# Variance

In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean. Informally, it measures how far a set of (random) numbers are spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by ${\displaystyle \sigma ^{2}}$, ${\displaystyle s^{2}}$, or ${\displaystyle \operatorname {Var} (X)}$.

Example of samples from two populations with the same mean but different variances. The red population has mean 100 and variance 100 (SD=10) while the blue population has mean 100 and variance 2500 (SD=50).

## Definition

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

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

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:

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

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

{\displaystyle {\begin{aligned}\operatorname {Var} (X)&=\operatorname {E} \left[(X-\operatorname {E} [X])^{2}\right]\\[4pt]&=\operatorname {E} \left[X^{2}-2X\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]-\operatorname {E} [X]^{2}\end{aligned}}}

In other words, the variance of X is equal to the mean of the square of X minus the square of the mean of 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. There exist numerically stable alternatives.

### Discrete random variable

If the generator of random variable ${\displaystyle X}$ is discrete with probability mass function ${\displaystyle x_{1}\mapsto p_{1},x_{2}\mapsto p_{2},\ldots ,x_{n}\mapsto p_{n}}$ then

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

or equivalently

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

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

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

(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 set of ${\displaystyle n}$ equally likely values can be written as

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

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

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

The variance of a set of ${\displaystyle n}$ 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]

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

### Continuous random variable

If the random variable ${\displaystyle X}$ represents samples generated by a continuous distribution with probability density function ${\displaystyle f(x)}$, and ${\displaystyle F(x)}$ is the corresponding cumulative distribution function, then the population variance is given by

{\displaystyle {\begin{aligned}\operatorname {Var} (X)=\sigma ^{2}&=\int (x-\mu )^{2}f(x)\,dx\\[4pt]&=\int x^{2}f(x)\,dx-2\mu \int xf(x)\,dx+\int \mu ^{2}f(x)\,dx\\[4pt]&=\int x^{2}\,dF(x)-2\mu \int x\,dF(x)+\mu ^{2}\int \,dF(x)\\[4pt]&=\int x^{2}\,dF(x)-2\mu \cdot \mu +\mu ^{2}\cdot 1\\[4pt]&=\int x^{2}\,dF(x)-\mu ^{2},\end{aligned}}}

or equivalently and conventionally,

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

where ${\displaystyle \mu }$ is the expected value of ${\displaystyle X}$ given by

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

with the integrals being definite integrals taken for ${\displaystyle x}$ ranging over the range of ${\displaystyle X.}$

If a continuous distribution does not have a finite 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 ${\displaystyle k}$ satisfies ${\displaystyle 1

## Examples

### Normal distribution

The normal distribution with parameters ${\displaystyle \mu }$ and ${\displaystyle \sigma }$ is a continuous distribution (also known as gaussian distribution) whose probability density function is given by

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

In this distribution, ${\displaystyle \operatorname {E} [X]=\mu }$ and the variance ${\displaystyle \operatorname {Var} (X)}$ is related with ${\displaystyle \sigma }$ via

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

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.

### Exponential distribution

The exponential distribution with parameter ${\displaystyle \lambda }$ is a continuous distribution whose support is the semi-infinite interval ${\displaystyle [0,\infty )}$. Its probability density function is given by

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

and it has expected value ${\displaystyle \mu =\lambda ^{-1}}$. The variance is equal to

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

So for an exponentially distributed random variable, ${\displaystyle \sigma ^{2}=\mu ^{2}.}$

### Poisson distribution

The Poisson distribution with parameter ${\displaystyle \lambda }$ is a discrete distribution for ${\displaystyle k=0,1,2,\ldots }$. Its probability mass function is given by

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

and it has expected value ${\displaystyle \mu =\lambda }$. The variance is equal to

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

So for a Poisson-distributed random variable, ${\displaystyle \sigma ^{2}=\mu }$.

### Binomial distribution

The binomial distribution with parameters ${\displaystyle n}$ and ${\displaystyle p}$ is a discrete distribution for ${\displaystyle k=0,1,2,\ldots ,n}$. Its probability mass function is given by

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

and it has expected value ${\displaystyle \mu =np}$. The variance is equal to

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

As a simple example, the binomial distribution with ${\displaystyle p=1/2}$ describes the probability of getting ${\displaystyle k}$ heads in ${\displaystyle n}$ tosses of a fair coin. Thus the expected value of the number of heads is ${\displaystyle n/2,}$ and the variance is ${\displaystyle n/4.}$

### Fair die

A fair six-sided die can be modeled as a discrete random variable, X, with outcomes 1 through 6, each with equal probability 1/6. The expected value of X is ${\displaystyle (1+2+3+4+5+6)/6=7/2.}$ Therefore, the variance of X is

{\displaystyle {\begin{aligned}\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{aligned}}}

The general formula for the variance of the outcome, X, of an n-sided die is

{\displaystyle {\begin{aligned}\operatorname {Var} (X)&=\operatorname {E} (X^{2})-(\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{aligned}}}

## Properties

### Basic properties

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:

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

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

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

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:

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

This statement is called the Bienaymé formula[2] and was discovered in 1853.[3][4] 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

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

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

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

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

{\displaystyle {\begin{aligned}\operatorname {Var} (X+Y)&=\operatorname {E} [(X+Y)^{2}]-(\operatorname {E} [X+Y])^{2}\\[5pt]&=\operatorname {E} [X^{2}+2XY+Y^{2}]-(\operatorname {E} [X]+\operatorname {E} [Y])^{2}.\end{aligned}}}

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

{\displaystyle {\begin{aligned}\operatorname {Var} (X+Y)&=\operatorname {E} [X^{2}]+2\operatorname {E} [XY]+\operatorname {E} [Y^{2}]-(\operatorname {E} [X]^{2}+2\operatorname {E} [X]\operatorname {E} [Y]+\operatorname {E} [Y]^{2})\\[5pt]&=\operatorname {E} [X^{2}]+\operatorname {E} [Y^{2}]-\operatorname {E} [X]^{2}-\operatorname {E} [Y]^{2}\\[5pt]&=\operatorname {Var} (X)+\operatorname {Var} (Y).\end{aligned}}}

### Sum of correlated variables

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

(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. 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

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

This implies that the variance of the mean increases with the average of the correlations. 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

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

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

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

Define ${\displaystyle X}$ as a column vector of ${\displaystyle n}$ random variables ${\displaystyle X_{1},\ldots ,X_{n}}$, and ${\displaystyle c}$ as a column vector of ${\displaystyle n}$ scalars ${\displaystyle c_{1},\ldots ,c_{n}}$. Therefore, ${\displaystyle c^{T}X}$ is a linear combination of these random variables, where ${\displaystyle c^{T}}$ denotes the transpose of ${\displaystyle c}$. Also let ${\displaystyle \Sigma }$ be the covariance matrix of ${\displaystyle X}$. The variance of ${\displaystyle c^{T}X}$ is then given by:[5]

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

### Weighted sum of variables

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

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

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 _{

### Product of independent variables

If two variables X and Y are independent, the variance of their product is given by[6]

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

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

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

### Product of statistically dependent variables

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

{\displaystyle {\begin{aligned}\operatorname {Var} (XY)={}&\operatorname {E} [X^{2}Y^{2}]-[\operatorname {E} (XY)]^{2}\\[5pt]={}&\operatorname {Cov} (X^{2},Y^{2})+\operatorname {E} (X^{2})\operatorname {E} (Y^{2})-[\operatorname {E} (XY)]^{2}\\[5pt]={}&\operatorname {Cov} (X^{2},Y^{2})+(\operatorname {Var} (X)+[\operatorname {E} (X)]^{2})(\operatorname {Var} (Y)+[\operatorname {E} (Y)]^{2})\\[5pt]&{}-[\operatorname {Cov} (X,Y)+\operatorname {E} (X)\operatorname {E} (Y)]^{2}\end{aligned}}}

### Decomposition

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

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

The conditional expectation ${\displaystyle \operatorname {E} (X\mid Y)}$ of ${\displaystyle X}$ given ${\displaystyle Y}$, and the conditional variance ${\displaystyle \operatorname {Var} (X\mid Y)}$ may be understood as follows. Given any particular value y of the random variable Y, there is a conditional expectation ${\displaystyle \operatorname {E} (X\mid Y=y)}$ given the event Y = y. This quantity depends on the particular value y; it is a function ${\displaystyle g(y)=\operatorname {E} (X\mid Y=y)}$. That same function evaluated at the random variable Y is the conditional expectation ${\displaystyle \operatorname {E} (X\mid Y)=g(Y).}$

In particular, if ${\displaystyle Y}$ is a discrete random variable assuming possible values ${\displaystyle y_{1},y_{2},y_{3}\ldots }$ with corresponding probabilities ${\displaystyle p_{1},p_{2},p_{3}\ldots ,}$, then in the formula for total variance, the first term on the right-hand side becomes

${\displaystyle \operatorname {E} (\operatorname {Var} [X\mid Y])=\sum _{i}p_{i}\sigma _{i}^{2},}$

where ${\displaystyle \sigma _{i}^{2}=\operatorname {Var} [X\mid Y=y_{i}]}$. Similarly, the second term on the right-hand side becomes

${\displaystyle \operatorname {Var} (\operatorname {E} [X\mid Y])=\sum _{i}p_{i}\mu _{i}^{2}-\left(\sum _{i}p_{i}\mu _{i}\right)^{2}=\sum _{i}p_{i}\mu _{i}^{2}-\mu ^{2},}$

where ${\displaystyle \mu _{i}=\operatorname {E} [X\mid Y=y_{i}]}$ and ${\displaystyle \mu =\sum _{i}p_{i}\mu _{i}}$. Thus the total variance is given by

${\displaystyle \operatorname {Var} [X]=\sum _{i}p_{i}\sigma _{i}^{2}+\left(\sum _{i}p_{i}\mu _{i}^{2}-\mu ^{2}\right).}$

A similar formula is 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. In linear regression analysis 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.

### Calculation from the CDF

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. ${\displaystyle \mathrm {argmin} _{m}\,\mathrm {E} \left(\left(X-m\right)^{2}\right)=\mathrm {E} (X)}$. 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.[7]

### 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 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 √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

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 that would have been calculated from an omniscient 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 that sample would be the set of actual measurements of yesterday's rainfall from available rain gauges within the geography of interest.

The simplest estimators for population mean and population variance are simply the mean and variance of the sample, the sample mean and (uncorrected) sample variance – these are consistent estimators (they converge to the correct value as the number of samples increases), but can be improved. Estimating the population variance by taking the sample's variance is close to optimal in general, but can be improved in two ways. Most simply, the sample variance is computed as an average of squared deviations about the (sample) mean, by dividing by n. However, using values other than n improves the estimator in various ways. Four common values for the denominator are n, n − 1, n + 1, and n − 1.5: n is the simplest (population variance of the sample), n − 1 eliminates bias, n + 1 minimizes mean squared error for the normal distribution, and n − 1.5 mostly eliminates bias in unbiased estimation of standard deviation for the normal distribution.

Firstly, if the omniscient 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 population variance. If the mean is determined in some other way than from the same samples used to estimate the variance then this bias does not arise and the variance can safely be estimated as that of the samples about the (independently known) mean.

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

### Population variance

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

{\displaystyle {\begin{aligned}\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]&=\left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}^{2}\right)-\mu ^{2}\end{aligned}}}

where the population mean is

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

The population variance can also be computed using

${\displaystyle \sigma ^{2}={\frac {1}{N^{2}}}\sum _{i

This is true because

{\displaystyle {\begin{aligned}{\frac {1}{2N^{2}}}\sum _{i,j=1}^{N}\left(x_{i}-x_{j}\right)^{2}&={\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)\\[5pt]&\quad +{\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{aligned}}}

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

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.[8] 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.[9] Directly taking the variance of the sample data 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}=\left({\frac {1}{n}}\sum _{i=1}^{n}Y_{i}^{2}\right)-{\overline {Y}}^{2}={\frac {1}{n^{2}}}\sum _{i,j\,:\,i

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 ${\displaystyle {\overline {Y}}}$ and ${\displaystyle \sigma _{Y}^{2}}$ are random variables. Their expected values can be evaluated by averaging over the ensemble of all possible samples {Yi} of size n from the population. For ${\displaystyle \sigma _{Y}^{2}}$ this gives:

{\displaystyle {\begin{aligned}\operatorname {E} [\sigma _{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 {1}{n}}\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 {1}{n}}\sum _{i=1}^{n}\left[{\frac {n-2}{n}}\operatorname {E} [Y_{i}^{2}]-{\frac {2}{n}}\sum _{j\neq i}\operatorname {E} [Y_{i}Y_{j}]+{\frac {1}{n^{2}}}\sum _{j=1}^{n}\sum _{k\neq j}^{n}\operatorname {E} [Y_{j}Y_{k}]+{\frac {1}{n^{2}}}\sum _{j=1}^{n}\operatorname {E} [Y_{j}^{2}]\right]\\[5pt]&={\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]\\[5pt]&={\frac {n-1}{n}}\sigma ^{2}.\end{aligned}}}

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

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

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:[10]

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

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[11]

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

If the Yi are independent and identically distributed, but not necessarily normally distributed, then[12][13]

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

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.).[14][15][16]

### 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.[17] 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[18] that for a sample {yi} of real numbers,

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

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

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

where ymin is the minimum of the sample.[19]

## 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–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.

## History

The term variance was first introduced by Ronald Fisher in his 1918 paper The Correlation Between Relatives on the Supposition of Mendelian Inheritance:[20]

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...

Geometric visualisation of the variance of an arbitrary distribution (2, 4, 4, 4, 5, 5, 7, 9):
1. A frequency distribution is constructed.
2. The centroid of the distribution gives its mean.
3. A square with sides equal to the difference of each value from the mean is formed for each value.
4. 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, σ².

## Moment of inertia

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

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

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. Physicists would consider this to have a low moment about the x axis so the moment-of-inertia tensor is

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

## 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. It is sometimes described as a measure of downside risk in an investments context. For skewed distributions, the semivariance can provide additional information that a variance does not.

For inequalities associated with the semivariance, see Chebyshev's inequality § Semivariances.

## Generalizations

### For complex variables

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

### For vector-valued random variables

#### As a matrix

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

#### As a scalar

Another natural generalization of variance for such vector-valued random variables ${\displaystyle X,}$ which results in a scalar value rather than in a matrix, is obtained by interpreting the deviation between the random variable and its mean as the Euclidean distance. This results in ${\displaystyle \operatorname {E} \left[(X-\mu )^{\operatorname {T} }(X-\mu )\right]=\operatorname {tr} (C),}$ which is the trace of the covariance matrix.

## Notes

1. ^ Yuli Zhang, Huaiyu Wu, Lei Cheng (June 2012). Some new deformation formulas about variance and covariance. Proceedings of 4th International Conference on Modelling, Identification and Control(ICMIC2012). pp. 987–992.
2. ^ Loève, M. (1977) "Probability Theory", Graduate Texts in Mathematics, Volume 45, 4th edition, Springer-Verlag, p. 12.
3. ^ 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 [1]
4. ^ 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 [2][3]
5. ^ Johnson, Richard; Wichern, Dean (2001). Applied Multivariate Statistical Analysis. Prentice Hall. p. 76. ISBN 0-13-187715-1
6. ^ Goodman, Leo A. (December 1960). "On the Exact Variance of Products". Journal of the American Statistical Association. 55 (292): 708. doi:10.2307/2281592. JSTOR 2281592.
7. ^ Kagan, A.; Shepp, L. A. (1998). "Why the variance?". Statistics & Probability Letters. 38 (4): 329–333. doi:10.1016/S0167-7152(98)00041-8.
8. ^ Navidi, William (2006) Statistics for Engineers and Scientists, McGraw-Hill, pg 14.
9. ^ Montgomery, D. C. and Runger, G. C. (1994) Applied statistics and probability for engineers, page 201. John Wiley & Sons New York
10. ^ Knight K. (2000), Mathematical Statistics, Chapman and Hall, New York. (proposition 2.11)
11. ^ Casella and Berger (2002) Statistical Inference, Example 7.3.3, p. 331
12. ^ Cho, Eungchun; Cho, Moon Jung; Eltinge, John (2005) The Variance of Sample Variance From a Finite Population. International Journal of Pure and Applied Mathematics 21 (3): 387-394. http://www.ijpam.eu/contents/2005-21-3/10/10.pdf
13. ^ Cho, Eungchun; Cho, Moon Jung (2009) Variance of Sample Variance With Replacement. International Journal of Pure and Applied Mathematics 52 (1): 43–47. http://www.ijpam.eu/contents/2009-52-1/5/5.pdf
14. ^ Kenney, John F.; Keeping, E.S. (1951) Mathematics of Statistics. Part Two. 2nd ed. D. Van Nostrand Company, Inc. Princeton: New Jersey. http://krishikosh.egranth.ac.in/bitstream/1/2025521/1/G2257.pdf
15. ^ Rose, Colin; Smith, Murray D. (2002) Mathematical Statistics with Mathematica. Springer-Verlag, New York. http://www.mathstatica.com/book/Mathematical_Statistics_with_Mathematica.pdf
16. ^ Weisstein, Eric W. (n.d.) Sample Variance Distribution. MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/SampleVarianceDistribution.html
17. ^ Samuelson, Paul (1968). "How Deviant Can You Be?". Journal of the American Statistical Association. 63 (324): 1522–1525. doi:10.1080/01621459.1968.10480944. JSTOR 2285901.
18. ^ Mercer, A. McD. (2000). "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 (1): 163–173. doi:10.1006/jmaa.1999.6688.
19. ^ Sharma, R. (2008). "Some more inequalities for arithmetic mean, harmonic mean and variance". J. Math. Inequalities. 2 (1): 109–114. CiteSeerX 10.1.1.551.9397. doi:10.7153/jmi-02-11.
20. ^
Analysis of covariance

Analysis of covariance (ANCOVA) is a general linear model which blends ANOVA and regression. ANCOVA evaluates whether the means of a dependent variable (DV) are equal across levels of a categorical independent variable (IV) often called a treatment, while statistically controlling for the effects of other continuous variables that are not of primary interest, known as covariates (CV) or nuisance variables. Mathematically, ANCOVA decomposes the variance in the DV into variance explained by the CV(s), variance explained by the categorical IV, and residual variance. Intuitively, ANCOVA can be thought of as 'adjusting' the DV by the group means of the CV(s).

The ANCOVA model assumes a linear relationship between the response (DV) and covariate (CV):

${\displaystyle y_{ij}=\mu +\tau _{i}+\mathrm {B} (x_{ij}-{\overline {x}})+\epsilon _{ij}.}$

In this equation, the DV, ${\displaystyle y_{ij}}$ is the jth observation under the ith categorical group; the CV, ${\displaystyle x_{ij}}$ is the jth observation of the covariate under the ith group. Variables in the model that are derived from the observed data are ${\displaystyle \mu }$ (the grand mean) and ${\displaystyle {\overline {x}}}$ (the global mean for covariate ${\displaystyle x}$). The variables to be fitted are ${\displaystyle \tau _{i}}$ (the effect of the ith level of the IV), ${\displaystyle B}$ (the slope of the line) and ${\displaystyle \epsilon _{ij}}$ (the associated unobserved error term for the jth observation in the ith group).

Under this specification, the a categorical treatment effects sum to zero ${\displaystyle \left(\sum _{i}^{a}\tau _{i}=0\right).}$ The standard assumptions of the linear regression model are also assumed to hold, as discussed below.

Analysis of variance

Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among group means in a sample. ANOVA was developed by statistician and evolutionary biologist Ronald Fisher. In the ANOVA setting, the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether the population means of several groups are equal, and therefore generalizes the t-test to more than two groups. ANOVA is useful for comparing (testing) three or more group means for statistical significance. It is conceptually similar to multiple two-sample t-tests, but is more conservative, resulting in fewer type I errors, and is therefore suited to a wide range of practical problems.

In statistics and machine learning, the bias–variance tradeoff is the property of a set of predictive models whereby models with a lower bias in parameter estimation have a higher variance of the parameter estimates across samples, and vice versa. The bias–variance dilemma or problem is the conflict in trying to simultaneously minimize these two sources of error that prevent supervised learning algorithms from generalizing beyond their training set:

The bias is an error from erroneous assumptions in the learning algorithm. High bias can cause an algorithm to miss the relevant relations between features and target outputs (underfitting).

The variance is an error from sensitivity to small fluctuations in the training set. High variance can cause an algorithm to model the random noise in the training data, rather than the intended outputs (overfitting).The bias–variance decomposition is a way of analyzing a learning algorithm's expected generalization error with respect to a particular problem as a sum of three terms, the bias, variance, and a quantity called the irreducible error, resulting from noise in the problem itself.

This tradeoff applies to all forms of supervised learning: classification, regression (function fitting), and structured output learning. It has also been invoked to explain the effectiveness of heuristics in human learning.

Chi-squared test

A chi-squared test, also written as χ2 test, is any statistical hypothesis test where the sampling distribution of the test statistic is a chi-squared distribution when the null hypothesis is true. Without other qualification, 'chi-squared test' often is used as short for Pearson's chi-squared test. The chi-squared test is used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories.

In the standard applications of the test, the observations are classified into mutually exclusive classes, and there is some theory, or say null hypothesis, which gives the probability that any observation falls into the corresponding class. The purpose of the test is to evaluate how likely the observations that are made would be, assuming the null hypothesis is true.

Chi-squared tests are often constructed from a sum of squared errors, or through the sample variance. Test statistics that follow a chi-squared distribution arise from an assumption of independent normally distributed data, which is valid in many cases due to the central limit theorem. A chi-squared test can be used to attempt rejection of the null hypothesis that the data are independent.

Also considered a chi-squared test is a test in which this is asymptotically true, meaning that the sampling distribution (if the null hypothesis is true) can be made to approximate a chi-squared distribution as closely as desired by making the sample size large enough.

Coefficient of variation

In probability theory and statistics, the coefficient of variation (CV), also known as relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is often expressed as a percentage, and is defined as the ratio of the standard deviation ${\displaystyle \ \sigma }$ to the mean ${\displaystyle \ \mu }$ (or its absolute value, ${\displaystyle |\mu |}$). The CV or RSD is widely used in analytical chemistry to express the precision and repeatability of an assay. It is also commonly used in fields such as engineering or physics when doing quality assurance studies and ANOVA gauge R&R.[citation needed] In addition, CV is utilized by economists and investors in economic models and in determining the volatility of a security[citation needed].

Covariance matrix

In probability theory and statistics, a covariance matrix, also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix, is a matrix whose element in the i, j position is the covariance between the i-th and j-th elements of a random vector. A random vector is a random variable with multiple dimensions. Each element of the vector is a scalar random variable. Each element has either a finite number of observed empirical values or a finite or infinite number of potential values. The potential values are specified by a theoretical joint probability distribution.

Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions. As an example, the variation in a collection of random points in two-dimensional space cannot be characterized fully by a single number, nor would the variances in the ${\displaystyle x}$ and ${\displaystyle y}$ directions contain all of the necessary information; a ${\displaystyle 2\times 2}$ matrix would be necessary to fully characterize the two-dimensional variation.

Because the covariance of the i-th random variable with itself is simply that random variable's variance, each element on the principal diagonal of the covariance matrix is the variance of one of the random variables. Because the covariance of the i-th random variable with the j-th one is the same thing as the covariance of the j-th random variable with the i-th random variable, every covariance matrix is symmetric. Also, every covariance matrix is positive semi-definite.

The auto-covariance matrix of a random vector ${\displaystyle \mathbf {X} }$ is typically denoted by ${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}$ or ${\displaystyle \Sigma }$.

F-test

An F-test is any statistical test in which the test statistic has an F-distribution under the null hypothesis.

It is most often used when comparing statistical models that have been fitted to a data set, in order to identify the model that best fits the population from which the data were sampled. Exact "F-tests" mainly arise when the models have been fitted to the data using least squares. The name was coined by George W. Snedecor, in honour of Sir Ronald A. Fisher. Fisher initially developed the statistic as the variance ratio in the 1920s.

Gender variance

Gender variance, or gender nonconformity, is behavior or gender expression by an individual that does not match masculine or feminine gender norms. People who exhibit gender variance may be called gender variant, gender non-conforming, gender diverse, gender atypical or genderqueer, and may be transgender or otherwise variant in their gender identity. In the case of transgender people, they may be perceived, or perceive themselves as, gender nonconforming before transitioning, but might not be perceived as such after transitioning. Some intersex people may also exhibit gender variance.

Heteroscedasticity

In statistics, a collection of random variables is heteroscedastic (or heteroskedastic; from Ancient Greek hetero “different” and skedasis “dispersion”) if there are sub-populations that have different variabilities from others. Here "variability" could be quantified by the variance or any other measure of statistical dispersion. Thus heteroscedasticity is the absence of homoscedasticity.

The existence of heteroscedasticity is a major concern in the application of regression analysis, including the analysis of variance, as it can invalidate statistical tests of significance that assume that the modelling errors are uncorrelated and uniform—hence that their variances do not vary with the effects being modeled. For instance, while the ordinary least squares estimator is still unbiased in the presence of heteroscedasticity, it is inefficient because the true variance and covariance are underestimated. Similarly, in testing for differences between sub-populations using a location test, some standard tests assume that variances within groups are equal.

Because heteroscedasticity concerns expectations of the second moment of the errors, its presence is referred to as misspecification of the second order.

Homoscedasticity

In statistics, a sequence or a vector of random variables is homoscedastic if all random variables in the sequence or vector have the same finite variance. This is also known as homogeneity of variance. The complementary notion is called heteroscedasticity. The spellings homoskedasticity and heteroskedasticity are also frequently used.The assumption of homoscedasticity simplifies mathematical and computational treatment. Serious violations in homoscedasticity (assuming a distribution of data is homoscedastic when in reality it is heteroscedastic ) may result in overestimating the goodness of fit as measured by the Pearson coefficient.

Index of dispersion

In probability theory and statistics, the index of dispersion, dispersion index, coefficient of dispersion, relative variance, or variance-to-mean ratio (VMR), like the coefficient of variation, is a normalized measure of the dispersion of a probability distribution: it is a measure used to quantify whether a set of observed occurrences are clustered or dispersed compared to a standard statistical model.

It is defined as the ratio of the variance ${\displaystyle \sigma ^{2}}$ to the mean ${\displaystyle \mu }$,

${\displaystyle D={\sigma ^{2} \over \mu }.}$

It is also known as the Fano factor, though this term is sometimes reserved for windowed data (the mean and variance are computed over a subpopulation), where the index of dispersion is used in the special case where the window is infinite. Windowing data is frequently done: the VMR is frequently computed over various intervals in time or small regions in space, which may be called "windows", and the resulting statistic called the Fano factor.

It is only defined when the mean ${\displaystyle \mu }$ is non-zero, and is generally only used for positive statistics, such as count data or time between events, or where the underlying distribution is assumed to be the exponential distribution or Poisson distribution.

Kruskal–Wallis one-way analysis of variance

The Kruskal–Wallis test by ranks, Kruskal–Wallis H test (named after William Kruskal and W. Allen Wallis), or one-way ANOVA on ranks is a non-parametric method for testing whether samples originate from the same distribution. It is used for comparing two or more independent samples of equal or different sample sizes. It extends the Mann–Whitney U test, which is used for comparing only two groups. The parametric equivalent of the Kruskal–Wallis test is the one-way analysis of variance (ANOVA).

A significant Kruskal–Wallis test indicates that at least one sample stochastically dominates one other sample. The test does not identify where this stochastic dominance occurs or for how many pairs of groups stochastic dominance obtains. For analyzing the specific sample pairs for stochastic dominance, Dunn's test, pairwise Mann-Whitney tests without Bonferroni correction, or the more powerful but less well known Conover–Iman test are sometimes used.

Since it is a non-parametric method, the Kruskal–Wallis test does not assume a normal distribution of the residuals, unlike the analogous one-way analysis of variance. If the researcher can make the assumptions of an identically shaped and scaled distribution for all groups, except for any difference in medians, then the null hypothesis is that the medians of all groups are equal, and the alternative hypothesis is that at least one population median of one group is different from the population median of at least one other group.

Minimum-variance unbiased estimator

In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

For practical statistics problems, it is important to determine the MVUE if one exists, since less-than-optimal procedures would naturally be avoided, other things being equal. This has led to substantial development of statistical theory related to the problem of optimal estimation.

While combining the constraint of unbiasedness with the desirability metric of least variance leads to good results in most practical settings—making MVUE a natural starting point for a broad range of analyses—a targeted specification may perform better for a given problem; thus, MVUE is not always the best stopping point.

Multivariate analysis of variance

In statistics, multivariate analysis of variance (MANOVA) is a procedure for comparing multivariate sample means. As a multivariate procedure, it is used when there are two or more dependent variables, and is typically followed by significance tests involving individual dependent variables separately. It helps to answer:

Do changes in the independent variable(s) have significant effects on the dependent variables?

What are the relationships among the dependent variables?

What are the relationships among the independent variables?

Normal distribution

In probability theory, the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.

The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, they become normally distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal. Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed.

The normal distribution is sometimes informally called the bell curve. However, many other distributions are bell-shaped (such as the Cauchy, Student's t, and logistic distributions).

The probability density of the normal distribution is

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

where

Principal component analysis

Principal component analysis (PCA) is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables (entities each of which takes on various numerical values) into a set of values of linearly uncorrelated variables called principal components. If there are ${\displaystyle n}$ observations with ${\displaystyle p}$ variables, then the number of distinct principal components is ${\displaystyle \min(n-1,p)}$. This transformation is defined in such a way that the first principal component has the largest possible variance (that is, accounts for as much of the variability in the data as possible), and each succeeding component in turn has the highest variance possible under the constraint that it is orthogonal to the preceding components. The resulting vectors (each being a linear combination of the variables and containing n observations) are an uncorrelated orthogonal basis set. PCA is sensitive to the relative scaling of the original variables.

PCA was invented in 1901 by Karl Pearson, as an analogue of the principal axis theorem in mechanics; it was later independently developed and named by Harold Hotelling in the 1930s. Depending on the field of application, it is also named the discrete Karhunen–Loève transform (KLT) in signal processing, the Hotelling transform in multivariate quality control, proper orthogonal decomposition (POD) in mechanical engineering, singular value decomposition (SVD) of X (Golub and Van Loan, 1983), eigenvalue decomposition (EVD) of XTX in linear algebra, factor analysis (for a discussion of the differences between PCA and factor analysis see Ch. 7 of Jolliffe's Principal Component Analysis), Eckart–Young theorem (Harman, 1960), or empirical orthogonal functions (EOF) in meteorological science, empirical eigenfunction decomposition (Sirovich, 1987), empirical component analysis (Lorenz, 1956), quasiharmonic modes (Brooks et al., 1988), spectral decomposition in noise and vibration, and empirical modal analysis in structural dynamics.

PCA is mostly used as a tool in exploratory data analysis and for making predictive models. It is often used to visualize genetic distance and relatedness between populations. PCA can be done by eigenvalue decomposition of a data covariance (or correlation) matrix or singular value decomposition of a data matrix, usually after a normalization step of the initial data. The normalization of each attribute consists of mean centering – subtracting each data value from its variable's measured mean so that its empirical mean (average) is zero – and, possibly, normalizing each variable's variance to make it equal to 1; see Z-scores. The results of a PCA are usually discussed in terms of component scores, sometimes called factor scores (the transformed variable values corresponding to a particular data point), and loadings (the weight by which each standardized original variable should be multiplied to get the component score). If component scores are standardized to unit variance, loadings must contain the data variance in them (and that is the magnitude of eigenvalues). If component scores are not standardized (therefore they contain the data variance) then loadings must be unit-scaled, ("normalized") and these weights are called eigenvectors; they are the cosines of orthogonal rotation of variables into principal components or back.

PCA is the simplest of the true eigenvector-based multivariate analyses. Often, its operation can be thought of as revealing the internal structure of the data in a way that best explains the variance in the data. If a multivariate dataset is visualised as a set of coordinates in a high-dimensional data space (1 axis per variable), PCA can supply the user with a lower-dimensional picture, a projection of this object when viewed from its most informative viewpoint[citation needed]. This is done by using only the first few principal components so that the dimensionality of the transformed data is reduced.

PCA is closely related to factor analysis. Factor analysis typically incorporates more domain specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix.

PCA is also related to canonical correlation analysis (CCA). CCA defines coordinate systems that optimally describe the cross-covariance between two datasets while PCA defines a new orthogonal coordinate system that optimally describes variance in a single dataset.

Resampling (statistics)

In statistics, resampling is any of a variety of methods for doing one of the following:

Estimating the precision of sample statistics (medians, variances, percentiles) by using subsets of available data (jackknifing) or drawing randomly with replacement from a set of data points (bootstrapping)

Exchanging labels on data points when performing significance tests (permutation tests, also called exact tests, randomization tests, or re-randomization tests)

Validating models by using random subsets (bootstrapping, cross validation)Common resampling techniques include bootstrapping, jackknifing and permutation tests.

Standard deviation

In statistics, the standard deviation (SD, also represented by the lower case Greek letter sigma σ or the Latin letter s) is a measure that is used to quantify the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.

The standard deviation of a random variable, statistical population, data set, or probability distribution is the square root of its variance. It is algebraically simpler, though in practice less robust, than the average absolute deviation.

A useful property of the standard deviation is that, unlike the variance, it is expressed in the same units as the data.

In addition to expressing the variability of a population, the standard deviation is commonly used to measure confidence in statistical conclusions. For example, the margin of error in polling data is determined by calculating the expected standard deviation in the results if the same poll were to be conducted multiple times. This derivation of a standard deviation is often called the "standard error" of the estimate or "standard error of the mean" when referring to a mean. It is computed as the standard deviation of all the means that would be computed from that population if an infinite number of samples were drawn and a mean for each sample were computed.

It is very important to note that the standard deviation of a population and the standard error of a statistic derived from that population (such as the mean) are quite different but related (related by the inverse of the square root of the number of observations). The reported margin of error of a poll is computed from the standard error of the mean (or alternatively from the product of the standard deviation of the population and the inverse of the square root of the sample size, which is the same thing) and is typically about twice the standard deviation—the half-width of a 95 percent confidence interval.

In science, many researchers report the standard deviation of experimental data, and only effects that fall much farther than two standard deviations away from what would have been expected are considered statistically significant—normal random error or variation in the measurements is in this way distinguished from likely genuine effects or associations. The standard deviation is also important in finance, where the standard deviation on the rate of return on an investment is a measure of the volatility of the investment.

When only a sample of data from a population is available, the term standard deviation of the sample or sample standard deviation can refer to either the above-mentioned quantity as applied to those data or to a modified quantity that is an unbiased estimate of the population standard deviation (the standard deviation of the entire population).

Weighted arithmetic mean

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics.

If all the weights are equal, then the weighted mean is the same as the arithmetic mean. While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counterintuitive properties, as captured for instance in Simpson's paradox.

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