Variance#Addition and multiplication by a constant

The variance of a random variable $X$ is the expected value of the squared deviation from the mean of $X$, $\backslash mu\; =\; \backslash operatorname\{E\}[X]$:

$$\backslash operatorname\{Var\}(X)\; =\; \backslash operatorname\{E\}\backslash left[(X\; -\; \backslash mu)^2\; \backslash right].$$

This definition encompasses random variables that are generated by processes that are discrete, continuous, neither, or mixed. The variance can also be thought of as the covariance of a random variable with itself:

$$\backslash operatorname\{Var\}(X)\; =\; \backslash operatorname\{Cov\}(X,\; X).$$

The variance is also equivalent to the second cumulant of a probability distribution that generates $X$. The variance is typically designated as $\backslash operatorname\{Var\}(X)$, or sometimes as $V(X)$ or $\backslash mathbb\{V\}(X)$, or symbolically as $\backslash sigma^2\_X$ or simply $\backslash sigma^2$ (pronounced "sigma squared"). The expression for the variance can be expanded as follows:

$$\backslash begin\{align\}$$

\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}

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, 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.

If the generator of random variable $X$ is discrete with probability mass function $x\_1\; \backslash mapsto\; p\_1,\; x\_2\; \backslash mapsto\; p\_2,\; \backslash ldots,\; x\_n\; \backslash mapsto\; p\_n$, then

$$\backslash operatorname\{Var\}(X)\; =\; \backslash sum\_\{i=1\}^n\; p\_i\; \backslash cdot\; \{\backslash left(x\_i\; -\; \backslash mu\backslash right)\}^2,$$

where $\backslash mu$ is the expected value. That is,

$$\backslash mu\; =\; \backslash sum\_\{i=1\}^n\; p\_i\; x\_i\; .$$

(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

$$\backslash operatorname\{Var\}(X)\; =\; \backslash frac\{1\}\{n\}\; \backslash sum\_\{i=1\}^n\; (x\_i\; -\; \backslash mu)^2$$

where $\backslash mu$ is the average value. That is,

$$\backslash mu\; =\; \backslash frac\{1\}\{n\}\backslash sum\_\{i=1\}^n\; x\_i\; .$$

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:{{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}}

$$\backslash operatorname\{Var\}(X)\; =\; \backslash frac\{1\}\{n^2\}\; \backslash sum\_\{i=1\}^n\; \backslash sum\_\{j=1\}^n\; \backslash frac\{1\}\{2\}\; \{\backslash left(x\_i\; -\; x\_j\backslash right)\}^2\; =\; \backslash frac\{1\}\{n^2\}\; \backslash sum\_i\; \backslash sum\_\{j>i\}\; \{\backslash left(x\_i\; -\; x\_j\backslash right)\}^2.$$

If the random variable $X$ has a probability density function $f(x)$, and $F(x)$ is the corresponding cumulative distribution function, then

$$\backslash begin\{align\}$$

\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}

or equivalently,

$$\backslash operatorname\{Var\}(X)\; =\; \backslash int\_\{\backslash R\}\; x^2\; f(x)\; \backslash ,dx\; -\; \backslash mu^2\; ,$$

where $\backslash mu$ is the expected value of $X$ given by

$$\backslash mu\; =\; \backslash int\_\{\backslash R\}\; x\; f(x)\; \backslash ,\; dx\; =\; \backslash int\_\{\backslash R\}\; x\; \backslash ,\; dF(x).$$

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 $[a,b]\backslash subset\backslash R,$ then

$$\backslash operatorname\{Var\}(X)\; =\; \backslash int^\{+\backslash infty\}\_\{-\backslash infty\}\; x^2\; f(x)\; \backslash ,\; dx\; -\; \backslash mu^2,$$

where the integral is an improper Riemann integral.

The exponential distribution with parameter {{mvar|λ}} > 0 is a continuous distribution whose probability density function is given by

$$f(x)\; =\; \backslash lambda\; e^\{-\backslash lambda\; x\}$$

on the interval {{closed-open|0, ∞}}. Its mean can be shown to be

$$\backslash operatorname\{E\}[X]\; =\; \backslash int\_0^\backslash infty\; x\; \backslash lambda\; e^\{-\backslash lambda\; x\}\; \backslash ,\; dx\; =\; \backslash frac\{1\}\{\backslash lambda\}.$$

Using integration by parts and making use of the expected value already calculated, we have:

$$\backslash begin\{align\}$$

\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}

Thus, the variance of {{mvar|X}} is given by

$$\backslash operatorname\{Var\}(X)\; =\; \backslash operatorname\{E\}\backslash left[X^2\backslash right]\; -\; \backslash operatorname\{E\}[X]^2\; =\; \backslash frac\{2\}\{\backslash lambda^2\}\; -\; \backslash left(\backslash frac\{1\}\{\backslash lambda\}\backslash right)^2\; =\; \backslash frac\{1\}\{\backslash lambda^2\}.$$

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 $(1\; +\; 2\; +\; 3\; +\; 4\; +\; 5\; +\; 6)/6\; =\; 7/2.$ Therefore, the variance of {{mvar|X}} is

$$\backslash begin\{align\}$$

\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}

The general formula for the variance of the outcome, {{mvar|X}}, of an {{nowrap|{{mvar|n}}-sided}} die is

$$\backslash begin\{align\}$$

\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}

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

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

$$\backslash operatorname\{Var\}(X)\backslash ge\; 0.$$

The variance of a constant is zero.

$$\backslash operatorname\{Var\}(a)\; =\; 0.$$

Conversely, if the variance of a random variable is 0, then it is almost surely a constant. That is, it always has the same value:

$$\backslash operatorname\{Var\}(X)=\; 0\; \backslash iff\; \backslash exists\; a\; :\; P(X=a)\; =\; 1.$$

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 $k$ satisfies

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

$$\backslash operatorname\{Var\}[X]\; =\; \backslash operatorname\{E\}(\backslash operatorname\{Var\}[X\backslash mid\; Y])\; +\; \backslash operatorname\{Var\}(\backslash operatorname\{E\}[X\backslash mid\; Y]).$$

The conditional expectation $\backslash operatorname\; E(X\backslash mid\; Y)$ of $X$ given $Y$, and the conditional variance $\backslash operatorname\{Var\}(X\backslash mid\; Y)$ 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

$$\backslash operatorname\{E\}(\backslash operatorname\{Var\}[X\; \backslash mid\; Y])\; =\; \backslash sum\_i\; p\_i\; \backslash sigma^2\_i,$$

where $\backslash sigma^2\_i\; =\; \backslash operatorname\{Var\}[X\; \backslash mid\; Y\; =\; y\_i]$. Similarly, the second term on the right-hand side becomes

$$\backslash operatorname\{Var\}(\backslash operatorname\{E\}[X\; \backslash mid\; Y])\; =\; \backslash sum\_i\; p\_i\; \backslash mu\_i^2\; -\; \backslash left(\backslash sum\_i\; p\_i\; \backslash mu\_i\backslash right)^2\; =\; \backslash sum\_i\; p\_i\; \backslash mu\_i^2\; -\; \backslash mu^2,$$

where $\backslash mu\_i\; =\; \backslash operatorname\{E\}[X\; \backslash mid\; Y\; =\; y\_i]$ and $\backslash mu\; =\; \backslash sum\_i\; p\_i\; \backslash mu\_i$. Thus the total variance is given by

$$\backslash operatorname\{Var\}[X]\; =\; \backslash sum\_i\; p\_i\; \backslash sigma^2\_i\; +\; \backslash left(\; \backslash sum\_i\; p\_i\; \backslash mu\_i^2\; -\; \backslash mu^2\; \backslash right).$$

A similar formula is applied in analysis of variance, where the corresponding formula is

$$\backslash mathit\{MS\}\_\backslash text\{total\}\; =\; \backslash mathit\{MS\}\_\backslash text\{between\}\; +\; \backslash mathit\{MS\}\_\backslash text\{within\};$$

here $\backslash mathit\{MS\}$ refers to the Mean of the Squares. In linear regression analysis the corresponding formula is

$$\backslash mathit\{MS\}\_\backslash text\{total\}\; =\; \backslash mathit\{MS\}\_\backslash text\{regression\}\; +\; \backslash mathit\{MS\}\_\backslash text\{residual\}.$$

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, $\backslash mathit\{SS\}$):

$$\backslash mathit\{SS\}\_\backslash text\{total\}\; =\; \backslash mathit\{SS\}\_\backslash text\{between\}\; +\; \backslash mathit\{SS\}\_\backslash text\{within\},$$

$$\backslash mathit\{SS\}\_\backslash text\{total\}\; =\; \backslash mathit\{SS\}\_\backslash text\{regression\}\; +\; \backslash mathit\{SS\}\_\backslash text\{residual\}.$$

The population variance for a non-negative random variable can be expressed in terms of the cumulative distribution function 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, can be conveniently expressed.

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. $\backslash mathrm\{argmin\}\_m\; \backslash ,\; \backslash mathrm\{E\}\backslash left(\backslash left(X\; -\; m\backslash right)^2\backslash right)\; =\; \backslash mathrm\{E\}(X)$. Conversely, if a continuous function $\backslash varphi$ satisfies $\backslash mathrm\{argmin\}\_m\backslash ,\backslash mathrm\{E\}(\backslash varphi(X\; -\; m))\; =\; \backslash mathrm\{E\}(X)$ for all random variables X, then it is necessarily of the form $\backslash varphi(x)\; =\; a\; x^2\; +\; b$, where {{nowrap|a > 0}}. This also holds in the multidimensional case.{{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 }}

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 their standard deviation or 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 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, is used frequently in theoretical statistics; however the expected absolute deviation tends to be more robust as it is less sensitive to outliers arising from measurement anomalies or an unduly heavy-tailed distribution.

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

$$\backslash operatorname\{Var\}(X+a)=\backslash operatorname\{Var\}(X).$$

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

$$\backslash operatorname\{Var\}(aX)=a^2\backslash operatorname\{Var\}(X).$$

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

$$\backslash begin\{align\}$$

\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}

where $\backslash operatorname\{Cov\}(X,Y)$ is the covariance.

In general, for the sum of $N$ random variables $\backslash \{X\_1,\backslash dots,X\_N\backslash \}$, the variance becomes:

$$\backslash operatorname\{Var\}\backslash left(\backslash sum\_\{i=1\}^N\; X\_i\backslash right)\; =\; \backslash sum\_\{i,j=1\}^N\backslash operatorname\{Cov\}(X\_i,X\_j)\; =\; \backslash sum\_\{i=1\}^N\backslash operatorname\{Var\}(X\_i)\; +\; \backslash sum\_\{i,j=1,i\backslash ne\; j\}^N\backslash operatorname\{Cov\}(X\_i,X\_j),$$

see also general Bienaymé's identity.

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

$$\backslash begin\{align\}$$

\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}

If the random variables $X\_1,\backslash dots,X\_N$ are such that

$$\backslash operatorname\{Cov\}(X\_i,X\_j)=0\backslash \; ,\backslash \; \backslash forall\backslash \; (i\backslash ne\; j)\; ,$$

then they are said to be uncorrelated. 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:

$$\backslash operatorname\{Var\}\backslash left(\backslash sum\_\{i=1\}^N\; X\_i\backslash right)\; =\; \backslash sum\_\{i=1\}^N\backslash operatorname\{Var\}(X\_i).$$

Since independent random variables are always uncorrelated (see {{Section link|Covariance|Uncorrelatedness and independence}}), 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.

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, $c^\backslash mathsf\{T\}\; X$ is a linear combination of these random variables, where $c^\backslash mathsf\{T\}$ denotes the transpose of $c$. Also let $\backslash Sigma$ be the covariance matrix of $X$. The variance of $c^\backslash mathsf\{T\}X$ is then given by:{{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 }}

$$\backslash operatorname\{Var\}\backslash left(c^\backslash mathsf\{T\}\; X\backslash right)\; =\; c^\backslash mathsf\{T\}\; \backslash Sigma\; c\; .$$

This implies that the variance of the mean can be written as (with a column vector of ones)

$$\backslash operatorname\{Var\}\backslash left(\backslash bar\{x\}\backslash right)\; =\; \backslash operatorname\{Var\}\backslash left(\backslash frac\{1\}\{n\}\; 1\text{'}X\backslash right)\; =\; \backslash frac\{1\}\{n^2\}\; 1\text{'}\backslash Sigma\; 1.$$

{{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:

$$\backslash operatorname\{Var\}\backslash left(\backslash sum\_\{i=1\}^n\; X\_i\backslash right)\; =\; \backslash sum\_\{i=1\}^n\; \backslash operatorname\{Var\}(X\_i).$$

This statement is called the Bienaymé formulaLoève, M. (1977) "Probability Theory", Graduate Texts in Mathematics, Volume 45, 4th edition, Springer-Verlag, p. 12. and was discovered in 1853.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. 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}}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. 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] 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 σ², then, since division by n is a linear transformation, this formula immediately implies that the variance of their mean is

\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}.

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.

To prove the initial statement, it suffices to show that

$$\backslash operatorname\{Var\}(X\; +\; Y)\; =\; \backslash operatorname\{Var\}(X)\; +\; \backslash operatorname\{Var\}(Y).$$

The general result then follows by induction. Starting with the definition,

$$\backslash begin\{align\}$$

\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}

Using the linearity of the expectation operator and the assumption of independence (or uncorrelatedness) of X and Y, this further simplifies as follows:

$$\backslash begin\{align\}$$

\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}

{{main article|Bienaymé's identity}}

In general, the variance of the sum of {{math|n}} variables is the sum of their covariances:

(Note: The second equality comes from the fact that {{math|1=Cov(X_i,X_i) = Var(X_i)}}.)

Here, $\backslash operatorname\{Cov\}(\backslash cdot,\backslash cdot)$ is the covariance, which is zero for independent random variables (if it exists). The formula states that the variance of a sum is equal to the sum of all elements in the covariance matrix of the components. 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 σ² and the average correlation of distinct variables is ρ, then the variance of their mean is

$$\backslash operatorname\{Var\}\backslash left(\backslash overline\{X\}\backslash right)\; =\; \backslash frac\{\backslash sigma^2\}\{n\}\; +\; \backslash frac\{n\; -\; 1\}\{n\}\backslash rho\backslash sigma^2.$$

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

$$\backslash operatorname\{Var\}\backslash left(\backslash overline\{X\}\backslash right)\; =\; \backslash frac\{1\}\{n\}\; +\; \backslash frac\{n\; -\; 1\}\{n\}\backslash rho.$$

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

$$\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash operatorname\{Var\}\backslash left(\backslash overline\{X\}\backslash right)\; =\; \backslash rho.$$

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 states that the sample mean will converge for independent variables.

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,Cornell, J R, and Benjamin, C A, Probability, Statistics, and Decisions for Civil Engineers, McGraw-Hill, NY, 1970, pp.178-9.

$$\backslash operatorname\{Var\}\backslash left(\backslash sum\_\{i=1\}^\{N\}X\_i\backslash right)=\backslash operatorname\{E\}\backslash left[N\backslash right]\backslash operatorname\{Var\}(X)+\backslash operatorname\{Var\}(N)(\backslash operatorname\{E\}\backslash left[X\backslash right])^2$$

which follows from the law of total variance.

If {{math|N}} has a Poisson distribution, then $\backslash operatorname\{E\}[N]=\backslash operatorname\{Var\}(N)$ with estimator {{math|n}} = {{math|N}}. So, the estimator of $\backslash operatorname\{Var\}\backslash left(\backslash sum\_\{i=1\}^\{n\}X\_i\backslash right)$ becomes $n\{S\_x\}^2+n\backslash bar\{X\}^2$, giving $\backslash operatorname\{SE\}(\backslash bar\{X\})=\backslash sqrt\{\backslash frac\{\{S\_x\}^2+\backslash bar\{X\}^2\}\{n\}\}$

(see standard error of the sample mean).

{{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 {{math|Cov(aX, bY) {{=}} ab Cov(X, Y)}} jointly imply that

$$\backslash operatorname\{Var\}(aX\; \backslash pm\; bY)\; =a^2\; \backslash operatorname\{Var\}(X)\; +\; b^2\; \backslash operatorname\{Var\}(Y)\; \backslash pm\; 2ab\backslash ,\; \backslash operatorname\{Cov\}(X,\; Y).$$

This implies that in a weighted sum of variables, the variable with the largest weight will have a disproportionally large weight in the variance of the total. For example, if X and Y are uncorrelated and the weight of X is two times the weight of Y, then the weight of the variance of X will be four times the weight of the variance of Y.

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

If two variables X and Y are independent, the variance of their product is given by{{Cite journal |last=Goodman |first=Leo A. |author-link=Leo Goodman |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}}

$$\backslash operatorname\{Var\}(XY)\; =\; [\backslash operatorname\{E\}(X)]^2\; \backslash operatorname\{Var\}(Y)\; +\; [\backslash operatorname\{E\}(Y)]^2\; \backslash operatorname\{Var\}(X)\; +\; \backslash operatorname\{Var\}(X)\backslash operatorname\{Var\}(Y).$$

Equivalently, using the basic properties of expectation, it is given by

$$\backslash operatorname\{Var\}(XY)\; =\; \backslash operatorname\{E\}\backslash left(X^2\backslash right)\; \backslash operatorname\{E\}\backslash left(Y^2\backslash right)\; -\; [\backslash operatorname\{E\}(X)]^2\; [\backslash operatorname\{E\}(Y)]^2.$$

In general, if two variables are statistically dependent, then the variance of their product is given by:

$$\backslash begin\{align\}$$

\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}

{{main|Uncertainty propagation}}

The 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

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

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

{{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 estimates the mean and variance from a limited set of observations by using an estimator equation. The estimator is a function of the sample of n observations drawn without observational bias from the whole 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 estimators (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 n as the number of samples. 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 (the variance of the sample), n − 1 eliminates bias, n + 1 minimizes mean squared error for the normal distribution, and n − 1.5 mostly eliminates bias in unbiased estimation of standard deviation for the normal distribution.{{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 }}

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 this factor, resulting in the sum of squared deviations about the sample mean divided by n -1 instead of n, is called Bessel's correction.{{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 }} 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 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.{{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}} The resulting estimator is biased, however, and is known as the biased sample variation.

In general, the population variance of a finite population of size {{mvar|N}} with values {{math|x_i}} is given by

$$\backslash begin\{align\}$$

\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}

where the population mean is $\backslash mu\; =\; \backslash operatorname\{E\}[x\_i]\; =\; \backslash frac\; 1N\; \backslash sum\_\{i=1\}^N\; x\_i$ and $\backslash operatorname\{E\}[x\_i^2]\; =\; \backslash left(\backslash frac\{1\}\{N\}\; \backslash sum\_\{i=1\}^N\; x\_i^2\backslash right)$, where $\backslash operatorname\{E\}$ is the expectation value operator.

The population variance can also be computed using{{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}}

(The right side has duplicate terms in the sum while the middle side has only unique terms to sum.) This is true because

$$\backslash begin\{align\}$$

&\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}

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.

{{see also|Sample standard deviation}}

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.{{cite book | last = Navidi | first = William | year = 2006 | title = Statistics for Engineers and Scientists | publisher = McGraw-Hill | page = 14 }} 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|Y₁, ..., Y_n}} from the population of size {{mvar|N}}, where {{math|n < N}}, and estimate the variance on the basis of this sample.Montgomery, D. C. and Runger, G. C. (1994) Applied statistics and probability for engineers, page 201. John Wiley & Sons New York Directly taking the variance of the sample data gives the average of the squared deviations:{{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 }}

$$\backslash tilde\{S\}\_Y^2\; =$$

\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

(See the section Population variance for the derivation of this formula.) Here, $\backslash overline\{Y\}$ denotes the sample mean:

$$\backslash overline\{Y\}\; =\; \backslash frac\{1\}\{n\}\; \backslash sum\_\{i=1\}^n\; Y\_i\; .$$

Since the {{math|Y_i}} are selected randomly, both $\backslash overline\{Y\}$ and $\backslash tilde\{S\}\_Y^2$ are random variables. Their expected values can be evaluated by averaging over the ensemble of all possible samples {{math|{Y_i}}} of size {{mvar|n}} from the population. For $\backslash tilde\{S\}\_Y^2$ this gives:

$$\backslash begin\{align\}$$

\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}

Here $\backslash sigma^2\; =\; \backslash operatorname\{E\}[Y\_i^2]\; -\; \backslash mu^2$ derived in the section is population variance and $\backslash operatorname\{E\}[Y\_i\; Y\_j]\; =\; \backslash operatorname\{E\}[Y\_i]\; \backslash operatorname\{E\}[Y\_j]\; =\; \backslash mu^2$ due to independency of $Y\_i$ and $Y\_j$.

Hence $\backslash tilde\{S\}\_Y^2$ gives an estimate of the population variance $\backslash sigma^2$ that is biased by a factor of $\backslash frac\{n\; -\; 1\}\{n\}$ because the expectation value of $\backslash tilde\{S\}\_Y^2$ is smaller than the population variance (true variance) by that factor. For this reason, $\backslash tilde\{S\}\_Y^2$ is referred to as the biased sample variance.

Correcting for this bias yields the unbiased sample variance, denoted $S^2$:

$$S^2\; =\; \backslash frac\{n\}\{n\; -\; 1\}\; \backslash tilde\{S\}\_Y^2\; =\; \backslash frac\{n\}\{n\; -\; 1\}\; \backslash left[\; \backslash frac\{1\}\{n\}\; \backslash sum\_\{i=1\}^n\; \backslash left(Y\_i\; -\; \backslash overline\{Y\}\backslash right)^2\; \backslash right]\; =\; \backslash frac\{1\}\{n\; -\; 1\}\; \backslash sum\_\{i=1\}^n\; \backslash left(Y\_i\; -\; \backslash overline\{Y\}\; \backslash right)^2$$

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 (the square root of variance). The square root is a 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 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(y₁, y₂) = (y₁ − y₂)²/2}}, meaning that it is obtained by averaging a 2-sample statistic over 2-element subsets of the population.

For a set of numbers {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 Microsoft Excel gives the unbiased sample variance while VAR.P is for population variance.

{{multiple image

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| image1 = Scaled chi squared.svg

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| image2 = Scaled chi squared cdf.svg

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| caption2 = Distribution and cumulative distribution of S²/σ², for various values of ν = n − 1, when the y_i are independent normally distributed.

}}

Being a function of random variables, the sample variance is itself a random variable, and it is natural to study its distribution. In the case that Y_i are independent observations from a normal distribution, Cochran's theorem shows that the unbiased sample variance S² follows a scaled chi-squared distribution (see also: asymptotic properties and an elementary proof):{{cite book | last = Knight | first = K. | year = 2000 | title = Mathematical Statistics | publisher = Chapman and Hall | location = New York | at = proposition 2.11 }}

(n - 1) \frac{S^2}{\sigma^2} \sim \chi^2_{n-1}

where {{math|σ²}} is the population variance. As a direct consequence, it follows that

\operatorname{E}\left(S^2\right) = \operatorname{E}\left(\frac{\sigma^2}{n - 1} \chi^2_{n-1}\right) = \sigma^2 ,

and{{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. 331 | edition = 2nd | isbn = 0-534-24312-6 }}

\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}.

If Y_i are independent and identically distributed, but not necessarily normally distributed, thenMood, A. M., Graybill, F. A., and Boes, D.C. (1974) Introduction to the Theory of Statistics, 3rd Edition, McGraw-Hill, New York, p. 229

\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),

where κ is the kurtosis of the distribution and μ₄ is the fourth central moment.

If the conditions of the law of large numbers hold for the squared observations, S² is a consistent estimator of σ². 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.).{{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 }}Rose, Colin; Smith, Murray D. (2002). "[http://www.mathstatica.com/book/Mathematical_Statistics_with_Mathematica.pdf Mathematical Statistics with Mathematica]". Springer-Verlag, New York.Weisstein, Eric W. "[http://mathworld.wolfram.com/SampleVarianceDistribution.html Sample Variance Distribution]". MathWorld Wolfram.

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

It has been shown{{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 }} that for a sample {y_i} of positive real numbers,

$$\backslash sigma\_y^2\; \backslash le\; 2y\_\{\backslash max\}\; (A\; -\; H),$$

where {{math|y_max}} is the maximum of the sample, {{mvar|A}} is the arithmetic mean, {{mvar|H}} is the harmonic mean 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

$$\backslash begin\{align\}$$

\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}

where {{math|y_min}} is the minimum of the sample.{{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 }}

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 medians 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.

{{see also|Moment (physics)#Examples}}

The variance of a probability distribution is analogous to the moment of inertia in classical mechanics of a corresponding mass distribution along a line, with respect to rotation about its center of mass.{{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}} It is because of this analogy that such things as the variance are called moments of probability distributions. 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{{Citation needed|date=February 2012}}

$$I\; =\; n\backslash left(\backslash mathbf\{1\}\_\{3\backslash times\; 3\}\; \backslash operatorname\{tr\}(\backslash Sigma)\; -\; \backslash Sigma\backslash right).$$

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

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

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:

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.{{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}}

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

The term variance was first introduced by Ronald Fisher in his 1918 paper The Correlation Between Relatives on the Supposition of Mendelian Inheritance: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]

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 the standard deviation corresponding to the 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 $\backslash sigma\_1$ and $\backslash sigma\_2$, it is found that the distribution, when both causes act together, has a standard deviation $\backslash sqrt\{\backslash sigma\_1^2\; +\; \backslash sigma\_2^2\}$. 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...

If $x$ is a scalar complex-valued random variable, with values in $\backslash mathbb\{C\},$ then its variance is $\backslash operatorname\{E\}\backslash left[(x\; -\; \backslash mu)(x\; -\; \backslash mu)^*\backslash right],$ where $x^*$ is the complex conjugate of $x.$ This variance is a real scalar.

If $X$ is a vector-valued random variable, with values in $\backslash mathbb\{R\}^n,$ and thought of as a column vector, then a natural generalization of variance is $\backslash operatorname\{E\}\backslash left[(X\; -\; \backslash mu)\; \{(X\; -\; \backslash mu)\}^\{\backslash mathsf\{T\}\}\backslash right],$ where $\backslash mu\; =\; \backslash operatorname\{E\}(X)$ and $X^\{\backslash mathsf\{T\}\}$ 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 $X$ is a vector- and complex-valued random variable, with values in $\backslash mathbb\{C\}^n,$ then the covariance matrix is $\backslash operatorname\{E\}\backslash left[(X\; -\; \backslash mu)\{(X\; -\; \backslash mu)\}^\backslash dagger\backslash right],$ where $X^\backslash dagger$ is the conjugate transpose of $X.${{Citation needed|date=September 2016}} This matrix is also positive semi-definite and square.

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 $\backslash det(C)$, the determinant of the covariance matrix. The generalized variance can be shown to be related to the multidimensional scatter of points around their mean.{{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 }}

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

{{Portal|Mathematics}}

{{Wiktionary|variance}}

• Bhatia–Davis inequality
• Coefficient of variation
• Homoscedasticity
• Least-squares spectral analysis for computing a frequency spectrum with spectral magnitudes in % of variance or in dB
• Modern portfolio theory
• Popoviciu's inequality on variances
• Measures for statistical dispersion
• Variance-stabilizing transformation

• Correlation
• Distance variance
• Explained variance
• Pooled variance
• Pseudo-variance

{{Reflist}}

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{{Theory of probability distributions}}

{{Statistics|descriptive|state=collapsed}}

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Category:Moments (mathematics)

Category:Statistical deviation and dispersion

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