] - (\operatorname{E}[X + Y])^2 \\[5pt]
    &= \operatorname{E}\left[X^2 + 2XY + Y^2\right] - (\operatorname{E}[X] + \operatorname{E}[Y])^2.
\end{align}</math>

Using the linearity of the expectation operator and the assumption of independence (or uncorrelatedness) of ''X'' and ''Y'', this further simplifies as follows:
<math display="block">\begin{align}
   \operatorname{Var}(X + Y) &= \operatorname{E}{\left[X^2\right]} + 2\operatorname{E}[XY] + \operatorname{E}{\left[Y^2\right]} - \left(\operatorname{E}[X]^2 + 2\operatorname{E}[X] \operatorname{E}[Y] + \operatorname{E}[Y]^2\right) \\[5pt]
   &= \operatorname{E}\left[X^2\right] + \operatorname{E}\left[Y^2\right] - \operatorname{E}[X]^2 - \operatorname{E}[Y]^2 \\[5pt]
   &= \operatorname{Var}(X) + \operatorname{Var}(Y).
\end{align}</math>

==== Sum of correlated variables ====

===== Sum of correlated variables with fixed sample size =====
{{main article|Bienaymé's identity}}
In general, the variance of the sum of {{math|n}} variables is the sum of their [[Covariance|covariance]]s:
<math display="block">\operatorname{Var}\left(\sum_{i=1}^n X_i\right) = \sum_{i=1}^n \sum_{j=1}^n \operatorname{Cov}\left(X_i, X_j\right) = \sum_{i=1}^n \operatorname{Var}\left(X_i\right) + 2 \sum_{1 \leq i < j\leq n} \operatorname{Cov}\left(X_i, X_j\right).</math>
(Note: The second equality comes from the fact that {{math|1=Cov(''X''<sub>''i''</sub>, ''X''<sub>''i''</sub>) = Var(''X''<sub>''i''</sub>)}}.)

Here, <math>\operatorname{Cov}(\cdot,\cdot)</math> is the [[Covariance|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. 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|classical test theory]].

So, if the variables have equal variance {{math|''σ''<sup>2</sup>}} and the average [[Correlation|correlation]] of distinct variables is {{math|''ρ''}}, then the variance of their mean is
<math display="block">\operatorname{Var}\left(\overline{X}\right) = \frac{\sigma^2}{n} + \frac{n - 1}{n}\rho\sigma^2.</math>

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 [[Standard error|uncertainty of the mean]]. Moreover, if the variables have unit variance, for example if they are standardized, then this simplifies to
<math display="block">\operatorname{Var}\left(\overline{X}\right) = \frac{1}{n} + \frac{n - 1}{n}\rho.</math>

This formula is used in the [[Spearman–Brown prediction formula]] of classical test theory. This converges to {{math|''ρ''}} if {{math|''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
<math display="block">\lim_{n \to \infty} \operatorname{Var}\left(\overline{X}\right) = \rho.</math>

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|law of large numbers]] 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 {{math|''N''}} is a random variable whose variation adds to the variation of {{math|''X''}}, such that,<ref>Cornell, J R, and Benjamin, C A, ''Probability, Statistics, and Decisions for Civil Engineers'', McGraw-Hill, NY, 1970, pp. 178-9.</ref>
<math display="block">\operatorname{Var}\left(\sum_{i=1}^{N}X_i\right)=\operatorname{E}\left[N\right]\operatorname{Var}(X)+\operatorname{Var}(N)(\operatorname{E}\left[X\right])^2</math>
which follows from the [[Law of total variance|law of total variance]].

If {{math|''N''}} has a [[Poisson distribution]], then <math>\operatorname{E}[N]=\operatorname{Var}(N)</math> with estimator {{math|1=''n'' = ''N''}}. So, the estimator of <math>\textstyle \operatorname{Var}\left(\sum_{i=1}^{n}X_i\right)</math> becomes {{tmath| \textstyle n{S_x}^2+n\bar{X}^2 }}, giving <math>\textstyle \operatorname{SE}(\bar{X})=\sqrt{{({S_x}^2+\bar{X}^2)}/{n}}</math>
(see ''{{slink|Standard error#Standard error of the sample mean}}'').

==== Weighted sum of variables ====
{{see also|Weighted arithmetic mean#Variance{{!}}Variance of a weighted arithmetic mean}}
{{distinguish|Weighted variance}}

The scaling property and the Bienaymé formula, along with the property of the [[Covariance|covariance]] {{math|Cov(''aX'',&nbsp;''bY'') {{=}} ''ab'' Cov(''X'',&nbsp;''Y'')}}  jointly imply that
<math display="block">\operatorname{Var}(aX \pm bY) =a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y) \pm 2ab\, \operatorname{Cov}(X, Y).</math>

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:
<math display="block">\operatorname{Var}\left(\sum_{i}^n a_iX_i\right) = \sum_{i=1}^na_i^2 \operatorname{Var}(X_i) + 2\sum_{1\le i}\sum_{<j\le n}a_ia_j\operatorname{Cov}(X_i,X_j)</math>

=== Product of variables ===

==== Product of independent variables ====

If two variables X and Y are [[Independence (probability theory)|independent]], the variance of their product is given by<ref>{{Cite journal |last=Goodman |first=Leo A. |date=December 1960 |title=On the Exact Variance of Products |journal=Journal of the American Statistical Association |volume=55 |issue=292 |pages=708–713 |doi=10.2307/2281592 |jstor=2281592}}</ref>
<math display="block">\operatorname{Var}(XY) = [\operatorname{E}(X)]^2 \operatorname{Var}(Y) + [\operatorname{E}(Y)]^2 \operatorname{Var}(X) + \operatorname{Var}(X)\operatorname{Var}(Y).</math>

Equivalently, using the basic properties of expectation, it is given by
<math display="block">\operatorname{Var}(XY) = \operatorname{E}\left(X^2\right) \operatorname{E}\left(Y^2\right) - [\operatorname{E}(X)]^2 [\operatorname{E}(Y)]^2.</math>

==== Product of statistically dependent variables ====

In general, if two variables are statistically dependent, then the variance of their product is given by:
<math display="block">\begin{align}
  \operatorname{Var}(XY)
    ={} &\operatorname{E}\left[X^2 Y^2\right] - [\operatorname{E}(XY)]^2 \\[5pt]
    ={} &\operatorname{Cov}\left(X^2, Y^2\right) + \operatorname{E}(X^2)\operatorname{E}\left(Y^2\right) - [\operatorname{E}(XY)]^2 \\[5pt]
    ={} &\operatorname{Cov}\left(X^2, Y^2\right) + \left(\operatorname{Var}(X) + [\operatorname{E}(X)]^2\right) \left(\operatorname{Var}(Y) + [\operatorname{E}(Y)]^2\right) \\[5pt]
        &- [\operatorname{Cov}(X, Y) + \operatorname{E}(X)\operatorname{E}(Y)]^2
\end{align}</math>

=== Arbitrary functions ===

The [[Delta method|delta method]] uses second-order Taylor expansions 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
<math display="block">\operatorname{Var}\left[f(X)\right] \approx \left(f'(\operatorname{E}\left[X\right])\right)^2\operatorname{Var}\left[X\right]</math>
provided that {{math|''f''}} is twice differentiable and that the mean and variance of {{math|''X''}} are finite.

== Population variance and sample variance ==
{{anchor|Estimation}}
{{see also|Unbiased estimation of standard deviation}}
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 [[Estimation theory|estimates]] the mean and variance from a limited set of observations by using an [[Estimator|estimator]] equation. The estimator is a function of the [[Sample (statistics)|sample]] of {{math|''n''}} observations drawn without observational bias from the whole [[Statistical population|population]] of potential observations. In this example, the 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|consistent estimator]]s (they converge to the value of the whole population as the number of samples increases) but can be improved. Most simply, the sample variance is computed as the sum of squared deviations about the (sample) mean, divided by {{math|''n''}} as the number of samples. However, using values other than {{math|''n''}} improves the estimator in various ways. Four common values for the denominator are {{math|''n''}}, {{math|''n'' − 1}}, {{math|''n'' + 1}}, and {{math|''n'' − 1.5}}: {{math|''n''}} is the simplest (the variance of the sample), {{math|''n'' − 1}} eliminates bias,<ref name=bessel /> {{math|''n'' + 1}} minimizes [[Mean squared error|mean squared error]] for the normal distribution,<ref name=Kourouklis/> and {{math|''n'' − 1.5}} mostly eliminates bias in [[Unbiased estimation of standard deviation|unbiased estimation of standard deviation]] for the normal distribution.<ref>{{cite journal | last = Brugger | first = R. M. | year = 1969 | title = A Note on Unbiased Estimation of the Standard Deviation | journal = The American Statistician | volume = 23 | issue = 4 | pages = 32 | doi = 10.1080/00031305.1969.10481865 }}</ref>

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|biased estimator]]: it underestimates the variance by a factor of {{math|(''n'' − 1) / ''n''}}; correcting this factor, resulting in the sum of squared deviations about the sample mean divided by {{math|''n'' − 1}} instead of {{math|''n''}}, is called ''[[Bessel's correction]]''.<ref name=bessel>{{cite book | last = Reichmann | first = W. J. | title = Use and Abuse of Statistics | publisher = Methuen | year = 1961 | edition = Reprinted 1964–1970 by Pelican | location = London | chapter = Appendix 8 }}</ref> The resulting estimator is unbiased and is called the '''(corrected) sample variance''' or '''unbiased sample 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|mean squared error]] 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 ''{{slink|Mean squared error#Variance}}'') and introduces bias. This always consists of scaling down the unbiased estimator (dividing by a number larger than {{math|''n'' − 1}}) and is a simple example of a [[Shrinkage estimator|shrinkage estimator]]: one "shrinks" the unbiased estimator towards zero. For the normal distribution, dividing by {{math|''n'' + 1}} (instead of {{math|''n'' − 1}} or {{math|''n''}}) minimizes mean squared error.<ref name=Kourouklis>{{cite journal |last=Kourouklis |first=Stavros |date=2012 |title=A New Estimator of the Variance Based on Minimizing Mean Squared Error |url=https://www.jstor.org/stable/23339501 |journal=The American Statistician |volume=66 |issue=4 |pages=234–236 |doi=10.1080/00031305.2012.735209 |jstor=23339501 |issn=0003-1305|url-access=subscription }}</ref> The resulting estimator is biased, however, and is known as the '''biased sample variation'''.

=== Population variance ===
In general, the '''''population variance''''' of a ''finite'' [[Statistical population|population]] of size {{mvar|N}} with values {{math|''x''<sub>''i''</sub>}} is given by
<math display="block">\begin{align}
  \sigma^2 &= \frac{1}{N} \sum_{i=1}^N {\left(x_i - \mu\right)}^2
            = \frac{1}{N} \sum_{i=1}^N  \left(x_i^2 - 2 \mu x_i + \mu^2 \right) \\[5pt]
           &= \left(\frac{1}{N} \sum_{i=1}^N x_i^2\right) - 2\mu \left(\frac{1}{N} \sum_{i=1}^N x_i\right) + \mu^2 \\[5pt]
           &= \operatorname{E}[x_i^2] - \mu^2 
\end{align}</math>
where the population mean is <math display="inline">\mu = \operatorname{E}[x_i] = \frac 1N \sum_{i=1}^N x_i </math> and {{tmath|1= \textstyle \operatorname{E}[x_i^2] = \left(\frac{1}{N} \sum_{i=1}^N x_i^2\right) }}, where <math display="inline">\operatorname{E} </math> is the [[Expected value|expectation value]] operator.

The population variance can also be computed using<ref>{{cite conference |author=Yuli Zhang |author2=Huaiyu Wu |author3=Lei Cheng |date=June 2012 |title=Some new deformation formulas about variance and covariance|conference=Proceedings of 4th International Conference on Modelling, Identification and Control(ICMIC2012) |pages=987–992}}</ref>
<math display="block">\sigma^2 = \frac {1} {N^2}\sum_{i<j}\left( x_i-x_j \right)^2 = \frac{1}{2N^2} \sum_{i, j=1}^N\left( x_i-x_j \right)^2.</math>

(The right side has duplicate terms in the sum while the middle side has only unique terms to sum.) This is true because
<math display="block">\begin{align}
      &\frac{1}{2N^2} \sum_{i, j=1}^N {\left( x_i - x_j \right)}^2 \\[5pt]
  ={} &\frac{1}{2N^2} \sum_{i, j=1}^N \left( x_i^2 - 2x_i x_j  + x_j^2 \right) \\[5pt]
  ={} &\frac{1}{2N} \sum_{j=1}^N \left(\frac{1}{N} \sum_{i=1}^N x_i^2\right) - \left(\frac{1}{N} \sum_{i=1}^N x_i\right) \left(\frac{1}{N} \sum_{j=1}^N x_j\right) + \frac{1}{2N} \sum_{i=1}^N \left(\frac{1}{N} \sum_{j=1}^N x_j^2\right) \\[5pt]
  ={} &\frac{1}{2} \left( \sigma^2 + \mu^2 \right) - \mu^2 + \frac{1}{2} \left( \sigma^2 + \mu^2 \right) \\[5pt]
  ={} &\sigma^2.
\end{align}</math>

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.

=== Sample variance ===
{{see also|Sample standard deviation}}

==== 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 (statistics)|sample]] of the population.<ref>{{cite book | last = Navidi | first = William | year = 2006 | title = Statistics for Engineers and Scientists | publisher = McGraw-Hill | page = 14 }}</ref> This is generally referred to as '''sample variance''' or '''empirical variance'''. 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 {{mvar|n}} values {{math|''Y''<sub>1</sub>, ..., ''Y''<sub>''n''</sub>}} from the population of size {{mvar|N}}, where {{math|''n'' < ''N''}}, and estimate the variance on the basis of this sample.<ref>Montgomery, D. C. and Runger, G. C. (1994) ''Applied statistics and probability for engineers'', page 201. John Wiley & Sons New York</ref> Directly taking the variance of the sample data gives the average of the squared deviations:<ref>{{cite conference | author1 = Yuli Zhang | author2 = Huaiyu Wu | author3 = Lei Cheng | date = June 2012 | title = Some new deformation formulas about variance and covariance | conference = Proceedings of 4th International Conference on Modelling, Identification and Control(ICMIC2012) | pages = 987–992 }}</ref>
<math display="block">\tilde{S}_Y^2 =
  \frac{1}{n} \sum_{i=1}^n \left(Y_i - \overline{Y}\right)^2 = \left(\frac 1n \sum_{i=1}^n Y_i^2\right) - \overline{Y}^2 =
  \frac{1}{n^2} \sum_{i,j\,:\,i<j}\left(Y_i - Y_j\right)^2.
</math>
(See the section ''{{slink|#Population variance}}'' for the derivation of this formula.) Here, <math>\overline{Y}</math> denotes the [[Sample mean|sample mean]]: 
<math display="block">\overline{Y} = \frac{1}{n} \sum_{i=1}^n Y_i .</math>

Since the {{math|''Y''<sub>''i''</sub>}} are selected randomly, both <math>\overline{Y}</math> and <math>\tilde{S}_Y^2</math> are [[Random variable|random variables]]. Their expected values can be evaluated by averaging over the ensemble of all possible samples {{math|{{mset|''Y''<sub>''i''</sub>}}}} of size {{mvar|n}} from the population. For <math>\tilde{S}_Y^2</math> this gives:
<math display="block">\begin{align}
  \operatorname{E}[\tilde{S}_Y^2]
    &= \operatorname{E}\left[ \frac{1}{n} \sum_{i=1}^n {\left(Y_i - \frac{1}{n} \sum_{j=1}^n Y_j \right)}^2 \right] \\[5pt]
    &= \frac 1n \sum_{i=1}^n \operatorname{E}\left[ Y_i^2 - \frac{2}{n} Y_i \sum_{j=1}^n Y_j + \frac{1}{n^2} \sum_{j=1}^n Y_j \sum_{k=1}^n Y_k \right] \\[5pt]
    &= \frac 1n \sum_{i=1}^n \left( \operatorname{E}\left[Y_i^2\right] - \frac{2}{n} \left( \sum_{j \neq i} \operatorname{E}\left[Y_i Y_j\right] + \operatorname{E}\left[Y_i^2\right] \right) + \frac{1}{n^2} \sum_{j=1}^n \sum_{k \neq j}^n \operatorname{E}\left[Y_j Y_k\right] +\frac{1}{n^2} \sum_{j=1}^n \operatorname{E}\left[Y_j^2\right] \right) \\[5pt]
    &= \frac 1n \sum_{i=1}^n \left( \frac{n - 2}{n} \operatorname{E}\left[Y_i^2\right] - \frac{2}{n} \sum_{j \neq i} \operatorname{E}\left[Y_i Y_j\right] + \frac{1}{n^2} \sum_{j=1}^n \sum_{k \neq j}^n \operatorname{E}\left[Y_j Y_k\right] +\frac{1}{n^2} \sum_{j=1}^n \operatorname{E}\left[Y_j^2\right] \right) \\[5pt]
    &= \frac 1n \sum_{i=1}^n \left[ \frac{n - 2}{n} \left(\sigma^2 + \mu^2\right) - \frac{2}{n} (n - 1)\mu^2 + \frac{1}{n^2} n(n - 1)\mu^2 + \frac{1}{n} \left(\sigma^2 + \mu^2\right) \right] \\[5pt]
    &= \frac{n - 1}{n} \sigma^2.
\end{align}</math>

Here <math display="inline">\sigma^2 = \operatorname{E}[Y_i^2] - \mu^2 </math> derived in the section is [[Variance#Population variance|population variance]] and <math display="inline">\operatorname{E}[Y_i Y_j] = \operatorname{E}[Y_i] \operatorname{E}[Y_j] = \mu^2</math> due to independency of <math>Y_i</math> and {{tmath| Y_j }}.

Hence <math display="inline">\tilde{S}_Y^2</math> gives an estimate of the population variance <math display="inline">\sigma^2</math> that is biased by a factor of <math display="inline">\frac{n - 1}{n}</math> because the expectation value of <math display="inline">\tilde{S}_Y^2</math> is smaller than the population variance (true variance) by that factor. For this reason, <math display="inline">\tilde{S}_Y^2</math> is referred to as the ''biased sample variance''.

==== Unbiased sample variance ====
Correcting for this bias yields the ''unbiased sample variance'', denoted {{tmath| S^2 }}:
<math display="block">S^2 = \frac{n}{n - 1} \tilde{S}_Y^2 = \frac{n}{n - 1} \left[ \frac{1}{n} \sum_{i=1}^n \left(Y_i - \overline{Y}\right)^2 \right] = \frac{1}{n - 1} \sum_{i=1}^n \left(Y_i - \overline{Y} \right)^2</math>

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

The unbiased sample variance is a [[U-statistic]] for the function {{math|1=''f''(''y''<sub>1</sub>, ''y''<sub>2</sub>) = (''y''<sub>1</sub> − ''y''<sub>2</sub>)<sup>2</sup>/2}}, meaning that it is obtained by averaging a 2-sample statistic over 2-element subsets of the population.

===== Example =====
For a set of numbers {{mset|10, 15, 30, 45, 57, 52, 63, 72, 81, 93, 102, 105}}, if this set is the whole data population for some measurement, then variance is the population variance 932.743 as the sum of the squared deviations about the mean of this set, divided by 12 as the number of the set members. If the set is a sample from the whole population, then the unbiased sample variance can be calculated as 1017.538 that is the sum of the squared deviations about the mean of the sample, divided by 11 instead of 12. A function VAR.S in [[Software:Microsoft Excel|Microsoft Excel]] gives the unbiased sample variance while VAR.P is for population variance.

==== Distribution of the sample variance ====
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Being a function of [[Random variable|random variable]]s, the sample variance is itself a random variable, and it is natural to study its distribution. In the case that ''Y''<sub>''i''</sub> are independent observations from a [[Normal distribution|normal distribution]], [[Cochran's theorem]] shows that the [[Variance#Unbiased sample variance|unbiased sample variance]] ''S''<sup>2</sup> follows a scaled [[Chi-squared distribution|chi-squared distribution]] (see also: [[Chi-squared distribution#Asymptotic properties|asymptotic properties]] and an [[Chi-squared distribution#Cochran's theorem|elementary proof]]):<ref>{{cite book | last = Knight | first = K. | year = 2000 | title = Mathematical Statistics | publisher = Chapman and Hall | location = New York | at = proposition 2.11 }}</ref>
<math display="block">
(n - 1) \frac{S^2}{\sigma^2} \sim \chi^2_{n-1} 
</math>
where {{math|''σ''<sup>2</sup>}} is the [[Variance#Population variance|population variance]]. As a direct consequence, it follows that 
<math display="block">
\operatorname{E}\left(S^2\right) = \operatorname{E}\left(\frac{\sigma^2}{n - 1} \chi^2_{n-1}\right) = \sigma^2 ,
</math>
and<ref>{{cite book | author1-last = Casella | author1-first = George | author2-last = Berger | author2-first = Roger L. | year = 2002 | title = Statistical Inference | at = Example 7.3.3, p.&nbsp;331 | edition = 2nd | isbn = 0-534-24312-6 }}</ref>
<math display="block">
 \operatorname{Var}\left[S^2\right] = \operatorname{Var}\left(\frac{\sigma^2}{n - 1} \chi^2_{n-1}\right) = \frac{\sigma^4}{{\left(n - 1\right)}^2} \operatorname{Var}\left(\chi^2_{n-1}\right) = \frac{2\sigma^4}{n - 1}.
</math>

If {{math|''Y''<sub>''i''</sub>}} are independent and identically distributed, but not necessarily normally distributed, then<ref>Mood, A. M., Graybill, F. A., and Boes, D.C. (1974) ''Introduction to the Theory of Statistics'', 3rd Edition, McGraw-Hill, New York, p. 229</ref>
<math display="block">
    \operatorname{E}\left[S^2\right] = \sigma^2, \quad
    \operatorname{Var}\left[S^2\right] = \frac{\sigma^4}{n} \left(\kappa - 1 + \frac{2}{n - 1} \right) = \frac{1}{n} \left(\mu_4 - \frac{n - 3}{n - 1}\sigma^4\right),
</math>
where ''κ'' is the [[Kurtosis|kurtosis]] of the distribution and {{math|''μ''<sub>4</sub>}} is the fourth [[Central moment|central moment]].

If the conditions of the [[Law of large numbers|law of large numbers]] hold for the squared observations, {{math|''S''<sup>2</sup>}} is a [[Consistent estimator|consistent estimator]] of&nbsp;{{math|''σ''<sup>2</sup>}}. 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.).<ref>{{cite book |last1=Kenney |first1=John F. |last2=Keeping |first2=E.S. |date=1951 |title=Mathematics of Statistics. Part Two. |edition=2nd |publisher=D. Van Nostrand Company, Inc. |location=Princeton, New Jersey |url=http://krishikosh.egranth.ac.in/bitstream/1/2025521/1/G2257.pdf |via=KrishiKosh |url-status=dead |archive-url=https://web.archive.org/web/20181117022434/http://krishikosh.egranth.ac.in/bitstream/1/2025521/1/G2257.pdf |archive-date= Nov 17, 2018 }}</ref><ref>Rose, Colin; Smith, Murray D. (2002). "[http://www.mathstatica.com/book/Mathematical_Statistics_with_Mathematica.pdf Mathematical Statistics with Mathematica]". Springer-Verlag, New York.</ref><ref>Weisstein, Eric W. "[http://mathworld.wolfram.com/SampleVarianceDistribution.html Sample Variance Distribution]". MathWorld Wolfram.</ref>

==== 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.<ref>{{cite journal |last=Samuelson |first=Paul |title=How Deviant Can You Be? |journal=[[Organization:Journal of the American Statistical Association|Journal of the American Statistical Association]] |volume=63 |issue=324 |year=1968 |pages=1522–1525 |jstor=2285901 |doi=10.1080/01621459.1968.10480944}}</ref> Values must lie within the limits {{tmath| \bar y \pm \sigma_Y (n-1)^{1/2} }}.


==== Effect of adding one observation on variance ====

When a single new observation <math>x_{n+1}</math> is added to a set of <math>n</math> observations with mean <math>\bar{x}_n</math> and variance <math>s_n^2</math>, the new variance <math>s_{n+1}^2</math> can be expressed using a recursive updating formula. Based on the identity for the sum of squares provided by Chan et al. (1983)<ref>{{cite journal |last1=Chan |first1=Tony F. |last2=Golub |first2=Gene H. |last3=LeVeque |first3=Randall J. |date=1983 |title=Algorithms for Computing the Sample Variance: Analysis and Recommendations |journal=The American Statistician |volume=37 |issue=3 |pages=242–247 |doi=10.2307/2683386 |url=https://www.jstor.org/stable/2683386}} formula 1.3(b [https://www.cs.yale.edu/publications/techreports/tr222.pdf page 1]</ref>:

:<math>n s_{n+1}^2 = (n-1)s_n^2 + \frac{n}{n+1}(x_{n+1} - \bar{x}_n)^2</math>

From this relationship, the impact of the new observation on the variance depends on its distance from the current mean. If <math>x_{n+1} = \bar{x}_n \pm s_n \sqrt{\frac{n+1}{n}}</math> then the '''variance will remain unchanged'''. Accordingly, if the new observation is closer to the mean (<math>|x_{n+1}| < \bar{x}_n + s_n \sqrt{\frac{n+1}{n}}</math>) then the variance will decrease - and if is further from the mean (<math>|x_{n+1}| > \bar{x}_n + s_n \sqrt{\frac{n+1}{n}}</math>) then the variance will increase.

{{hidden begin|style=width:100%|ta1=center|border=1px #aaa solid|title=[Proof]}}
'''Derivation for Sample Variance'''
Using the updating formula for the sum of squares (<math>SS</math>):
:<math>SS_{n+1} = SS_n + \frac{n}{n+1}(x_{n+1} - \bar{x}_n)^2</math>

Substituting the relationship for sample variance (<math>SS = (n-1)s^2</math>):
:<math>n s^2_{n+1} = (n-1)s^2_n + \frac{n}{n+1}(x_{n+1} - \bar{x}_n)^2</math>

Setting <math>s^2_{n+1} = s^2_n</math>:
:<math>n s^2_n = (n-1)s^2_n + \frac{n}{n+1}(x_{n+1} - \bar{x}_n)^2</math>
:<math>s^2_n = \frac{n}{n+1}(x_{n+1} - \bar{x}_n)^2</math>

Solving for <math>x_{n+1}</math> yields:
:<math>x_{n+1} = \bar{x}_n \pm s_n \sqrt{\frac{n+1}{n}}</math>

'''Derivation for Population Variance'''
For population variance (<math>\sigma^2 = \frac{SS}{n}</math>), the updating formula is:
:<math>(n+1)\sigma^2_{n+1} = n\sigma^2_n + \frac{n}{n+1}(x_{n+1} - \mu_n)^2</math>

Setting <math>\sigma^2_{n+1} = \sigma^2_n</math>:
:<math>(n+1)\sigma^2_n = n\sigma^2_n + \frac{n}{n+1}(x_{n+1} - \mu_n)^2</math>
:<math>\sigma^2_n = \frac{n}{n+1}(x_{n+1} - \mu_n)^2</math>

Solving for <math>x_{n+1}</math> yields:
:<math>x_{n+1} = \mu_n \pm \sigma_n \sqrt{\frac{n+1}{n}}</math>

{{hidden end}}

=== Relations with the harmonic and arithmetic means ===

It has been shown<ref>{{cite journal |first=A. McD. |last=Mercer |title=Bounds for A–G, A–H, G–H, and a family of inequalities of Ky Fan's type, using a general method |journal=J. Math. Anal. Appl. |volume=243 |issue=1 |pages=163–173 |year=2000 |doi=10.1006/jmaa.1999.6688 |doi-access=free }}</ref> that for a sample {{math|{{mset|''y''<sub>''i''</sub>}}}} of positive real numbers,
<math display="block">  \sigma_y^2 \le 2y_{\max} (A - H), </math>
where {{math|''y''<sub>max</sub>}} is the maximum of the sample, {{mvar|A}} is the arithmetic mean, {{mvar|H}} is the [[Harmonic mean|harmonic mean]] of the sample and <math>\sigma_y^2</math> is the (biased) variance of the sample.

This bound has been improved, and it is known that variance is bounded by
<math display="block">\begin{align}
 \sigma_y^2 &\le \frac{y_{\max} (A - H)(y_\max - A)}{y_\max - H}, \\[1ex]
 \sigma_y^2 &\ge \frac{y_{\min} (A - H)(A - y_\min)}{H - y_\min},
\end{align} </math>
where {{math|''y''<sub>min</sub>}} is the minimum of the sample.<ref name=Sharma2008>{{cite journal |first=R. |last=Sharma |title= Some more inequalities for arithmetic mean, harmonic mean and variance|journal= Journal of Mathematical Inequalities|volume=2 |issue=1 |pages=109–114 |year=2008 |doi=10.7153/jmi-02-11|citeseerx=10.1.1.551.9397 }}</ref>

== 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 both [[Median|median]]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 [[Bootstrapping (statistics)|bootstrap]] and the [[Resampling (statistics)|jackknife]], may be used to test the equality of variances.

== Moment of inertia ==
The variance of a probability distribution is analogous to the [[Physics:Moment of inertia|moment of inertia]] in [[Physics:Classical mechanics|classical mechanics]] of a corresponding mass distribution along a line, with respect to rotation about its center of mass.<ref name=pearson>{{Cite web |last=Magnello |first=M. Eileen |title=Karl Pearson and the Origins of Modern Statistics: An Elastician becomes a Statistician | url=https://rutherfordjournal.org/article010107.html |website=The Rutherford Journal}}</ref> It is because of this analogy that such things as the variance are called ''[[Moment (mathematics)|moment]]s'' of [[Probability distribution|probability distribution]]s.<ref name=pearson/> 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 <math>\Sigma</math> is given by <math display="block">I = n\left(\mathbf{1}_{3\times 3} \operatorname{tr}(\Sigma) - \Sigma\right).</math>

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
<math display="block">\Sigma = \begin{bmatrix} 10 & 0 & 0 \\ 0 & 0.1 & 0 \\ 0 & 0 & 0.1 \end{bmatrix}.</math>

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
<math display="block">I = n\begin{bmatrix} 0.2 & 0 & 0 \\ 0 & 10.1 & 0 \\ 0 & 0 & 10.1 \end{bmatrix}.</math>

== 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:
<math display="block">\text{Semivariance} = \frac{1}{n} \sum_{i:x_i < \mu} {\left(x_i - \mu\right)}^2</math>
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.<ref>{{cite web |url=https://famafrench.dimensional.com/questions-answers/qa-semi-variance-a-better-risk-measure.aspx |title=Q&A: Semi-Variance: A Better Risk Measure? |last1=Fama |first1=Eugene F. |last2=French |first2=Kenneth R. |date=2010-04-21 |website=Fama/French Forum }}</ref>

For inequalities associated with the semivariance, see ''{{slink|Chebyshev's inequality|Semivariances}}''.

== Etymology ==
The term ''variance'' was first introduced by [[Biography:Ronald Fisher|Ronald Fisher]] in his 1918 paper ''The Correlation Between Relatives on the Supposition of Mendelian Inheritance'':<ref>[[Biography:Ronald Fisher|Ronald Fisher]] (1918) [http://digital.library.adelaide.edu.au/dspace/bitstream/2440/15097/1/9.pdf The correlation between relatives on the supposition of Mendelian Inheritance]</ref>

<blockquote>The great body of available statistics show us that the deviations of a human measurement from its mean follow very closely the [[Normal distribution|Normal Law of Errors]], and, therefore, that the variability may be uniformly measured by the [[Standard deviation|standard deviation]] corresponding to the [[Square root|square root]] of the mean square error. When there are two independent causes of variability capable of producing in an otherwise uniform population distributions with standard deviations <math>\sigma_1</math> and <math>\sigma_2</math>, it is found that the distribution, when both causes act together, has a standard deviation <math>\sqrt{\sigma_1^2 + \sigma_2^2}</math>.  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...</blockquote>

== Generalizations ==

=== For complex variables ===
If <math>x</math> is a scalar [[Complex number|complex]]-valued random variable, with values in {{tmath| \mathbb{C} }}, then its variance is {{tmath| \operatorname{E}\left[(x - \mu)(x - \mu)^*\right] }}, where <math>x^*</math> is the [[Complex conjugate|complex conjugate]] of {{tmath| x }}.  This variance is a real scalar.

=== For vector-valued random variables ===

==== As a matrix ====
If <math>X</math> is a [[Vector space|vector]]-valued random variable, with values in <math>\mathbb{R}^n,</math> and thought of as a column vector, then a natural generalization of variance is <math>\operatorname{E}\left[(X - \mu) {(X - \mu)}^{\mathsf{T}}\right],</math> where <math>\mu = \operatorname{E}(X)</math> and <math>X^{\mathsf{T}}</math> is the transpose of {{mvar|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 <math>X</math> is a vector- and complex-valued random variable, with values in {{tmath| \mathbb{C}^n }}, then the [[Covariance matrix#Complex random vectors|covariance matrix is]] {{tmath| \operatorname{E}\left[(X - \mu){(X - \mu)}^\dagger\right] }}, where <math>X^\dagger</math> is the [[Conjugate transpose|conjugate transpose]] of {{tmath| X }}. This matrix is also positive semi-definite and square.

==== As a scalar ====
Another generalization of variance for vector-valued random variables <math>X</math>, which results in a scalar value rather than in a matrix, is the [[Generalized variance|generalized variance]] {{tmath| \det(C) }}, the [[Determinant|determinant]] of the covariance matrix. The generalized variance can be shown to be related to the multidimensional scatter of points around their mean.<ref>{{cite book |last1=Kocherlakota |first1=S. |title=Encyclopedia of Statistical Sciences |last2=Kocherlakota |first2=K. |chapter=Generalized Variance |publisher=Wiley Online Library |doi=10.1002/0471667196.ess0869 |year=2004 |isbn=0-471-66719-6 }}</ref>

A different generalization is obtained by considering the equation for the scalar variance, <math> \operatorname{Var}(X) = \operatorname{E}\left[(X - \mu)^2 \right] </math>, and reinterpreting <math>(X - \mu)^2</math> as the squared [[Euclidean distance]] between the random variable and its mean, or, simply as the scalar product of the vector <math>X - \mu</math> with itself. This results in <math>\operatorname{E}\left[(X - \mu)^{\mathsf{T}}(X - \mu)\right] = \operatorname{tr}(C),</math> which is the [[Trace (linear algebra)|trace]] of the covariance matrix.

== See also ==
* [[Bhatia–Davis inequality]]
* [[Coefficient of variation]]
* [[Homoscedasticity]]
* [[Least-squares spectral analysis]] for computing a [[Physics:Frequency spectrum|frequency spectrum]] with spectral magnitudes in % of variance or in [[Physics:Decibel|dB]]
* [[Finance:Modern portfolio theory|Modern portfolio theory]]
* [[Popoviciu's inequality on variances]]
* [[Statistical dispersion|Measures for statistical dispersion]]
* [[Variance-stabilizing transformation]]

=== Types of variance ===
* [[Correlation]]
* Distance variance
* Explained variance
* [[Pooled variance]]
* Pseudo-variance

== References ==
{{reflist}}

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{{theory of probability distributions}}
{{statistics|descriptive|state=collapsed}}


[[Category:Statistical deviation and dispersion]]
[[Category:Articles containing proofs]]

{{Sourceattribution|Variance}}