# 變異數

## 定義

The variance of a random variable X is its second central moment, the expected value of the squared deviation from the mean μ = E[X]:

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

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

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

The variance is also equivalent to the second cumulant of the probability distribution for X. The variance is typically designated as Var(X), ${\displaystyle \scriptstyle \sigma _{X}^{2}}$, or simply σ2 (pronounced "sigma squared"). The expression for the variance can be expanded:

{\displaystyle {\begin{aligned}\operatorname {Var} (X)&=\operatorname {E} \left[X^{2}-2X\operatorname {E} [X]+(\operatorname {E} [X])^{2}\right]\\&=\operatorname {E} \left[X^{2}\right]-2\operatorname {E} [X]\operatorname {E} [X]+(\operatorname {E} [X])^{2}\\&=\operatorname {E} \left[X^{2}\right]-(\operatorname {E} [X])^{2}\end{aligned}}}

A mnemonic for the above expression is "mean of square minus square of mean".

### 連續隨機變數

If the random variable X is continuous with probability density function f(x), then the variance is given by

${\displaystyle \operatorname {Var} (X)=\sigma ^{2}=\int (x-\mu )^{2}\,f(x)\,dx\,=\int x^{2}\,f(x)\,dx\,-\mu ^{2}}$

where ${\displaystyle \mu }$ is the expected value,

${\displaystyle \mu =\int x\,f(x)\,dx\,}$

and where the integrals are definite integrals taken for x ranging over the range of X.

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

### 離散隨機變數

If the random variable X is discrete with probability mass function x1 ↦ p1, ..., xn ↦ pn, then

${\displaystyle \operatorname {Var} (X)=\sum _{i=1}^{n}p_{i}\cdot (x_{i}-\mu )^{2}=\sum _{i=1}^{n}(p_{i}\cdot x_{i}^{2})-\mu ^{2}}$

where ${\displaystyle \mu }$ is the expected value, i.e.

${\displaystyle \mu =\sum _{i=1}^{n}p_{i}\cdot x_{i}}$ .

(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 set of n equally likely values can be written as

${\displaystyle \operatorname {Var} (X)={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}.}$

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 points from each other:[1]

${\displaystyle \operatorname {Var} (X)={\frac {1}{n^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}{\frac {1}{2}}(x_{i}-x_{j})^{2}.}$

## 例子

### 正態分佈

The normal distribution with parameters μ and σ is a continuous distribution whose probability density function is given by:

${\displaystyle f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}.}$

It has mean μ and variance equal to:

${\displaystyle \operatorname {Var} (X)=\int _{-\infty }^{\infty }{\frac {(x-\mu )^{2}}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\,dx=\sigma ^{2}.}$

The role of the normal distribution in the central limit theorem is in part responsible for the prevalence of the variance in probability and statistics.

### 指數分佈

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

${\displaystyle f(x)=\lambda e^{-\lambda x},\,}$

and it has expected value μ = λ−1. The variance is equal to:

${\displaystyle \operatorname {Var} (X)=\int _{0}^{\infty }(x-\lambda ^{-1})^{2}\,\lambda e^{-\lambda x}dx=\lambda ^{-2}.\,}$

So for an exponentially distributed random variable σ2 = μ2.

### Poisson分佈

The Poisson分佈 with parameter λ is a discrete distribution for k = 0, 1, 2, ... Its probability mass function is given by:

${\displaystyle p(k)={\frac {\lambda ^{k}}{k!}}e^{-\lambda },}$

and it has expected value μ = λ. The variance is equal to:

${\displaystyle \operatorname {Var} (X)=\sum _{k=0}^{\infty }{\frac {\lambda ^{k}}{k!}}e^{-\lambda }(k-\lambda )^{2}=\lambda ,}$

So for a Poisson-distributed random variable σ2 = μ.

### 二項分佈

The binomial distribution with parameters n and p is a discrete distribution for k = 0, 1, 2, ..., n. Its probability mass function is given by:

${\displaystyle p(k)={n \choose k}p^{k}(1-p)^{n-k},}$

and it has expected value μ = np. The variance is equal to:

${\displaystyle \operatorname {Var} (X)=\sum _{k=0}^{n}{n \choose k}p^{k}(1-p)^{n-k}(k-np)^{2}=np(1-p),}$

#### 擲銀仔

The binomial distribution with ${\displaystyle p=0.5}$ describes the probability of getting ${\displaystyle k}$ heads in ${\displaystyle n}$ tosses. Thus the expected value of the number of heads is ${\displaystyle {\frac {n}{2}}}$, and the variance is ${\displaystyle {\frac {n}{4}}}$.

### 擲骰仔

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

{\displaystyle {\begin{aligned}\sum _{i=1}^{6}{\tfrac {1}{6}}(i-3.5)^{2}={\tfrac {1}{6}}\sum _{i=1}^{6}(i-3.5)^{2}&={\tfrac {1}{6}}\left((-2.5)^{2}{+}(-1.5)^{2}{+}(-0.5)^{2}{+}0.5^{2}{+}1.5^{2}{+}2.5^{2}\right)\\&={\tfrac {1}{6}}\cdot 17.50={\tfrac {35}{12}}\approx 2.92.\end{aligned}}}

The general formula for the variance of the outcome X of a die of n sides is:

{\displaystyle {\begin{aligned}\sigma ^{2}=E(X^{2})-(E(X))^{2}&={\frac {1}{n}}\sum _{i=1}^{n}i^{2}-\left({\frac {1}{n}}\sum _{i=1}^{n}i\right)^{2}\\&={\tfrac {1}{6}}(n+1)(2n+1)-{\tfrac {1}{4}}(n+1)^{2}\\&={\frac {n^{2}-1}{12}}.\end{aligned}}}

## 特性

### 基本特性

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

${\displaystyle \operatorname {Var} (X)\geq 0.}$

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

${\displaystyle P(X=a)=1\Leftrightarrow \operatorname {Var} (X)=0.}$

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.

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

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

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

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

${\displaystyle \operatorname {Var} (aX+bY)=a^{2}\operatorname {Var} (X)+b^{2}\operatorname {Var} (Y)+2ab\,\operatorname {Cov} (X,Y),}$
${\displaystyle \operatorname {Var} (X-Y)=\operatorname {Var} (X)+\operatorname {Var} (Y)-2\,\operatorname {Cov} (X,Y),}$

where Cov(., .) is the covariance. In general we have for the sum of ${\displaystyle N}$ random variables ${\displaystyle \{X_{1},\dots ,X_{N}\}}$:

${\displaystyle \operatorname {Var} \left(\sum _{i=1}^{N}X_{i}\right)=\sum _{i,j=1}^{N}\operatorname {Cov} (X_{i},X_{j})=\sum _{i=1}^{N}\operatorname {Var} (X_{i})+\sum _{i\neq j}\operatorname {Cov} (X_{i},X_{j}).}$

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

{\displaystyle {\begin{aligned}\operatorname {Var} \left(\sum _{i=1}^{N}a_{i}X_{i}\right)&=\sum _{i,j=1}^{N}a_{i}a_{j}\operatorname {Cov} (X_{i},X_{j})\\&=\sum _{i=1}^{N}a_{i}^{2}\operatorname {Var} (X_{i})+\sum _{i\not =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

If the random variables ${\displaystyle X_{1},\dots ,X_{N}}$ are such that

${\displaystyle \operatorname {Cov} (X_{i},X_{j})=0\ ,\ \forall \ (i\neq j),}$

they are said to be uncorrelated. It follows immediately from the expression given earlier that if the random variables ${\displaystyle X_{1},\dots ,X_{N}}$ are uncorrelated, then the variance of their sum is equal to the sum of their variances, or, expressed symbolically:

${\displaystyle \operatorname {Var} \left(\sum _{i=1}^{N}X_{i}\right)=\sum _{i=1}^{N}\operatorname {Var} (X_{i}).}$

Since independent random variables are always uncorrelated, the equation above holds in particular when the random variables ${\displaystyle X_{1},\dots ,X_{n}}$ are independent. Thus independence is sufficient but not necessary for the variance of the sum to equal the sum of the variances.

### Sum of uncorrelated variables (Bienaymé formula)

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:

${\displaystyle \operatorname {Var} {\Big (}\sum _{i=1}^{n}X_{i}{\Big )}=\sum _{i=1}^{n}\operatorname {Var} (X_{i}).}$

This statement is called the Bienaymé formula[2] and was discovered in 1853.[未記出處或無根據] It is often made with the stronger condition that the variables are independent, but uncorrelatedness 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

${\displaystyle \operatorname {Var} \left({\overline {X}}\right)=\operatorname {Var} \left({\frac {1}{n}}\sum _{i=1}^{n}X_{i}\right)={\frac {1}{n^{2}}}\sum _{i=1}^{n}\operatorname {Var} \left(X_{i}\right)={\frac {\sigma ^{2}}{n}}.}$

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.

### 獨立變數嘅乘積

If two variables X and Y are independent, the variance of their product is given by[3][4]

{\displaystyle {\begin{aligned}\operatorname {Var} (XY)&=[E(X)]^{2}\operatorname {Var} (Y)+[E(Y)]^{2}\operatorname {Var} (X)+\operatorname {Var} (X)\operatorname {Var} (Y)\\&=E(X^{2})E(Y^{2})-[E(X)]^{2}[E(Y)]^{2}.\end{aligned}}}

### 相關變數嘅乘積

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

${\displaystyle \operatorname {Var} \left(\sum _{i=1}^{n}X_{i}\right)=\sum _{i=1}^{n}\sum _{j=1}^{n}\operatorname {Cov} (X_{i},X_{j})=\sum _{i=1}^{n}\operatorname {Var} (X_{i})+2\sum _{1\leq i}\sum _{

(Note: The second equality comes from the fact that Cov(Xi,Xi) = Var(Xi).)

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

So if the variables have equal variance σ2 and the average correlation of distinct variables is ρ, then the variance of their mean is

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

This implies that the variance of the mean increases with the average of the correlations. Moreover, if the variables have unit variance, for example if they are standardized, then this simplifies to

${\displaystyle \operatorname {Var} ({\overline {X}})={\frac {1}{n}}+{\frac {n-1}{n}}\rho .}$

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

${\displaystyle \lim _{n\to \infty }\operatorname {Var} ({\overline {X}})=\rho .}$

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 generally not converge to the population mean, even though the Law of large numbers states that the sample mean will converge for independent variables.

### 變數嘅加權平均

The scaling property and the Bienaymé formula, along with this property from the covariance page: Cov(aXbY) = ab Cov(XY) jointly imply that

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

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:

${\displaystyle \operatorname {Var} \left(\sum _{i}^{n}a_{i}X_{i}\right)=\sum _{i=1}^{n}a_{i}^{2}\operatorname {Var} (X_{i})+2\sum _{1\leq i}\sum _{

### 拆項

The general formula for variance decomposition or the law of total variance is: If ${\displaystyle X}$ and ${\displaystyle Y}$ are two random variables, and the variance of ${\displaystyle X}$ exists, then

${\displaystyle \operatorname {Var} (X)=\operatorname {Var} (\operatorname {E} (X|Y))+\operatorname {E} (\operatorname {Var} (X|Y)).}$

Here, ${\displaystyle \operatorname {E} (X|Y)}$ is the conditional expectation of ${\displaystyle X}$ given ${\displaystyle Y}$, and ${\displaystyle \operatorname {Var} (X|Y)}$ is the conditional variance of ${\displaystyle X}$ given ${\displaystyle Y}$. (A more intuitive explanation is that given a particular value of ${\displaystyle Y}$, then ${\displaystyle X}$ follows a distribution with mean ${\displaystyle \operatorname {E} (X|Y)}$ and variance ${\displaystyle \operatorname {Var} (X|Y)}$. The above formula tells how to find ${\displaystyle \operatorname {Var} (X)}$ based on the distributions of these two quantities when ${\displaystyle Y}$ is allowed to vary.) This formula is often applied in analysis of variance, where the corresponding formula is

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

here ${\displaystyle {\mathit {MS}}}$ refers to the Mean of the Squares. It is also used in linear regression analysis, where the corresponding formula is

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

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

${\displaystyle {\mathit {SS}}_{\text{total}}={\mathit {SS}}_{\text{between}}+{\mathit {SS}}_{\text{within}},}$
${\displaystyle {\mathit {SS}}_{\text{total}}={\mathit {SS}}_{\text{regression}}+{\mathit {SS}}_{\text{residual}}.}$

### Formulae for the variance

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

${\displaystyle \operatorname {Var} (X)=\operatorname {E} (X^{2})-(\operatorname {E} (X))^{2}.}$

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

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

### CDF嘅計算

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

${\displaystyle 2\int _{0}^{\infty }u(1-F(u))\,du-{\Big (}\int _{0}^{\infty }1-F(u)\,du{\Big )}^{2}.}$

This expression can be used to calculate the variance in situations where the CDF, but not the density, can be conveniently expressed.

### Characteristic property

The second moment of a random variable attains the minimum value when taken around the first moment (i.e., mean) of the random variable, i.e. ${\displaystyle \mathrm {argmin} _{m}\,\mathrm {E} ((X-m)^{2})=\mathrm {E} (X)\,}$. Conversely, if a continuous function ${\displaystyle \varphi }$ satisfies ${\displaystyle \mathrm {argmin} _{m}\,\mathrm {E} (\varphi (X-m))=\mathrm {E} (X)\,}$ for all random variables X, then it is necessarily of the form ${\displaystyle \varphi (x)=ax^{2}+b}$, where a > 0. This also holds in the multidimensional case.[5]

### Matrix notation for the variance of a linear combination

Let's define ${\displaystyle X}$ as a column vector of n random variables ${\displaystyle X_{1},...,X_{n}}$, and c as a column vector of N scalars ${\displaystyle c_{1},...,c_{n}}$. Therefore ${\displaystyle c^{T}X}$ is a linear combination of these random variables, where ${\displaystyle c^{T}}$ denotes the transpose of vector ${\displaystyle c}$. Let also be ${\displaystyle \Sigma }$ the variance-covariance matrix of the vector X. The variance of ${\displaystyle c^{T}X}$ is given by:[6]

${\displaystyle \operatorname {Var} (c^{T}X)=c^{T}\Sigma c.}$

### Units of measurement

Unlike expected absolute deviation, the variance of a variable has units that are the square of the units of the variable itself. For example, a variable measured in meters will have a variance measured in square meters. 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 √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.

## Approximating the variance of a function

The delta method uses second-order Taylor expansions to approximate the variance of a function of one or more random variables: 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

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

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

## Population variance and sample variance

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Real-world distributions such as the distribution of yesterday's rain throughout the day are typically not fully known, unlike the behavior of perfect dice or an ideal distribution such as the normal distribution, because it is impractical to account for every raindrop. Instead one estimates the mean and variance of the whole distribution by using an estimator, a function of the sample of n observations drawn suitably randomly from the whole sample space, in this example the set of all measurements of yesterday's rainfall in all available rain gauges. 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 correct value as the number of samples increases), but can be improved. Estimating the population variance as the variance of the sample is close to optimal in general, but can be improved in two incompatible ways. The sample variance is computed as an average of squared deviations about the (sample) mean, most simply dividing by n. However, using other values than n improves the estimator in various ways. Four common values for the denominator are n, n − 1, n + 1, and n − 1.5: n is the simplest (population variance of the sample), n − 1 eliminates bias, n + 1 minimizes mean squared error for the normal distribution, and n − 1.5 mostly eliminates bias in unbiased estimation of standard deviation for the normal distribution.

Firstly, if the mean is unknown (and is computed as the sample mean), then the sample variance is a biased estimator: it underestimates the variance by a factor of (n − 1) / n; correcting by this factor (dividing by n − 1 instead of n) is called Bessel's correction. The resulting estimator is unbiased, and is called the (corrected) sample variance or unbiased sample variance. For example, when n = 1 the variance of a single observation about the sample mean (itself) is obviously zero regardless of the true 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 (already known) mean.

Secondly, the sample variance does not generally minimize mean squared error, and correcting for bias often makes this worse: one can always choose a scale factor that performs better than the corrected sample variance, though the optimal scale factor depends on the excess kurtosis of the population (see mean squared error: variance), and introduces bias. This always consists of scaling down the unbiased estimator (dividing by a number larger than n − 1), and is a simple example of a shrinkage estimator: one "shrinks" the unbiased estimator towards zero. For the normal distribution, dividing by n + 1 (instead of n − 1 or n) minimizes mean squared error. The resulting estimator is biased, however, and is known as the biased sample variation.

### Population variance

In general, the population variance of a finite population of size N with values xi is given by

${\displaystyle \sigma ^{2}={\frac {1}{N}}\sum _{i=1}^{N}\left(x_{i}-\mu \right)^{2}=\left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}^{2}\right)-\mu ^{2}}$

where

${\displaystyle \mu ={\frac {1}{N}}\sum _{i=1}^{N}x_{i}}$

is the population mean. The population variance therefore is the variance of the underlying probability distribution. In this sense, the concept of population can be extended to continuous random variables with infinite populations.

### Sample variance

In many practical situations, the true variance of a population is not known a priori and must be computed somehow. When dealing with extremely large populations, it is not possible to count every object in the population, so the computation must be performed on a sample of the population.[7] Sample variance can also be applied to the estimation of the variance of a continuous distribution from a sample of that distribution.

We take a sample with replacement of n values y1, ..., yn from the population, where n < N, and estimate the variance on the basis of this sample.[8] Directly taking the variance of the sample gives the average of the squared deviations:

${\displaystyle \sigma _{y}^{2}={\frac {1}{n}}\sum _{i=1}^{n}\left(y_{i}-{\overline {y}}\right)^{2}}$

Here, ${\displaystyle {\overline {y}}}$ denotes the sample mean:

${\displaystyle {\overline {y}}={\frac {1}{n}}\sum _{i=1}^{n}y_{i}.}$

Since the yi are selected randomly, both ${\displaystyle \scriptstyle {\overline {y}}}$ and ${\displaystyle \scriptstyle \sigma _{y}^{2}}$ are random variables. Their expected values can be evaluated by summing over the ensemble of all possible samples {yi} from the population. For ${\displaystyle \scriptstyle \sigma _{y}^{2}}$ this gives:

{\displaystyle {\begin{aligned}E[\sigma _{y}^{2}]&=E\left[{\frac {1}{n}}\sum _{i=1}^{n}\left(y_{i}-{\frac {1}{n}}\sum _{j=1}^{n}y_{j}\right)^{2}\right]\\&={\frac {1}{n}}\sum _{i=1}^{n}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]\\&={\frac {1}{n}}\sum _{i=1}^{n}\left[{\frac {n-2}{n}}E[y_{i}^{2}]-{\frac {2}{n}}\sum _{j\neq i}E[y_{i}y_{j}]+{\frac {1}{n^{2}}}\sum _{j=1}^{n}\sum _{k\neq j}E[y_{j}y_{k}]+{\frac {1}{n^{2}}}\sum _{j=1}^{n}E[y_{j}^{2}]\right]\\&={\frac {1}{n}}\sum _{i=1}^{n}\left[{\frac {n-2}{n}}(\sigma ^{2}+\mu ^{2})-{\frac {2}{n}}(n-1)\mu ^{2}+{\frac {1}{n^{2}}}n(n-1)\mu ^{2}+{\frac {1}{n}}(\sigma ^{2}+\mu ^{2})\right]\\&={\frac {n-1}{n}}\sigma ^{2}.\end{aligned}}}

Hence ${\displaystyle \scriptstyle \sigma _{y}^{2}}$ gives an estimate of the population variance that is biased by a factor of (n-1)/n. For this reason, ${\displaystyle \scriptstyle \sigma _{y}^{2}}$ is referred to as the biased sample variance. Correcting for this bias yields the unbiased sample variance:

${\displaystyle s^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}\left(y_{i}-{\overline {y}}\right)^{2}}$

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 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 n − 1.5 yields an almost unbiased estimator.

The unbiased sample variance is a U-statistic for the function ƒ(y1y2) = (y1 − y2)2/2, meaning that it is obtained by averaging a 2-sample statistic over 2-element subsets of the population.

### Distribution of the sample variance

Distribution and cumulative distribution of s22, for various values of ν = n − 1, when the yi 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 yi are independent observations from a normal distribution, Cochran's theorem shows that s2 follows a scaled chi-squared distribution:[9]

${\displaystyle (n-1){\frac {s^{2}}{\sigma ^{2}}}\sim \chi _{n-1}^{2}.}$

As a direct consequence, it follows that

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

and[10]

${\displaystyle \operatorname {Var} [s^{2}]=\operatorname {Var} \left({\frac {\sigma ^{2}}{n-1}}\chi _{n-1}^{2}\right)={\frac {\sigma ^{4}}{(n-1)^{2}}}\operatorname {Var} \left(\chi _{n-1}^{2}\right)={\frac {2\sigma ^{4}}{n-1}}.}$

If the yi are independent and identically distributed, but not necessarily normally distributed, then[11]

${\displaystyle \operatorname {E} [s^{2}]=\sigma ^{2},\quad \operatorname {Var} [s^{2}]=\sigma ^{4}\left({\frac {2}{n-1}}+{\frac {\kappa }{n}}\right)={\frac {1}{n}}\left(\mu _{4}-{\frac {n-3}{n-1}}\sigma ^{4}\right),}$

where κ is the excess kurtosis of the distribution and μ4 is the fourth moment about the mean.

If the conditions of the law of large numbers hold for the squared observations, s2 is a consistent estimator of σ2.[未記出處或無根據]. One can see indeed that the variance of the estimator tends asymptotically to zero.

### 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.[12] Values must lie within the limits ${\displaystyle {\bar {y}}\pm \sigma _{y}(n-1)^{1/2}.}$

### Relations with the harmonic and arithmetic means

It has been shown[13] that for a sample {yi} of real numbers,

${\displaystyle \sigma _{y}^{2}\leq 2y_{\text{max}}(A-H),}$

where ymax is the maximum of the sample, A is the arithmetic mean, H is the harmonic mean of the sample and ${\displaystyle \sigma _{y}^{2}}$ is the (biased) variance of the sample.

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

${\displaystyle \sigma _{y}^{2}\leq {\frac {y_{\text{max}}(A-H)(y_{\text{max}}-A)}{y_{\text{max}}-H}},}$
${\displaystyle \sigma _{y}^{2}\geq {\frac {y_{\text{min}}(A-H)(A-y_{\text{min}})}{H-y_{\text{min}}}},}$

where ymin is the minimum of the sample.[14]

## Generalizations

If ${\displaystyle X}$ is a vector-valued random variable, with values in ${\displaystyle \mathbb {R} ^{n}}$, and thought of as a column vector, then the natural generalization of variance is ${\displaystyle \operatorname {E} ((X-\mu )(X-\mu )^{\operatorname {T} })}$, where ${\displaystyle \mu =\operatorname {E} (X)}$ and ${\displaystyle X^{\operatorname {T} }}$ is the transpose of ${\displaystyle X}$, and so is a row vector. This variance is a positive semi-definite square matrix, commonly referred to as the covariance matrix.

If ${\displaystyle X}$ is a complex-valued random variable, with values in ${\displaystyle \mathbb {C} }$, then its variance is ${\displaystyle \operatorname {E} ((X-\mu )(X-\mu )^{\dagger })}$, where ${\displaystyle X^{\dagger }}$ is the conjugate transpose of ${\displaystyle X}$. This variance is also a positive semi-definite square matrix.

## Tests of equality of variances

Testing for the equality of two or more variances is difficult. The F test and chi square tests are both adversely affected by non-normality and are not recommended for this purpose.

Several non parametric tests have been proposed: these include the Barton-David-Ansari-Fruend-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-Fruend-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 Lehman 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.

## History

The term variance was first introduced by Ronald Fisher in his 1918 paper The Correlation Between Relatives on the Supposition of Mendelian Inheritance:[15]

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 ${\displaystyle \sigma _{1}}$ and ${\displaystyle \sigma _{2}}$, it is found that the distribution, when both causes act together, has a standard deviation ${\displaystyle {\sqrt {\sigma _{1}^{2}+\sigma _{2}^{2}}}}$. 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...

## Moment of inertia

The variance of a probability distribution is analogous to the moment of inertia in classical mechanics of a corresponding mass distribution along a line, with respect to rotation about its center of mass.[未記出處或無根據] It is because of this analogy that such things as the variance are called moments of probability distributions.[未記出處或無根據] 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 ${\displaystyle \Sigma }$ is given by[未記出處或無根據]

${\displaystyle I=n(\mathbf {1} _{3\times 3}\operatorname {tr} (\Sigma )-\Sigma ).}$

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

${\displaystyle \Sigma ={\begin{bmatrix}10&0&0\\0&0.1&0\\0&0&0.1\end{bmatrix}}.}$

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

${\displaystyle I=n{\begin{bmatrix}0.2&0&0\\0&10.1&0\\0&0&10.1\end{bmatrix}}.}$

## Notes

1. Template:Cite conference
2. Loeve, M. (1977) "Probability Theory", Graduate Texts in Mathematics, Volume 45, 4th edition, Springer-Verlag, p. 12.
3. Goodman, Leo A., "On the exact variance of products," Journal of the American Statistical Association, December 1960, 708–713.
4. Goodman, Leo A., "The variance of the product of K random variables," Journal of the American Statistical Association, March 1962, 54ff.

5. 本引用來源將會喺幾十分鐘後自動完成。您可以檢查英文對應模手動擴充
6. Johnson, Richard; Wichern, Dean (2001). Applied Multivariate Statistical Analysis. Prentice Hall. 第 76頁. ISBN 0-13-187715-1Template:Inconsistent citations
7. Navidi, William (2006) Statistics for Engineers and Scientists, McGraw-Hill, pg 14.
8. Montgomery, D. C. and Runger, G. C. (1994) Applied statistics and probability for engineers, page 201. John Wiley & Sons New York
9. Knight K. (2000), Mathematical Statistics, Chapman and Hall, New York. (proposition 2.11)
10. Casella and Berger (2002) Statistical Inference, Example 7.3.3, p. 331 Template:Full
11. Neter, Wasserman, and Kutner (1990) Applied Linear Statistical Models, 3rd edition, pp. 622-623 Template:Full
12. Samuelson, Paul (1968)"How Deviant Can You Be?", Journal of the American Statistical Association, 63, number 324 (December, 1968), pp. 1522–1525 Template:JSTOR
13. A. McD. Mercer. Bounds for A-G, A-H, G-H, and a family of inequalities of Ky Fan’s type, using a general method. J. Math. Anal. Appl. 243, 163–173 (2000)
14. R. Sharma. Some more inequalities for arithmetic mean, harmonic mean and variance. J. Math. Inequalities, 2(1), 109–114 (2008).