# Variance

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Statistical measure of how far values spread from their average

This article is about the mathematical concept. For other uses, see [Variance (disambiguation)](/source/Variance_(disambiguation)).

Example of samples from two populations with the same mean but different variances. The red population has mean *μ* = 100 and variance *σ*2 = 100 (*σ* = 10), while the blue population has mean *μ* = 100 and variance *σ*2 = 2500 (*σ* = 50).

In [probability theory](/source/Probability_theory) and [statistics](/source/Statistics), **variance** is the [expected value](/source/Expected_value) of the [squared deviation from the mean](/source/Squared_deviations_from_the_mean) of a [random variable](/source/Random_variable). The [standard deviation](/source/Standard_deviation) is obtained as the square root of the variance. Variance is a measure of [dispersion](/source/Statistical_dispersion), meaning it is a measure of how far a set of numbers are spread out from their average value. It is the second [central moment](/source/Central_moment) of a [distribution](/source/Probability_distribution), and the [covariance](/source/Covariance) of the random variable with itself, and it is often represented by ⁠ σ 2 {\displaystyle \sigma ^{2}} ⁠, ⁠ s 2 {\displaystyle s^{2}} ⁠, ⁠ Var ⁡ ( X ) {\displaystyle \operatorname {Var} (X)} ⁠, ⁠ V ( X ) {\displaystyle V(X)} ⁠, or ⁠ V ( X ) {\displaystyle \mathbb {V} (X)} ⁠.[1]

An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the [expected absolute deviation](/source/Average_absolute_deviation); for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard deviation is more commonly reported as a measure of dispersion once the calculation is finished. Another disadvantage is that the variance is not finite for many distributions.

There are two distinct concepts that are both called "variance". One, as discussed above, is part of a theoretical [probability distribution](/source/Probability_distribution) and is defined by an equation. The other variance is a characteristic of a set of observations. When variance is calculated from observations, those observations are typically measured from a real-world system. If all possible observations of the system are present, then the calculated variance is called the population variance. Normally, however, only a subset is available, and the variance calculated from this is called the sample variance. The variance calculated from a sample is considered an estimate of the full population variance. There are multiple ways to estimate the population variance on the basis of the sample variance, as discussed in the section below.

The two kinds of variance are closely related. To see how, consider that a theoretical probability distribution can be used as a generator of hypothetical observations. If an infinite number of observations are generated using a distribution, then the sample variance calculated from that infinite set will match the value calculated using the distribution's equation for variance. Variance has a central role in statistics, where some ideas that use it include [descriptive statistics](/source/Descriptive_statistics), [statistical inference](/source/Statistical_inference), [hypothesis testing](/source/Hypothesis_testing), [goodness of fit](/source/Goodness_of_fit), and [Monte Carlo sampling](/source/Monte_Carlo_method).

Geometric visualisation of the variance of an arbitrary distribution (2, 4, 4, 4, 5, 5, 7, 9):

1. A frequency distribution is constructed.
1. The centroid of the distribution gives its mean.
1. A square with sides equal to the difference of each value from the mean is formed for each value.
1. Arranging the squares into a rectangle with one side equal to the number of values, *n*, results in the other side being the distribution's variance, *σ*2.

## Definition

The variance of a random variable X {\displaystyle X} is the [expected value](/source/Expected_value) of the [squared deviation from the mean](/source/Squared_deviations_from_the_mean) of ⁠ X {\displaystyle X} ⁠, ⁠ μ = E ⁡ [ X ] {\displaystyle \mu =\operatorname {E} [X]} ⁠: Var ⁡ ( X ) = E ⁡ [ ( X − μ ) 2 ] . {\displaystyle \operatorname {Var} (X)=\operatorname {E} \left[(X-\mu )^{2}\right].} This definition encompasses random variables that are generated by processes that are [discrete](/source/Discrete_random_variable), [continuous](/source/Continuous_random_variable), [neither](/source/Cantor_distribution), or mixed. The variance can also be thought of as the [covariance](/source/Covariance) of a random variable with itself: Var ⁡ ( X ) = Cov ⁡ ( X , X ) . {\displaystyle \operatorname {Var} (X)=\operatorname {Cov} (X,X).}

The variance is also equivalent to the second [cumulant](/source/Cumulant) of a probability distribution that generates ⁠ X {\displaystyle X} ⁠. The variance is typically designated as ⁠ Var ⁡ ( X ) {\displaystyle \operatorname {Var} (X)} ⁠, or sometimes as V ( X ) {\displaystyle V(X)} or ⁠ V ( X ) {\displaystyle \mathbb {V} (X)} ⁠, or symbolically as ⁠ σ X 2 {\displaystyle \sigma _{X}^{2}} ⁠ or simply σ 2 {\displaystyle \sigma ^{2}} (pronounced "[sigma](/source/Sigma) squared"). The expression for the variance can be expanded as follows: Var ⁡ ( X ) = E ⁡ [ ( X − E ⁡ [ X ] ) 2 ] = E ⁡ [ X 2 − 2 X E ⁡ [ X ] + E ⁡ [ X ] 2 ] = E ⁡ [ X 2 ] − 2 E ⁡ [ X ] E ⁡ [ X ] + E ⁡ [ X ] 2 = E ⁡ [ X 2 ] − 2 E ⁡ [ X ] 2 + E ⁡ [ X ] 2 = E ⁡ [ X 2 ] − E ⁡ [ X ] 2 {\displaystyle {\begin{aligned}\operatorname {Var} (X)&=\operatorname {E} \left[{\left(X-\operatorname {E} [X]\right)}^{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]-2\operatorname {E} [X]^{2}+\operatorname {E} [X]^{2}\\[4pt]&=\operatorname {E} \left[X^{2}\right]-\operatorname {E} [X]^{2}\end{aligned}}}

In other words, the variance of ⁠ X {\displaystyle X} ⁠ is equal to the mean of the square of ⁠ X {\displaystyle X} ⁠ minus the square of the mean of ⁠ X {\displaystyle X} ⁠. This equation should not be used for computations using [floating-point arithmetic](/source/Floating-point_arithmetic), because it suffers from [catastrophic cancellation](/source/Catastrophic_cancellation) if the two components of the equation are similar in magnitude. For other numerically stable alternatives, see *[Algorithms for calculating variance](/source/Algorithms_for_calculating_variance)*.

### Discrete random variable

If the generator of random variable X {\displaystyle X} is [discrete](/source/Discrete_probability_distribution) with [probability mass function](/source/Probability_mass_function) x 1 ↦ p 1 , x 2 ↦ p 2 , … , x n ↦ p n {\displaystyle x_{1}\mapsto p_{1},x_{2}\mapsto p_{2},\ldots ,x_{n}\mapsto p_{n}} , then Var ⁡ ( X ) = ∑ i = 1 n p i ⋅ ( x i − μ ) 2 , {\displaystyle \operatorname {Var} (X)=\sum _{i=1}^{n}p_{i}\cdot {\left(x_{i}-\mu \right)}^{2},} where μ {\displaystyle \mu } is the expected value. That is, μ = ∑ i = 1 n p i x i . {\displaystyle \mu =\sum _{i=1}^{n}p_{i}x_{i}.} (When such a discrete [weighted variance](/source/Weighted_variance) is specified by weights whose sum is not 1, then one divides by the sum of the weights.)

The variance of a collection of n {\displaystyle n} equally likely values can be written as Var ⁡ ( X ) = 1 n ∑ i = 1 n ( x i − μ ) 2 , {\displaystyle \operatorname {Var} (X)={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-\mu )^{2},} where μ {\displaystyle \mu } is the average value. That is, μ = 1 n ∑ i = 1 n x i . {\displaystyle \mu ={\frac {1}{n}}\sum _{i=1}^{n}x_{i}.}

The variance of a set of n {\displaystyle n} equally likely values can be equivalently expressed, without directly referring to the mean, in terms of squared deviations of all pairwise squared distances of points from each other:[2] Var ⁡ ( X ) = 1 n 2 ∑ i = 1 n ∑ j = 1 n 1 2 ( x i − x j ) 2 = 1 n 2 ∑ i ∑ j > i ( x i − x j ) 2 . {\displaystyle \operatorname {Var} (X)={\frac {1}{n^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}{\frac {1}{2}}{\left(x_{i}-x_{j}\right)}^{2}={\frac {1}{n^{2}}}\sum _{i}\sum _{j>i}{\left(x_{i}-x_{j}\right)}^{2}.}

### Absolutely continuous random variable

If the random variable X {\displaystyle X} has a [probability density function](/source/Probability_density_function) ⁠ f ( x ) {\displaystyle f(x)} ⁠, and F ( x ) {\displaystyle F(x)} is the corresponding [cumulative distribution function](/source/Cumulative_distribution_function), then Var ⁡ ( X ) = σ 2 = ∫ R ( x − μ ) 2 f ( x ) d x = ∫ R x 2 f ( x ) d x − 2 μ ∫ R x f ( x ) d x + μ 2 ∫ R f ( x ) d x = ∫ R x 2 d F ( x ) − 2 μ ∫ R x d F ( x ) + μ 2 ∫ R d F ( x ) = ∫ R x 2 d F ( x ) − 2 μ ⋅ μ + μ 2 ⋅ 1 = ∫ R x 2 d F ( x ) − μ 2 , {\displaystyle {\begin{aligned}\operatorname {Var} (X)=\sigma ^{2}&=\int _{\mathbb {R} }{\left(x-\mu \right)}^{2}f(x)\,dx\\[4pt]&=\int _{\mathbb {R} }x^{2}f(x)\,dx-2\mu \int _{\mathbb {R} }xf(x)\,dx+\mu ^{2}\int _{\mathbb {R} }f(x)\,dx\\[4pt]&=\int _{\mathbb {R} }x^{2}\,dF(x)-2\mu \int _{\mathbb {R} }x\,dF(x)+\mu ^{2}\int _{\mathbb {R} }\,dF(x)\\[4pt]&=\int _{\mathbb {R} }x^{2}\,dF(x)-2\mu \cdot \mu +\mu ^{2}\cdot 1\\[4pt]&=\int _{\mathbb {R} }x^{2}\,dF(x)-\mu ^{2},\end{aligned}}} or equivalently, Var ⁡ ( X ) = ∫ R x 2 f ( x ) d x − μ 2 , {\displaystyle \operatorname {Var} (X)=\int _{\mathbb {R} }x^{2}f(x)\,dx-\mu ^{2},} where μ {\displaystyle \mu } is the expected value of X {\displaystyle X} given by μ = ∫ R x f ( x ) d x = ∫ R x d F ( x ) . {\displaystyle \mu =\int _{\mathbb {R} }xf(x)\,dx=\int _{\mathbb {R} }x\,dF(x).}

In these formulas, the integrals with respect to d x {\displaystyle dx} and d F ( x ) {\displaystyle dF(x)} are [Lebesgue](/source/Lebesgue_integral) and [Lebesgue–Stieltjes](/source/Lebesgue%E2%80%93Stieltjes_integration) integrals, respectively.

If the function x 2 f ( x ) {\displaystyle x^{2}f(x)} is [Riemann-integrable](/source/Riemann-integrable) on every finite interval [ a , b ] ⊂ R , {\displaystyle [a,b]\subset \mathbb {R} ,} then Var ⁡ ( X ) = ∫ − ∞ + ∞ x 2 f ( x ) d x − μ 2 , {\displaystyle \operatorname {Var} (X)=\int _{-\infty }^{+\infty }x^{2}f(x)\,dx-\mu ^{2},} where the integral is an [improper Riemann integral](/source/Improper_Riemann_integral).

## Examples

### Exponential distribution

The [exponential distribution](/source/Exponential_distribution) with parameter ⁠ λ > 0 {\displaystyle \lambda >0} ⁠ is a continuous distribution whose [probability density function](/source/Probability_density_function) is given by f ( x ) = λ e − λ x {\displaystyle f(x)=\lambda e^{-\lambda x}} on the interval [0, ∞). Its mean can be shown to be E ⁡ [ X ] = ∫ 0 ∞ x λ e − λ x d x = 1 λ . {\displaystyle \operatorname {E} [X]=\int _{0}^{\infty }x\lambda e^{-\lambda x}\,dx={\frac {1}{\lambda }}.}

Using [integration by parts](/source/Integration_by_parts) and making use of the expected value already calculated, we have: E ⁡ [ X 2 ] = ∫ 0 ∞ x 2 λ e − λ x d x = [ − x 2 e − λ x ] 0 ∞ + ∫ 0 ∞ 2 x e − λ x d x = 0 + 2 λ E ⁡ [ X ] = 2 λ 2 . {\displaystyle {\begin{aligned}\operatorname {E} \left[X^{2}\right]&=\int _{0}^{\infty }x^{2}\lambda e^{-\lambda x}\,dx\\&={\left[-x^{2}e^{-\lambda x}\right]}_{0}^{\infty }+\int _{0}^{\infty }2xe^{-\lambda x}\,dx\\&=0+{\frac {2}{\lambda }}\operatorname {E} [X]\\&={\frac {2}{\lambda ^{2}}}.\end{aligned}}}

Thus, the variance of ⁠ X {\displaystyle X} ⁠ is given by Var ⁡ ( X ) = E ⁡ [ X 2 ] − E ⁡ [ X ] 2 = 2 λ 2 − ( 1 λ ) 2 = 1 λ 2 . {\displaystyle \operatorname {Var} (X)=\operatorname {E} \left[X^{2}\right]-\operatorname {E} [X]^{2}={\frac {2}{\lambda ^{2}}}-\left({\frac {1}{\lambda }}\right)^{2}={\frac {1}{\lambda ^{2}}}.}

### Fair die

A fair [six-sided die](/source/Dice) can be modeled as a discrete random variable, ⁠ X {\displaystyle X} ⁠, with outcomes 1 through 6, each with equal probability 1/6. The expected value of ⁠ X {\displaystyle X} ⁠ is ( 1 + 2 + 3 + 4 + 5 + 6 ) / 6 = 7 / 2. {\displaystyle (1+2+3+4+5+6)/6=7/2.} Therefore, the variance of ⁠ X {\displaystyle X} ⁠ is Var ⁡ ( X ) = ∑ i = 1 6 1 6 ( i − 7 2 ) 2 = 1 6 ( ( − 5 / 2 ) 2 + ( − 3 / 2 ) 2 + ( − 1 / 2 ) 2 + ( 1 / 2 ) 2 + ( 3 / 2 ) 2 + ( 5 / 2 ) 2 ) = 35 12 ≈ 2.92. {\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 {\displaystyle X} ⁠, of an ⁠ n {\displaystyle n} ⁠-sided die is Var ⁡ ( X ) = E ⁡ ( X 2 ) − ( E ⁡ ( X ) ) 2 = 1 n ∑ i = 1 n i 2 − ( 1 n ∑ i = 1 n i ) 2 = ( n + 1 ) ( 2 n + 1 ) 6 − ( n + 1 2 ) 2 = n 2 − 1 12 . {\displaystyle {\begin{aligned}\operatorname {Var} (X)&=\operatorname {E} \left(X^{2}\right)-(\operatorname {E} (X))^{2}\\[5pt]&={\frac {1}{n}}\sum _{i=1}^{n}i^{2}-\left({\frac {1}{n}}\sum _{i=1}^{n}i\right)^{2}\\[5pt]&={\frac {(n+1)(2n+1)}{6}}-\left({\frac {n+1}{2}}\right)^{2}\\[4pt]&={\frac {n^{2}-1}{12}}.\end{aligned}}}

### Commonly used probability distributions

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

Name of the probability distribution Probability distribution function Mean Variance Binomial distribution Pr ( X = k ) = ( n k ) p k ( 1 − p ) n − k {\displaystyle \Pr \,(X=k)={\binom {n}{k}}p^{k}(1-p)^{n-k}} n p {\displaystyle np} n p ( 1 − p ) {\displaystyle np(1-p)} Geometric distribution Pr ( X = k ) = ( 1 − p ) k − 1 p {\displaystyle \Pr \,(X=k)=(1-p)^{k-1}p} 1 p {\displaystyle {\frac {1}{p}}} ( 1 − p ) p 2 {\displaystyle {\frac {(1-p)}{p^{2}}}} Normal distribution f ( x ∣ μ , σ 2 ) = 1 2 π σ 2 e − 1 2 ( x − μ σ ) 2 {\displaystyle f\left(x\mid \mu ,\sigma ^{2}\right)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {1}{2}}{\left({\frac {x-\mu }{\sigma }}\right)}^{2}}} μ {\displaystyle \mu } σ 2 {\displaystyle \sigma ^{2}} Uniform distribution (continuous) f ( x ∣ a , b ) = { 1 b − a for a ≤ x ≤ b , 0 for x < a or x > b {\displaystyle f(x\mid a,b)={\begin{cases}{\frac {1}{b-a}}&{\text{for }}a\leq x\leq b,\\[3pt]0&{\text{for }}x<a{\text{ or }}x>b\end{cases}}} a + b 2 {\displaystyle {\frac {a+b}{2}}} ( b − a ) 2 12 {\displaystyle {\frac {(b-a)^{2}}{12}}} Exponential distribution f ( x ∣ λ ) = λ e − λ x {\displaystyle f(x\mid \lambda )=\lambda e^{-\lambda x}} 1 λ {\displaystyle {\frac {1}{\lambda }}} 1 λ 2 {\displaystyle {\frac {1}{\lambda ^{2}}}} Poisson distribution f ( k ∣ λ ) = e − λ λ k k ! {\displaystyle f(k\mid \lambda )={\frac {e^{-\lambda }\lambda ^{k}}{k!}}} λ {\displaystyle \lambda } λ {\displaystyle \lambda }

## Properties

### Basic properties

Variance is non-negative because the squares are positive or zero: Var ⁡ ( X ) ≥ 0. {\displaystyle \operatorname {Var} (X)\geq 0.}

The variance of a constant is zero. Var ⁡ ( a ) = 0. {\displaystyle \operatorname {Var} (a)=0.}

Conversely, if the variance of a random variable is 0, then it is [almost surely](/source/Almost_surely) a constant. That is, it always has the same value: Var ⁡ ( X ) = 0 ⟺ ∃ a : P ( X = a ) = 1. {\displaystyle \operatorname {Var} (X)=0\iff \exists a:P(X=a)=1.}

### Issues of finiteness

If a distribution does not have a finite expected value, as is the case for the [Cauchy distribution](/source/Cauchy_distribution), then the variance cannot be finite either. However, some distributions may not have a finite variance, despite their expected value being finite. An example is a [Pareto distribution](/source/Pareto_distribution) whose [index](/source/Pareto_index) k {\displaystyle k} satisfies 1 < k ≤ 2. {\displaystyle 1<k\leq 2.}

### Decomposition

The general formula for variance decomposition or the [law of total variance](/source/Law_of_total_variance) is: If X {\displaystyle X} and Y {\displaystyle Y} are two random variables, and the variance of X {\displaystyle X} exists, then Var ⁡ [ X ] = E ⁡ ( Var ⁡ [ X ∣ Y ] ) + Var ⁡ ( E ⁡ [ X ∣ Y ] ) . {\displaystyle \operatorname {Var} [X]=\operatorname {E} (\operatorname {Var} [X\mid Y])+\operatorname {Var} (\operatorname {E} [X\mid Y]).}

The [conditional expectation](/source/Conditional_expectation) E ⁡ ( X ∣ Y ) {\displaystyle \operatorname {E} (X\mid Y)} of X {\displaystyle X} given Y {\displaystyle Y} , and the [conditional variance](/source/Conditional_variance) Var ⁡ ( X ∣ Y ) {\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 E ⁡ ( X ∣ Y = y ) {\displaystyle \operatorname {E} (X\mid Y=y)} given the event *Y* = *y*. This quantity depends on the particular value *y*; it is a function g ( y ) = E ⁡ ( X ∣ Y = y ) {\displaystyle g(y)=\operatorname {E} (X\mid Y=y)} . That same function evaluated at the random variable *Y* is the conditional expectation ⁠ E ⁡ ( X ∣ Y ) = g ( Y ) {\displaystyle \operatorname {E} (X\mid Y)=g(Y)} ⁠.

In particular, if Y {\displaystyle Y} is a discrete random variable assuming possible values y 1 , y 2 , y 3 … {\displaystyle y_{1},y_{2},y_{3}\ldots } with corresponding probabilities p 1 , p 2 , p 3 … , {\displaystyle p_{1},p_{2},p_{3}\ldots ,} , then in the formula for total variance, the first term on the right-hand side becomes E ⁡ ( Var ⁡ [ X ∣ Y ] ) = ∑ i p i σ i 2 , {\displaystyle \operatorname {E} (\operatorname {Var} [X\mid Y])=\sum _{i}p_{i}\sigma _{i}^{2},} where σ i 2 = Var ⁡ [ X ∣ Y = y i ] {\displaystyle \sigma _{i}^{2}=\operatorname {Var} [X\mid Y=y_{i}]} . Similarly, the second term on the right-hand side becomes Var ⁡ ( E ⁡ [ X ∣ Y ] ) = ∑ i p i μ i 2 − ( ∑ i p i μ i ) 2 = ∑ i p i μ i 2 − μ 2 , {\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 ⁠ 1 {\displaystyle {1}} ⁠ and ⁠ μ = ∑ i p i μ i {\displaystyle \textstyle \mu =\sum _{i}p_{i}\mu _{i}} ⁠. Thus the total variance is given by Var ⁡ [ X ] = ∑ i p i σ i 2 + ( ∑ i p i μ i 2 − μ 2 ) . {\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](/source/Analysis_of_variance), where the corresponding formula is M S total = M S between + M S within ; {\displaystyle {\mathit {MS}}_{\text{total}}={\mathit {MS}}_{\text{between}}+{\mathit {MS}}_{\text{within}};} here M S {\displaystyle {\mathit {MS}}} refers to the Mean of the Squares. In [linear regression](/source/Linear_regression) analysis the corresponding formula is M S total = M S regression + M S residual . {\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, S S {\displaystyle {\mathit {SS}}} ): S S total = S S between + S S within , {\displaystyle {\mathit {SS}}_{\text{total}}={\mathit {SS}}_{\text{between}}+{\mathit {SS}}_{\text{within}},} S S total = S S regression + S S residual . {\displaystyle {\mathit {SS}}_{\text{total}}={\mathit {SS}}_{\text{regression}}+{\mathit {SS}}_{\text{residual}}.}

### Calculation from the CDF

The population variance for a non-negative random variable can be expressed in terms of the [cumulative distribution function](/source/Cumulative_distribution_function) *F* using 2 ∫ 0 ∞ u ( 1 − F ( u ) ) d u − [ ∫ 0 ∞ ( 1 − F ( u ) ) d u ] 2 . {\displaystyle 2\int _{0}^{\infty }u(1-F(u))\,du-{\left[\int _{0}^{\infty }(1-F(u))\,du\right]}^{2}.}

This expression can be used to calculate the variance in situations where the CDF, but not the [density](/source/Probability_density_function), can be conveniently expressed.

### Characteristic property

The second [moment](/source/Moment_(mathematics)) of a random variable attains the minimum value when taken around the first moment (i.e., mean) of the random variable, i.e. ⁠ a r g m i n m E ( ( X − m ) 2 ) = E ( X ) {\displaystyle \mathrm {argmin} _{m}\,\mathrm {E} \left(\left(X-m\right)^{2}\right)=\mathrm {E} (X)} ⁠. Conversely, if a continuous function φ {\displaystyle \varphi } satisfies a r g m i n m E ( φ ( X − m ) ) = E ( X ) {\displaystyle \mathrm {argmin} _{m}\,\mathrm {E} (\varphi (X-m))=\mathrm {E} (X)} for all random variables *X*, then it is necessarily of the form ⁠ φ ( x ) = a x 2 + b {\displaystyle \varphi (x)=ax^{2}+b} ⁠, where *a* > 0. This also holds in the multidimensional case.[3]

### Units of measurement

Unlike the [expected absolute deviation](/source/Average_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](/source/Standard_deviation) or [root mean square deviation](/source/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](/source/Covariance), is used frequently in theoretical statistics; however the expected absolute deviation tends to be more [robust](/source/Robust_statistics) as it is less sensitive to [outliers](/source/Outlier) arising from [measurement anomalies](/source/Measurement_error) or an unduly [heavy-tailed distribution](/source/Heavy-tailed_distribution).

## Propagation

### Addition and multiplication by a constant

Variance is [invariant](/source/Invariant_(mathematics)) with respect to changes in a [location parameter](/source/Location_parameter). That is, if a constant is added to all values of the variable, the variance is unchanged: Var ⁡ ( X + a ) = Var ⁡ ( X ) . {\displaystyle \operatorname {Var} (X+a)=\operatorname {Var} (X).}

If all values are scaled by a constant, the variance is [scaled](/source/Homogeneous_function) by the square of that constant: Var ⁡ ( a X ) = a 2 Var ⁡ ( X ) . {\displaystyle \operatorname {Var} (aX)=a^{2}\operatorname {Var} (X).}

The variance of a sum of two random variables is given by Var ⁡ ( a X + b Y ) = a 2 Var ⁡ ( X ) + b 2 Var ⁡ ( Y ) + 2 a b Cov ⁡ ( X , Y ) Var ⁡ ( a X − b Y ) = a 2 Var ⁡ ( X ) + b 2 Var ⁡ ( Y ) − 2 a b Cov ⁡ ( X , Y ) {\displaystyle {\begin{aligned}\operatorname {Var} (aX+bY)&=a^{2}\operatorname {Var} (X)+b^{2}\operatorname {Var} (Y)+2ab\,\operatorname {Cov} (X,Y)\\[1ex]\operatorname {Var} (aX-bY)&=a^{2}\operatorname {Var} (X)+b^{2}\operatorname {Var} (Y)-2ab\,\operatorname {Cov} (X,Y)\end{aligned}}} where Cov ⁡ ( X , Y ) {\displaystyle \operatorname {Cov} (X,Y)} is the [covariance](/source/Covariance).

### Linear combinations

In general, for the sum of N {\displaystyle N} random variables ⁠ { X 1 , … , X N } {\displaystyle \{X_{1},\dots ,X_{N}\}} ⁠, the variance becomes: Var ⁡ ( ∑ i = 1 N X i ) = ∑ i , j = 1 N Cov ⁡ ( X i , X j ) = ∑ i = 1 N Var ⁡ ( X i ) + ∑ i , j = 1 , i ≠ j N Cov ⁡ ( X i , X j ) , {\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,j=1,i\neq j}^{N}\operatorname {Cov} (X_{i},X_{j}),} see also general [Bienaymé's identity](/source/Bienaym%C3%A9's_identity).

These results lead to the variance of a [linear combination](/source/Linear_combination) as: Var ⁡ ( ∑ i = 1 N a i X i ) = ∑ i , j = 1 N a i a j Cov ⁡ ( X i , X j ) = ∑ i = 1 N a i 2 Var ⁡ ( X i ) + ∑ i ≠ j a i a j Cov ⁡ ( X i , X j ) = ∑ i = 1 N a i 2 Var ⁡ ( X i ) + 2 ∑ 1 ≤ i < j ≤ N a i a j Cov ⁡ ( X i , X j ) . {\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\neq j}a_{i}a_{j}\operatorname {Cov} (X_{i},X_{j})\\&=\sum _{i=1}^{N}a_{i}^{2}\operatorname {Var} (X_{i})+2\sum _{1\leq i<j\leq N}a_{i}a_{j}\operatorname {Cov} (X_{i},X_{j}).\end{aligned}}}

If the random variables X 1 , … , X N {\displaystyle X_{1},\dots ,X_{N}} are such that Cov ⁡ ( X i , X j ) = 0 , ∀ ( i ≠ j ) , {\displaystyle \operatorname {Cov} (X_{i},X_{j})=0\ ,\ \forall \ (i\neq j),} then they are said to be [uncorrelated](/source/Covariance#Definition). It follows immediately from the expression given earlier that if the random variables X 1 , … , X N {\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: Var ⁡ ( ∑ i = 1 N X i ) = ∑ i = 1 N Var ⁡ ( X i ) . {\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 (see [Covariance § Uncorrelatedness and independence](/source/Covariance#Uncorrelatedness_and_independence)), the equation above holds in particular when the random variables X 1 , … , X n {\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.

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

Define X {\displaystyle X} as a column vector of n {\displaystyle n} random variables ⁠ X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} ⁠, and c {\displaystyle c} as a column vector of n {\displaystyle n} scalars ⁠ c 1 , … , c n {\displaystyle c_{1},\ldots ,c_{n}} ⁠. Therefore, c T X {\displaystyle c^{\mathsf {T}}X} is a [linear combination](/source/Linear_combination) of these random variables, where c T {\displaystyle c^{\mathsf {T}}} denotes the [transpose](/source/Transpose) of ⁠ c {\displaystyle c} ⁠. Also let Σ {\displaystyle \Sigma } be the [covariance matrix](/source/Covariance_matrix) of ⁠ X {\displaystyle X} ⁠. The variance of c T X {\displaystyle c^{\mathsf {T}}X} is then given by:[4] Var ⁡ ( c T X ) = c T Σ c . {\displaystyle \operatorname {Var} \left(c^{\mathsf {T}}X\right)=c^{\mathsf {T}}\Sigma c.}

This implies that the variance of the mean can be written as (with a column vector of ones) Var ⁡ ( x ¯ ) = Var ⁡ ( 1 n 1 ′ X ) = 1 n 2 1 ′ Σ 1. {\displaystyle \operatorname {Var} \left({\bar {x}}\right)=\operatorname {Var} \left({\frac {1}{n}}1'X\right)={\frac {1}{n^{2}}}1'\Sigma 1.}

### Sum of variables

#### Sum of uncorrelated variables

Main article: [Bienaymé's identity](/source/Bienaym%C3%A9's_identity)

See also: [Sum of normally distributed random variables](/source/Sum_of_normally_distributed_random_variables)

One reason for the use of the variance in preference to other measures of dispersion is that the variance of the sum (or the difference) of [uncorrelated](/source/Uncorrelated) random variables is the sum of their variances: Var ⁡ ( ∑ i = 1 n X i ) = ∑ i = 1 n Var ⁡ ( X i ) . {\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](/source/Ir%C3%A9n%C3%A9e-Jules_Bienaym%C3%A9) formula[5] and was discovered in 1853.[6][7] It is often made with the stronger condition that the variables are [independent](/source/Statistical_independence), but being uncorrelated suffices. So if all the variables have the same variance *σ*2, then, since division by *n* is a linear transformation, this formula immediately implies that the variance of their mean is Var ⁡ ( X ¯ ) = Var ⁡ ( 1 n ∑ i = 1 n X i ) = 1 n 2 ∑ i = 1 n Var ⁡ ( X i ) = 1 n 2 n σ 2 = σ 2 n . {\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](/source/Standard_error_(statistics)) of the sample mean, which is used in the [central limit theorem](/source/Central_limit_theorem).

To prove the initial statement, it suffices to show that Var ⁡ ( X + Y ) = Var ⁡ ( X ) + Var ⁡ ( Y ) . {\displaystyle \operatorname {Var} (X+Y)=\operatorname {Var} (X)+\operatorname {Var} (Y).}

The general result then follows by induction. Starting with the definition, Var ⁡ ( X + Y ) = E ⁡ [ ( X + Y ) 2 ] − ( E ⁡ [ X + Y ] ) 2 = E ⁡ [ X 2 + 2 X Y + Y 2 ] − ( E ⁡ [ X ] + E ⁡ [ Y ] ) 2 . {\displaystyle {\begin{aligned}\operatorname {Var} (X+Y)&=\operatorname {E} \left[(X+Y)^{2}\right]-(\operatorname {E} [X+Y])^{2}\\[5pt]&=\operatorname {E} \left[X^{2}+2XY+Y^{2}\right]-(\operatorname {E} [X]+\operatorname {E} [Y])^{2}.\end{aligned}}}

Using the linearity of the [expectation operator](/source/Expectation_Operator) and the assumption of independence (or uncorrelatedness) of *X* and *Y*, this further simplifies as follows: Var ⁡ ( X + Y ) = E ⁡ [ X 2 ] + 2 E ⁡ [ X Y ] + E ⁡ [ Y 2 ] − ( E ⁡ [ X ] 2 + 2 E ⁡ [ X ] E ⁡ [ Y ] + E ⁡ [ Y ] 2 ) = E ⁡ [ X 2 ] + E ⁡ [ Y 2 ] − E ⁡ [ X ] 2 − E ⁡ [ Y ] 2 = Var ⁡ ( X ) + Var ⁡ ( Y ) . {\displaystyle {\begin{aligned}\operatorname {Var} (X+Y)&=\operatorname {E} {\left[X^{2}\right]}+2\operatorname {E} [XY]+\operatorname {E} {\left[Y^{2}\right]}-\left(\operatorname {E} [X]^{2}+2\operatorname {E} [X]\operatorname {E} [Y]+\operatorname {E} [Y]^{2}\right)\\[5pt]&=\operatorname {E} \left[X^{2}\right]+\operatorname {E} \left[Y^{2}\right]-\operatorname {E} [X]^{2}-\operatorname {E} [Y]^{2}\\[5pt]&=\operatorname {Var} (X)+\operatorname {Var} (Y).\end{aligned}}}

#### Sum of correlated variables

#### Sum of correlated variables with fixed sample size

Main article: [Bienaymé's identity](/source/Bienaym%C3%A9's_identity)

In general, the variance of the sum of n variables is the sum of their [covariances](/source/Covariance): Var ⁡ ( ∑ i = 1 n X i ) = ∑ i = 1 n ∑ j = 1 n Cov ⁡ ( X i , X j ) = ∑ i = 1 n Var ⁡ ( X i ) + 2 ∑ 1 ≤ i < j ≤ n Cov ⁡ ( X i , X j ) . {\displaystyle \operatorname {Var} \left(\sum _{i=1}^{n}X_{i}\right)=\sum _{i=1}^{n}\sum _{j=1}^{n}\operatorname {Cov} \left(X_{i},X_{j}\right)=\sum _{i=1}^{n}\operatorname {Var} \left(X_{i}\right)+2\sum _{1\leq i<j\leq n}\operatorname {Cov} \left(X_{i},X_{j}\right).} (Note: The second equality comes from the fact that Cov(*X**i*, *X**i*) = Var(*X**i*).)

Here, Cov ⁡ ( ⋅ , ⋅ ) {\displaystyle \operatorname {Cov} (\cdot ,\cdot )} is the [covariance](/source/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](/source/Cronbach's_alpha) in [classical test theory](/source/Classical_test_theory).

So, if the variables have equal variance *σ*2 and the average [correlation](/source/Correlation) of distinct variables is *ρ*, then the variance of their mean is Var ⁡ ( X ¯ ) = σ 2 n + n − 1 n ρ σ 2 . {\displaystyle \operatorname {Var} \left({\overline {X}}\right)={\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](/source/Standard_error). Moreover, if the variables have unit variance, for example if they are standardized, then this simplifies to Var ⁡ ( X ¯ ) = 1 n + n − 1 n ρ . {\displaystyle \operatorname {Var} \left({\overline {X}}\right)={\frac {1}{n}}+{\frac {n-1}{n}}\rho .}

This formula is used in the [Spearman–Brown prediction formula](/source/Spearman%E2%80%93Brown_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 lim n → ∞ Var ⁡ ( X ¯ ) = ρ . {\displaystyle \lim _{n\to \infty }\operatorname {Var} \left({\overline {X}}\right)=\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](/source/Law_of_large_numbers) states that the sample mean will converge for independent variables.

#### Sum of uncorrelated variables with random sample size

There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. In such cases, the sample size *N* is a random variable whose variation adds to the variation of *X*, such that,[8] Var ⁡ ( ∑ i = 1 N X i ) = E ⁡ [ N ] Var ⁡ ( X ) + Var ⁡ ( N ) ( E ⁡ [ X ] ) 2 {\displaystyle \operatorname {Var} \left(\sum _{i=1}^{N}X_{i}\right)=\operatorname {E} \left[N\right]\operatorname {Var} (X)+\operatorname {Var} (N)(\operatorname {E} \left[X\right])^{2}} which follows from the [law of total variance](/source/Law_of_total_variance).

If *N* has a [Poisson distribution](/source/Poisson_distribution), then E ⁡ [ N ] = Var ⁡ ( N ) {\displaystyle \operatorname {E} [N]=\operatorname {Var} (N)} with estimator *n* = *N*. So, the estimator of Var ⁡ ( ∑ i = 1 n X i ) {\displaystyle \textstyle \operatorname {Var} \left(\sum _{i=1}^{n}X_{i}\right)} becomes ⁠ n S x 2 + n X ¯ 2 {\displaystyle \textstyle n{S_{x}}^{2}+n{\bar {X}}^{2}} ⁠, giving SE ⁡ ( X ¯ ) = ( S x 2 + X ¯ 2 ) / n {\displaystyle \textstyle \operatorname {SE} ({\bar {X}})={\sqrt {{({S_{x}}^{2}+{\bar {X}}^{2})}/{n}}}} (see *[Standard error § Standard error of the sample mean](/source/Standard_error#Standard_error_of_the_sample_mean)*).

#### Weighted sum of variables

See also: [Variance of a weighted arithmetic mean](/source/Weighted_arithmetic_mean#Variance)

Not to be confused with [Weighted variance](/source/Weighted_variance).

The scaling property and the Bienaymé formula, along with the property of the [covariance](/source/Covariance) Cov(*aX*, *bY*) = *ab* Cov(*X*, *Y*) jointly imply that Var ⁡ ( a X ± b Y ) = a 2 Var ⁡ ( X ) + b 2 Var ⁡ ( Y ) ± 2 a b Cov ⁡ ( X , Y ) . {\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: Var ⁡ ( ∑ i n a i X i ) = ∑ i = 1 n a i 2 Var ⁡ ( X i ) + 2 ∑ 1 ≤ i ∑ < j ≤ n a i a j Cov ⁡ ( X i , X j ) {\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 _{<j\leq n}a_{i}a_{j}\operatorname {Cov} (X_{i},X_{j})}

### Product of variables

#### Product of independent variables

If two variables X and Y are [independent](/source/Independence_(probability_theory)), the variance of their product is given by[9] Var ⁡ ( X Y ) = [ E ⁡ ( X ) ] 2 Var ⁡ ( Y ) + [ E ⁡ ( Y ) ] 2 Var ⁡ ( X ) + Var ⁡ ( X ) Var ⁡ ( Y ) . {\displaystyle \operatorname {Var} (XY)=[\operatorname {E} (X)]^{2}\operatorname {Var} (Y)+[\operatorname {E} (Y)]^{2}\operatorname {Var} (X)+\operatorname {Var} (X)\operatorname {Var} (Y).}

Equivalently, using the basic properties of expectation, it is given by Var ⁡ ( X Y ) = E ⁡ ( X 2 ) E ⁡ ( Y 2 ) − [ E ⁡ ( X ) ] 2 [ E ⁡ ( Y ) ] 2 . {\displaystyle \operatorname {Var} (XY)=\operatorname {E} \left(X^{2}\right)\operatorname {E} \left(Y^{2}\right)-[\operatorname {E} (X)]^{2}[\operatorname {E} (Y)]^{2}.}

#### Product of statistically dependent variables

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

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

### Arbitrary functions

Main article: [Uncertainty propagation](/source/Uncertainty_propagation)

The [delta method](/source/Delta_method) uses second-order [Taylor expansions](/source/Taylor_expansion) to approximate the variance of a function of one or more random variables (see *[Taylor expansions for the moments of functions of random variables](/source/Taylor_expansions_for_the_moments_of_functions_of_random_variables)*). For example, the approximate variance of a function of one variable is given by Var ⁡ [ f ( X ) ] ≈ ( f ′ ( E ⁡ [ X ] ) ) 2 Var ⁡ [ X ] {\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

See also: [Unbiased estimation of standard deviation](/source/Unbiased_estimation_of_standard_deviation)

Real-world observations such as the measurements of yesterday's rain throughout the day typically cannot be complete sets of all possible observations that could be made. As such, the variance calculated from the finite set will in general not match the variance that would have been calculated from the full population of possible observations. This means that one [estimates](/source/Estimation_theory) the mean and variance from a limited set of observations by using an [estimator](/source/Estimator) equation. The estimator is a function of the [sample](/source/Sample_(statistics)) of *n* [observations](/source/Observations) drawn without observational bias from the whole [population](/source/Statistical_population) of potential observations. In this example, the sample would be the set of actual measurements of yesterday's rainfall from available rain gauges within the geography of interest.

The simplest estimators for population mean and population variance are simply the mean and variance of the sample, the **sample mean** and **(uncorrected) sample variance** – these are [consistent estimators](/source/Consistent_estimator) (they converge to the value of the whole population as the number of samples increases) but can be improved. Most simply, the sample variance is computed as the sum of [squared deviations](/source/Squared_deviations) about the (sample) mean, divided by *n* as the number of samples. However, using values other than *n* improves the estimator in various ways. Four common values for the denominator are *n*, *n* − 1, *n* + 1, and *n* − 1.5: *n* is the simplest (the variance of the sample), *n* − 1 eliminates bias,[10] *n* + 1 minimizes [mean squared error](/source/Mean_squared_error) for the normal distribution,[11] and *n* − 1.5 mostly eliminates bias in [unbiased estimation of standard deviation](/source/Unbiased_estimation_of_standard_deviation) for the normal distribution.[12]

Firstly, if the true population mean is unknown, then the sample variance (which uses the sample mean in place of the true mean) is a [biased estimator](/source/Biased_estimator): it underestimates the variance by a factor of (*n* − 1) / *n*; correcting this factor, resulting in the sum of squared deviations about the sample mean divided by *n* − 1 instead of *n*, is called *[Bessel's correction](/source/Bessel's_correction)*.[10] The resulting estimator is unbiased and is called the **(corrected) sample variance** or **unbiased sample variance**. If the mean is determined in some other way than from the same samples used to estimate the variance, then this bias does not arise, and the variance can safely be estimated as that of the samples about the (independently known) mean.

Secondly, the sample variance does not generally minimize [mean squared error](/source/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](/source/Excess_kurtosis) of the population (see *[Mean squared error § Variance](/source/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](/source/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.[11] 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](/source/Statistical_population) of size ⁠ N {\displaystyle N} ⁠ with values *x**i* is given by σ 2 = 1 N ∑ i = 1 N ( x i − μ ) 2 = 1 N ∑ i = 1 N ( x i 2 − 2 μ x i + μ 2 ) = ( 1 N ∑ i = 1 N x i 2 ) − 2 μ ( 1 N ∑ i = 1 N x i ) + μ 2 = E ⁡ [ x i 2 ] − μ 2 , {\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]&=\operatorname {E} [x_{i}^{2}]-\mu ^{2},\end{aligned}}} where the population mean is μ = E ⁡ [ x i ] = 1 N ∑ i = 1 N x i {\textstyle \mu =\operatorname {E} [x_{i}]={\frac {1}{N}}\sum _{i=1}^{N}x_{i}} and ⁠ E ⁡ [ x i 2 ] = ( 1 N ∑ i = 1 N x i 2 ) {\displaystyle \textstyle \operatorname {E} [x_{i}^{2}]=\left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}^{2}\right)} ⁠, where E {\textstyle \operatorname {E} } is the [expectation value](/source/Expected_value) operator.

The population variance can also be computed using[13] σ 2 = 1 N 2 ∑ i < j ( x i − x j ) 2 = 1 2 N 2 ∑ i , j = 1 N ( x i − x j ) 2 . {\displaystyle \sigma ^{2}={\frac {1}{N^{2}}}\sum _{i<j}\left(x_{i}-x_{j}\right)^{2}={\frac {1}{2N^{2}}}\sum _{i,j=1}^{N}\left(x_{i}-x_{j}\right)^{2}.}

(The right side has duplicate terms in the sum while the middle side has only unique terms to sum.) This is true because 1 2 N 2 ∑ i , j = 1 N ( x i − x j ) 2 = 1 2 N 2 ∑ i , j = 1 N ( x i 2 − 2 x i x j + x j 2 ) = 1 2 N ∑ j = 1 N ( 1 N ∑ i = 1 N x i 2 ) − ( 1 N ∑ i = 1 N x i ) ( 1 N ∑ j = 1 N x j ) + 1 2 N ∑ i = 1 N ( 1 N ∑ j = 1 N x j 2 ) = 1 2 ( σ 2 + μ 2 ) − μ 2 + 1 2 ( σ 2 + μ 2 ) = σ 2 . {\displaystyle {\begin{aligned}&{\frac {1}{2N^{2}}}\sum _{i,j=1}^{N}{\left(x_{i}-x_{j}\right)}^{2}\\[5pt]={}&{\frac {1}{2N^{2}}}\sum _{i,j=1}^{N}\left(x_{i}^{2}-2x_{i}x_{j}+x_{j}^{2}\right)\\[5pt]={}&{\frac {1}{2N}}\sum _{j=1}^{N}\left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}^{2}\right)-\left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}\right)\left({\frac {1}{N}}\sum _{j=1}^{N}x_{j}\right)+{\frac {1}{2N}}\sum _{i=1}^{N}\left({\frac {1}{N}}\sum _{j=1}^{N}x_{j}^{2}\right)\\[5pt]={}&{\frac {1}{2}}\left(\sigma ^{2}+\mu ^{2}\right)-\mu ^{2}+{\frac {1}{2}}\left(\sigma ^{2}+\mu ^{2}\right)\\[5pt]={}&\sigma ^{2}.\end{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

See also: [Sample standard deviation](/source/Sample_standard_deviation)

#### Biased sample variance

In many practical situations, the true variance of a population is not known *a priori* and must be computed somehow. When dealing with extremely large populations, it is not possible to count every object in the population, so the computation must be performed on a [sample](/source/Sample_(statistics)) of the population.[14] This is generally referred to as **sample variance** or **empirical variance**. Sample variance can also be applied to the estimation of the variance of a continuous distribution from a sample of that distribution.

We take a [sample with replacement](/source/Statistical_sample) of ⁠ n {\displaystyle n} ⁠ values *Y*1, ..., *Y**n* from the population of size ⁠ N {\displaystyle N} ⁠, where *n* < *N*, and estimate the variance on the basis of this sample.[15] Directly taking the variance of the sample data gives the average of the [squared deviations](/source/Squared_deviations):[16] S ~ Y 2 = 1 n ∑ i = 1 n ( Y i − Y ¯ ) 2 = ( 1 n ∑ i = 1 n Y i 2 ) − Y ¯ 2 = 1 n 2 ∑ i , j : i < j ( Y i − Y j ) 2 . {\displaystyle {\tilde {S}}_{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<j}\left(Y_{i}-Y_{j}\right)^{2}.} (See the section *[§ Population variance](#Population_variance)* for the derivation of this formula.) Here, Y ¯ {\displaystyle {\overline {Y}}} denotes the [sample mean](/source/Sample_mean): Y ¯ = 1 n ∑ i = 1 n Y i . {\displaystyle {\overline {Y}}={\frac {1}{n}}\sum _{i=1}^{n}Y_{i}.}

Since the *Y**i* are selected randomly, both Y ¯ {\displaystyle {\overline {Y}}} and S ~ Y 2 {\displaystyle {\tilde {S}}_{Y}^{2}} are [random variables](/source/Random_variable). Their expected values can be evaluated by averaging over the ensemble of all possible samples {*Y**i*} of size ⁠ n {\displaystyle n} ⁠ from the population. For S ~ Y 2 {\displaystyle {\tilde {S}}_{Y}^{2}} this gives: E ⁡ [ S ~ Y 2 ] = E ⁡ [ 1 n ∑ i = 1 n ( Y i − 1 n ∑ j = 1 n Y j ) 2 ] = 1 n ∑ i = 1 n E ⁡ [ Y i 2 − 2 n Y i ∑ j = 1 n Y j + 1 n 2 ∑ j = 1 n Y j ∑ k = 1 n Y k ] = 1 n ∑ i = 1 n ( E ⁡ [ Y i 2 ] − 2 n ( ∑ j ≠ i E ⁡ [ Y i Y j ] + E ⁡ [ Y i 2 ] ) + 1 n 2 ∑ j = 1 n ∑ k ≠ j n E ⁡ [ Y j Y k ] + 1 n 2 ∑ j = 1 n E ⁡ [ Y j 2 ] ) = 1 n ∑ i = 1 n ( n − 2 n E ⁡ [ Y i 2 ] − 2 n ∑ j ≠ i E ⁡ [ Y i Y j ] + 1 n 2 ∑ j = 1 n ∑ k ≠ j n E ⁡ [ Y j Y k ] + 1 n 2 ∑ j = 1 n E ⁡ [ Y j 2 ] ) = 1 n ∑ i = 1 n [ n − 2 n ( σ 2 + μ 2 ) − 2 n ( n − 1 ) μ 2 + 1 n 2 n ( n − 1 ) μ 2 + 1 n ( σ 2 + μ 2 ) ] = n − 1 n σ 2 . {\displaystyle {\begin{aligned}\operatorname {E} [{\tilde {S}}_{Y}^{2}]&=\operatorname {E} \left[{\frac {1}{n}}\sum _{i=1}^{n}{\left(Y_{i}-{\frac {1}{n}}\sum _{j=1}^{n}Y_{j}\right)}^{2}\right]\\[5pt]&={\frac {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(\operatorname {E} \left[Y_{i}^{2}\right]-{\frac {2}{n}}\left(\sum _{j\neq i}\operatorname {E} \left[Y_{i}Y_{j}\right]+\operatorname {E} \left[Y_{i}^{2}\right]\right)+{\frac {1}{n^{2}}}\sum _{j=1}^{n}\sum _{k\neq j}^{n}\operatorname {E} \left[Y_{j}Y_{k}\right]+{\frac {1}{n^{2}}}\sum _{j=1}^{n}\operatorname {E} \left[Y_{j}^{2}\right]\right)\\[5pt]&={\frac {1}{n}}\sum _{i=1}^{n}\left({\frac {n-2}{n}}\operatorname {E} \left[Y_{i}^{2}\right]-{\frac {2}{n}}\sum _{j\neq i}\operatorname {E} \left[Y_{i}Y_{j}\right]+{\frac {1}{n^{2}}}\sum _{j=1}^{n}\sum _{k\neq j}^{n}\operatorname {E} \left[Y_{j}Y_{k}\right]+{\frac {1}{n^{2}}}\sum _{j=1}^{n}\operatorname {E} \left[Y_{j}^{2}\right]\right)\\[5pt]&={\frac {1}{n}}\sum _{i=1}^{n}\left[{\frac {n-2}{n}}\left(\sigma ^{2}+\mu ^{2}\right)-{\frac {2}{n}}(n-1)\mu ^{2}+{\frac {1}{n^{2}}}n(n-1)\mu ^{2}+{\frac {1}{n}}\left(\sigma ^{2}+\mu ^{2}\right)\right]\\[5pt]&={\frac {n-1}{n}}\sigma ^{2}.\end{aligned}}}

Here σ 2 = E ⁡ [ Y i 2 ] − μ 2 {\textstyle \sigma ^{2}=\operatorname {E} [Y_{i}^{2}]-\mu ^{2}} derived in the section is [population variance](#Population_variance) and E ⁡ [ Y i Y j ] = E ⁡ [ Y i ] E ⁡ [ Y j ] = μ 2 {\textstyle \operatorname {E} [Y_{i}Y_{j}]=\operatorname {E} [Y_{i}]\operatorname {E} [Y_{j}]=\mu ^{2}} due to independency of Y i {\displaystyle Y_{i}} and ⁠ Y j {\displaystyle Y_{j}} ⁠.

Hence S ~ Y 2 {\textstyle {\tilde {S}}_{Y}^{2}} gives an estimate of the population variance σ 2 {\textstyle \sigma ^{2}} that is biased by a factor of n − 1 n {\textstyle {\frac {n-1}{n}}} because the expectation value of S ~ Y 2 {\textstyle {\tilde {S}}_{Y}^{2}} is smaller than the population variance (true variance) by that factor. For this reason, S ~ Y 2 {\textstyle {\tilde {S}}_{Y}^{2}} is referred to as the *biased sample variance*.

#### Unbiased sample variance

Correcting for this bias yields the *unbiased sample variance*, denoted ⁠ S 2 {\displaystyle S^{2}} ⁠: S 2 = n n − 1 S ~ Y 2 = n n − 1 [ 1 n ∑ i = 1 n ( Y i − Y ¯ ) 2 ] = 1 n − 1 ∑ i = 1 n ( Y i − Y ¯ ) 2 {\displaystyle S^{2}={\frac {n}{n-1}}{\tilde {S}}_{Y}^{2}={\frac {n}{n-1}}\left[{\frac {1}{n}}\sum _{i=1}^{n}\left(Y_{i}-{\overline {Y}}\right)^{2}\right]={\frac {1}{n-1}}\sum _{i=1}^{n}\left(Y_{i}-{\overline {Y}}\right)^{2}}

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](/source/Bessel's_correction), and it is also used in [sample covariance](/source/Sample_covariance) and the [sample standard deviation](/source/Sample_standard_deviation) (the square root of variance). The square root is a [concave function](/source/Concave_function) and thus introduces negative bias (by [Jensen's inequality](/source/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](/source/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](/source/U-statistic) for the function *f*(*y*1, *y*2) = (*y*1 − *y*2)2/2, meaning that it is obtained by averaging a 2-sample statistic over 2-element subsets of the population.

#### Example

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

#### Distribution of the sample variance

Distribution and cumulative distribution of *S*2/σ2, for various values of *ν* = *n* − 1, when the *yi* are independent normally distributed.

Being a function of [random variables](/source/Random_variable), the sample variance is itself a random variable, and it is natural to study its distribution. In the case that *Y**i* are independent observations from a [normal distribution](/source/Normal_distribution), [Cochran's theorem](/source/Cochran's_theorem) shows that the [unbiased sample variance](#Unbiased_sample_variance) *S*2 follows a scaled [chi-squared distribution](/source/Chi-squared_distribution) (see also: [asymptotic properties](/source/Chi-squared_distribution#Asymptotic_properties) and an [elementary proof](/source/Chi-squared_distribution#Cochran's_theorem)):[17] ( n − 1 ) S 2 σ 2 ∼ χ n − 1 2 , {\displaystyle (n-1){\frac {S^{2}}{\sigma ^{2}}}\sim \chi _{n-1}^{2},} where *σ*2 is the [population variance](#Population_variance). As a direct consequence, it follows that E ⁡ ( S 2 ) = E ⁡ ( σ 2 n − 1 χ n − 1 2 ) = σ 2 , {\displaystyle \operatorname {E} \left(S^{2}\right)=\operatorname {E} \left({\frac {\sigma ^{2}}{n-1}}\chi _{n-1}^{2}\right)=\sigma ^{2},} and[18] Var ⁡ [ S 2 ] = Var ⁡ ( σ 2 n − 1 χ n − 1 2 ) = σ 4 ( n − 1 ) 2 Var ⁡ ( χ n − 1 2 ) = 2 σ 4 n − 1 . {\displaystyle \operatorname {Var} \left[S^{2}\right]=\operatorname {Var} \left({\frac {\sigma ^{2}}{n-1}}\chi _{n-1}^{2}\right)={\frac {\sigma ^{4}}{{\left(n-1\right)}^{2}}}\operatorname {Var} \left(\chi _{n-1}^{2}\right)={\frac {2\sigma ^{4}}{n-1}}.}

If *Y**i* are independent and identically distributed, but not necessarily normally distributed, then[19] E ⁡ [ S 2 ] = σ 2 ; Var ⁡ [ S 2 ] = σ 4 n ( κ − 1 + 2 n − 1 ) = 1 n ( μ 4 − n − 3 n − 1 σ 4 ) , {\displaystyle \operatorname {E} \left[S^{2}\right]=\sigma ^{2};\quad \operatorname {Var} \left[S^{2}\right]={\frac {\sigma ^{4}}{n}}\left(\kappa -1+{\frac {2}{n-1}}\right)={\frac {1}{n}}\left(\mu _{4}-{\frac {n-3}{n-1}}\sigma ^{4}\right),} where *κ* is the [kurtosis](/source/Kurtosis) of the distribution and *μ*4 is the fourth [central moment](/source/Central_moment).

If the conditions of the [law of large numbers](/source/Law_of_large_numbers) hold for the squared observations, *S*2 is a [consistent estimator](/source/Consistent_estimator) of *σ*2. One can see indeed that the variance of the estimator tends asymptotically to zero. An asymptotically equivalent formula was given in Kenney and Keeping (1951:164), Rose and Smith (2002:264), and Weisstein (n.d.).[20][21][22]

#### Samuelson's inequality

[Samuelson's inequality](/source/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.[23] Values must lie within the limits ⁠ y ¯ ± σ Y ( n − 1 ) 1 / 2 {\displaystyle {\bar {y}}\pm \sigma _{Y}(n-1)^{1/2}} ⁠.

#### Effect of adding one observation on variance

When a single new observation x n + 1 {\displaystyle x_{n+1}} is added to a set of n {\displaystyle n} observations with mean x ¯ n {\displaystyle {\bar {x}}_{n}} and variance s n 2 {\displaystyle s_{n}^{2}} , the new variance s n + 1 2 {\displaystyle s_{n+1}^{2}} can be expressed using a recursive updating formula. Based on the identity for the sum of squares provided by Chan et al. (1983)[24]:

- n s n + 1 2 = ( n − 1 ) s n 2 + n n + 1 ( x n + 1 − x ¯ n ) 2 {\displaystyle ns_{n+1}^{2}=(n-1)s_{n}^{2}+{\frac {n}{n+1}}(x_{n+1}-{\bar {x}}_{n})^{2}}

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

[Proof]

**Derivation for Sample Variance**

Using the updating formula for the sum of squares ( S S {\displaystyle SS} ):

- S S n + 1 = S S n + n n + 1 ( x n + 1 − x ¯ n ) 2 {\displaystyle SS_{n+1}=SS_{n}+{\frac {n}{n+1}}(x_{n+1}-{\bar {x}}_{n})^{2}}

Substituting the relationship for sample variance ( S S = ( n − 1 ) s 2 {\displaystyle SS=(n-1)s^{2}} ):

- n s n + 1 2 = ( n − 1 ) s n 2 + n n + 1 ( x n + 1 − x ¯ n ) 2 {\displaystyle ns_{n+1}^{2}=(n-1)s_{n}^{2}+{\frac {n}{n+1}}(x_{n+1}-{\bar {x}}_{n})^{2}}

Setting s n + 1 2 = s n 2 {\displaystyle s_{n+1}^{2}=s_{n}^{2}} :

- n s n 2 = ( n − 1 ) s n 2 + n n + 1 ( x n + 1 − x ¯ n ) 2 {\displaystyle ns_{n}^{2}=(n-1)s_{n}^{2}+{\frac {n}{n+1}}(x_{n+1}-{\bar {x}}_{n})^{2}}

- s n 2 = n n + 1 ( x n + 1 − x ¯ n ) 2 {\displaystyle s_{n}^{2}={\frac {n}{n+1}}(x_{n+1}-{\bar {x}}_{n})^{2}}

Solving for x n + 1 {\displaystyle x_{n+1}} yields:

- x n + 1 = x ¯ n ± s n n + 1 n {\displaystyle x_{n+1}={\bar {x}}_{n}\pm s_{n}{\sqrt {\frac {n+1}{n}}}}

**Derivation for Population Variance**

For population variance ( σ 2 = S S n {\displaystyle \sigma ^{2}={\frac {SS}{n}}} ), the updating formula is:

- ( n + 1 ) σ n + 1 2 = n σ n 2 + n n + 1 ( x n + 1 − μ n ) 2 {\displaystyle (n+1)\sigma _{n+1}^{2}=n\sigma _{n}^{2}+{\frac {n}{n+1}}(x_{n+1}-\mu _{n})^{2}}

Setting σ n + 1 2 = σ n 2 {\displaystyle \sigma _{n+1}^{2}=\sigma _{n}^{2}} :

- ( n + 1 ) σ n 2 = n σ n 2 + n n + 1 ( x n + 1 − μ n ) 2 {\displaystyle (n+1)\sigma _{n}^{2}=n\sigma _{n}^{2}+{\frac {n}{n+1}}(x_{n+1}-\mu _{n})^{2}}

- σ n 2 = n n + 1 ( x n + 1 − μ n ) 2 {\displaystyle \sigma _{n}^{2}={\frac {n}{n+1}}(x_{n+1}-\mu _{n})^{2}}

Solving for x n + 1 {\displaystyle x_{n+1}} yields:

- x n + 1 = μ n ± σ n n + 1 n {\displaystyle x_{n+1}=\mu _{n}\pm \sigma _{n}{\sqrt {\frac {n+1}{n}}}}

### Relations with the harmonic and arithmetic means

It has been shown[25] that for a sample {*y**i*} of positive real numbers, σ y 2 ≤ 2 y max ( A − H ) , {\displaystyle \sigma _{y}^{2}\leq 2y_{\max }(A-H),} where *y*max is the maximum of the sample, ⁠ A {\displaystyle A} ⁠ is the arithmetic mean, ⁠ H {\displaystyle H} ⁠ is the [harmonic mean](/source/Harmonic_mean) of the sample and σ y 2 {\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 σ y 2 ≤ y max ( A − H ) ( y max − A ) y max − H , σ y 2 ≥ y min ( A − H ) ( A − y min ) H − y min , {\displaystyle {\begin{aligned}\sigma _{y}^{2}&\leq {\frac {y_{\max }(A-H)(y_{\max }-A)}{y_{\max }-H}},\\[1ex]\sigma _{y}^{2}&\geq {\frac {y_{\min }(A-H)(A-y_{\min })}{H-y_{\min }}},\end{aligned}}} where *y*min is the minimum of the sample.[26]

## Tests of equality of variances

The [F-test of equality of variances](/source/F-test_of_equality_of_variances) and the [chi square tests](/source/Chi_square_test) are adequate when the sample is normally distributed. Non-normality makes testing for the equality of two or more variances more difficult.

Several non parametric tests have been proposed: these include the Barton–David–Ansari–Freund–Siegel–Tukey test, the [Capon test](https://en.wikipedia.org/w/index.php?title=Capon_test&action=edit&redlink=1), [Mood test](/source/Mood_test), the [Klotz test](https://en.wikipedia.org/w/index.php?title=Klotz_test&action=edit&redlink=1) and the [Sukhatme test](https://en.wikipedia.org/w/index.php?title=Sukhatme_test&action=edit&redlink=1). The Sukhatme test applies to two variances and requires that both [medians](/source/Median) 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](https://en.wikipedia.org/w/index.php?title=Lehmann_test&action=edit&redlink=1) 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](https://en.wikipedia.org/w/index.php?title=Box_test&action=edit&redlink=1), the [Box–Anderson test](https://en.wikipedia.org/w/index.php?title=Box%E2%80%93Anderson_test&action=edit&redlink=1) and the [Moses test](https://en.wikipedia.org/w/index.php?title=Moses_test&action=edit&redlink=1).

Resampling methods, which include the [bootstrap](/source/Bootstrapping_(statistics)) and the [jackknife](/source/Resampling_(statistics)), may be used to test the equality of variances.

## Moment of inertia

See also: [Moment (physics) § Examples](/source/Moment_(physics)#Examples)

The variance of a probability distribution is analogous to the [moment of inertia](/source/Moment_of_inertia) in [classical mechanics](/source/Classical_mechanics) of a corresponding mass distribution along a line, with respect to rotation about its center of mass.[27] It is because of this analogy that such things as the variance are called *[moments](/source/Moment_(mathematics))* of [probability distributions](/source/Probability_distribution).[27] The covariance matrix is related to the [moment of inertia tensor](/source/Moment_of_inertia_tensor) for multivariate distributions. The moment of inertia of a cloud of *n* points with a covariance matrix of Σ {\displaystyle \Sigma } is given by[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*] I = n ( 1 3 × 3 tr ⁡ ( Σ ) − Σ ) . {\displaystyle I=n\left(\mathbf {1} _{3\times 3}\operatorname {tr} (\Sigma )-\Sigma \right).}

This difference between moment of inertia in physics and in statistics is clear for points that are gathered along a line. Suppose many points are close to the *x* axis and distributed along it. The covariance matrix might look like Σ = [ 10 0 0 0 0.1 0 0 0 0.1 ] . {\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 I = n [ 0.2 0 0 0 10.1 0 0 0 10.1 ] . {\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: Semivariance = 1 n ∑ i : x i < μ ( x i − μ ) 2 {\displaystyle {\text{Semivariance}}={\frac {1}{n}}\sum _{i:x_{i}<\mu }{\left(x_{i}-\mu \right)}^{2}} It is also described as a specific measure in different fields of application. For skewed distributions, the semivariance can provide additional information that a variance does not.[28]

For inequalities associated with the semivariance, see *[Chebyshev's inequality § Semivariances](/source/Chebyshev's_inequality#Semivariances)*.

## Etymology

The term *variance* was first introduced by [Ronald Fisher](/source/Ronald_Fisher) in his 1918 paper *[The Correlation Between Relatives on the Supposition of Mendelian Inheritance](/source/The_Correlation_Between_Relatives_on_the_Supposition_of_Mendelian_Inheritance)*:[29]

The great body of available statistics show us that the deviations of a [human measurement](/source/Biometry) from its mean follow very closely the [Normal Law of Errors](/source/Normal_distribution), and, therefore, that the variability may be uniformly measured by the [standard deviation](/source/Standard_deviation) corresponding to the [square root](/source/Square_root) of the [mean square error](/source/Mean_square_error). When there are two independent causes of variability capable of producing in an otherwise uniform population distributions with standard deviations σ 1 {\displaystyle \sigma _{1}} and σ 2 {\displaystyle \sigma _{2}} , it is found that the distribution, when both causes act together, has a standard deviation σ 1 2 + σ 2 2 {\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...

## Generalizations

### For complex variables

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

### For vector-valued random variables

#### As a matrix

If X {\displaystyle X} is a [vector](/source/Vector_space)-valued random variable, with values in R n , {\displaystyle \mathbb {R} ^{n},} and thought of as a column vector, then a natural generalization of variance is E ⁡ [ ( X − μ ) ( X − μ ) T ] , {\displaystyle \operatorname {E} \left[(X-\mu ){(X-\mu )}^{\mathsf {T}}\right],} where μ = E ⁡ ( X ) {\displaystyle \mu =\operatorname {E} (X)} and X T {\displaystyle X^{\mathsf {T}}} is the transpose of ⁠ X {\displaystyle X} ⁠, and so is a row vector. The result is a [positive semi-definite square matrix](/source/Positive_definite_matrix), commonly referred to as the [variance-covariance matrix](/source/Variance-covariance_matrix) (or simply as the *covariance matrix*).

If X {\displaystyle X} is a vector- and complex-valued random variable, with values in ⁠ C n {\displaystyle \mathbb {C} ^{n}} ⁠, then the [covariance matrix is](/source/Covariance_matrix#Complex_random_vectors) ⁠ E ⁡ [ ( X − μ ) ( X − μ ) † ] {\displaystyle \operatorname {E} \left[(X-\mu ){(X-\mu )}^{\dagger }\right]} ⁠, where X † {\displaystyle X^{\dagger }} is the [conjugate transpose](/source/Conjugate_transpose) of ⁠ X {\displaystyle X} ⁠.[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*] This matrix is also positive semi-definite and square.

#### As a scalar

Another generalization of variance for vector-valued random variables X , {\displaystyle X,} which results in a scalar value rather than in a matrix, is the [generalized variance](/source/Generalized_variance) ⁠ det ( C ) {\displaystyle \det(C)} ⁠, the [determinant](/source/Determinant) of the covariance matrix. The generalized variance can be shown to be related to the multidimensional scatter of points around their mean.[30]

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

## See also

- [Mathematics portal](https://en.wikipedia.org/wiki/Portal:Mathematics)

Look up ***[variance](https://en.wiktionary.org/wiki/variance)*** in Wiktionary, the free dictionary.

- [Bhatia–Davis inequality](/source/Bhatia%E2%80%93Davis_inequality)

- [Coefficient of variation](/source/Coefficient_of_variation)

- [Homoscedasticity](/source/Homoscedasticity)

- [Least-squares spectral analysis](/source/Least-squares_spectral_analysis) for computing a [frequency spectrum](/source/Frequency_spectrum) with spectral magnitudes in % of variance or in [dB](/source/Decibel)

- [Modern portfolio theory](/source/Modern_portfolio_theory)

- [Popoviciu's inequality on variances](/source/Popoviciu's_inequality_on_variances)

- [Measures for statistical dispersion](/source/Statistical_dispersion)

- [Variance-stabilizing transformation](/source/Variance-stabilizing_transformation)

### Types of variance

- [Correlation](/source/Correlation)

- [Distance variance](/source/Distance_variance)

- [Explained variance](/source/Explained_variance)

- [Pooled variance](/source/Pooled_variance)

- [Pseudo-variance](/source/Pseudo-variance)

## References

1. **[^](#cite_ref-1)** Wasserman, Larry (2005). *All of Statistics: a concise course in statistical inference*. Springer texts in statistics. p. 51. [ISBN](/source/ISBN_(identifier)) [978-1-4419-2322-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4419-2322-6).

1. **[^](#cite_ref-2)** 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.

1. **[^](#cite_ref-3)** Kagan, A.; Shepp, L. A. (1998). "Why the variance?". *Statistics & Probability Letters*. **38** (4): 329–333. [doi](/source/Doi_(identifier)):[10.1016/S0167-7152(98)00041-8](https://doi.org/10.1016%2FS0167-7152%2898%2900041-8).

1. **[^](#cite_ref-4)** Johnson, Richard; Wichern, Dean (2001). [*Applied Multivariate Statistical Analysis*](https://archive.org/details/appliedmultivari00john_130). Prentice Hall. p. [76](https://archive.org/details/appliedmultivari00john_130/page/n96). [ISBN](/source/ISBN_(identifier)) [0-13-187715-1](https://en.wikipedia.org/wiki/Special:BookSources/0-13-187715-1).

1. **[^](#cite_ref-5)** [Loève, M.](/source/Michel_Lo%C3%A8ve) (1977) "Probability Theory", *Graduate Texts in Mathematics*, Volume 45, 4th edition, Springer-Verlag, p. 12.

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1. ^ [***a***](#cite_ref-bessel_10-0) [***b***](#cite_ref-bessel_10-1) Reichmann, W. J. (1961). "Appendix 8". *Use and Abuse of Statistics* (Reprinted 1964–1970 by Pelican ed.). London: Methuen.

1. ^ [***a***](#cite_ref-Kourouklis_11-0) [***b***](#cite_ref-Kourouklis_11-1) Kourouklis, Stavros (2012). ["A New Estimator of the Variance Based on Minimizing Mean Squared Error"](https://www.jstor.org/stable/23339501). *The American Statistician*. **66** (4): 234–236. [doi](/source/Doi_(identifier)):[10.1080/00031305.2012.735209](https://doi.org/10.1080%2F00031305.2012.735209). [ISSN](/source/ISSN_(identifier)) [0003-1305](https://search.worldcat.org/issn/0003-1305). [JSTOR](/source/JSTOR_(identifier)) [23339501](https://www.jstor.org/stable/23339501).

1. **[^](#cite_ref-12)** Brugger, R. M. (1969). "A Note on Unbiased Estimation of the Standard Deviation". *The American Statistician*. **23** (4): 32. [doi](/source/Doi_(identifier)):[10.1080/00031305.1969.10481865](https://doi.org/10.1080%2F00031305.1969.10481865).

1. **[^](#cite_ref-13)** 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.

1. **[^](#cite_ref-14)** Navidi, William (2006). *Statistics for Engineers and Scientists*. McGraw-Hill. p. 14.

1. **[^](#cite_ref-15)** Montgomery, D. C. and Runger, G. C. (1994) *Applied statistics and probability for engineers*, page 201. John Wiley & Sons New York

1. **[^](#cite_ref-16)** 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.

1. **[^](#cite_ref-17)** Knight, K. (2000). *Mathematical Statistics*. New York: Chapman and Hall. proposition 2.11.

1. **[^](#cite_ref-18)** [Casella, George](/source/George_Casella); [Berger, Roger L.](/source/Roger_Lee_Berger) (2002). *Statistical Inference* (2nd ed.). Example 7.3.3, p. 331. [ISBN](/source/ISBN_(identifier)) [0-534-24312-6](https://en.wikipedia.org/wiki/Special:BookSources/0-534-24312-6).

1. **[^](#cite_ref-19)** Mood, A. M., Graybill, F. A., and Boes, D.C. (1974) *Introduction to the Theory of Statistics*, 3rd Edition, McGraw-Hill, New York, p. 229

1. **[^](#cite_ref-20)** Kenney, John F.; Keeping, E.S. (1951). [*Mathematics of Statistics. Part Two*](https://web.archive.org/web/20181117022434/http://krishikosh.egranth.ac.in/bitstream/1/2025521/1/G2257.pdf) (PDF) (2nd ed.). Princeton, New Jersey: D. Van Nostrand Company, Inc. Archived from [the original](http://krishikosh.egranth.ac.in/bitstream/1/2025521/1/G2257.pdf) (PDF) on Nov 17, 2018 – via KrishiKosh.

1. **[^](#cite_ref-21)** Rose, Colin; Smith, Murray D. (2002). "[Mathematical Statistics with Mathematica](http://www.mathstatica.com/book/Mathematical_Statistics_with_Mathematica.pdf)". Springer-Verlag, New York.

1. **[^](#cite_ref-22)** Weisstein, Eric W. "[Sample Variance Distribution](http://mathworld.wolfram.com/SampleVarianceDistribution.html)". MathWorld Wolfram.

1. **[^](#cite_ref-23)** Samuelson, Paul (1968). "How Deviant Can You Be?". *[Journal of the American Statistical Association](/source/Journal_of_the_American_Statistical_Association)*. **63** (324): 1522–1525. [doi](/source/Doi_(identifier)):[10.1080/01621459.1968.10480944](https://doi.org/10.1080%2F01621459.1968.10480944). [JSTOR](/source/JSTOR_(identifier)) [2285901](https://www.jstor.org/stable/2285901).

1. **[^](#cite_ref-24)** Chan, Tony F.; Golub, Gene H.; LeVeque, Randall J. (1983). ["Algorithms for Computing the Sample Variance: Analysis and Recommendations"](https://www.jstor.org/stable/2683386). *The American Statistician*. **37** (3): 242–247. [doi](/source/Doi_(identifier)):[10.2307/2683386](https://doi.org/10.2307%2F2683386). formula 1.3(b [page 1](https://www.cs.yale.edu/publications/techreports/tr222.pdf)

1. **[^](#cite_ref-25)** 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"](https://doi.org/10.1006%2Fjmaa.1999.6688). *J. Math. Anal. Appl*. **243** (1): 163–173. [doi](/source/Doi_(identifier)):[10.1006/jmaa.1999.6688](https://doi.org/10.1006%2Fjmaa.1999.6688).

1. **[^](#cite_ref-Sharma2008_26-0)** Sharma, R. (2008). "Some more inequalities for arithmetic mean, harmonic mean and variance". *Journal of Mathematical Inequalities*. **2** (1): 109–114. [CiteSeerX](/source/CiteSeerX_(identifier)) [10.1.1.551.9397](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.551.9397). [doi](/source/Doi_(identifier)):[10.7153/jmi-02-11](https://doi.org/10.7153%2Fjmi-02-11).

1. ^ [***a***](#cite_ref-pearson_27-0) [***b***](#cite_ref-pearson_27-1) Magnello, M. Eileen. ["Karl Pearson and the Origins of Modern Statistics: An Elastician becomes a Statistician"](https://rutherfordjournal.org/article010107.html). *The Rutherford Journal*.

1. **[^](#cite_ref-28)** Fama, Eugene F.; French, Kenneth R. (2010-04-21). ["Q&A: Semi-Variance: A Better Risk Measure?"](https://famafrench.dimensional.com/questions-answers/qa-semi-variance-a-better-risk-measure.aspx). *Fama/French Forum*.

1. **[^](#cite_ref-29)** [Ronald Fisher](/source/Ronald_Fisher) (1918) [The correlation between relatives on the supposition of Mendelian Inheritance](http://digital.library.adelaide.edu.au/dspace/bitstream/2440/15097/1/9.pdf)

1. **[^](#cite_ref-30)** Kocherlakota, S.; Kocherlakota, K. (2004). "Generalized Variance". *Encyclopedia of Statistical Sciences*. Wiley Online Library. [doi](/source/Doi_(identifier)):[10.1002/0471667196.ess0869](https://doi.org/10.1002%2F0471667196.ess0869). [ISBN](/source/ISBN_(identifier)) [0-471-66719-6](https://en.wikipedia.org/wiki/Special:BookSources/0-471-66719-6).

v t e Theory of probability distributions probability mass function (pmf) probability density function (pdf) cumulative distribution function (cdf) quantile function raw moment central moment mean variance standard deviation skewness kurtosis L-moment moment generating function (mgf) characteristic function probability generating function (pgf) cumulant combinant

v t e Statistics Outline Index Descriptive statistics Continuous data Center Mean Arithmetic Arithmetic-Geometric Contraharmonic Cubic Generalized/power Geometric Harmonic Heronian Heinz Lehmer Median Mode Dispersion Average absolute deviation Coefficient of variation Interquartile range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary tables Contingency table Frequency distribution Grouped data Dependence Partial correlation Pearson product-moment correlation Rank correlation Kendall's τ Spearman's ρ Scatter plot Graphics Bar chart Biplot Box plot Control chart Correlogram Fan chart Forest plot Histogram Pie chart Q–Q plot Radar chart Run chart Scatter plot Stem-and-leaf display Violin plot Heatmap Scatter Plot Matrix ECDF plot Line chart Statistical data processing Transformations Data transformation Log transformation Power transform Box–Cox transformation Yeo–Johnson transformation Variance-stabilizing transformation Anscombe transform Fisher transformation Scaling and normalization Feature scaling Normalization Standardization (z-score) Min–max normalization Unit vector normalization Data cleaning Data cleaning Outlier Winsorizing Truncation Missing data Data reduction Dimensionality reduction Principal component analysis Factor analysis Time-series preprocessing Differencing Detrending Seasonal adjustment Stationarity transformation Data collection Study design Effect size Missing data Optimal design Population Replication Sample size determination Statistic Statistical power Survey methodology Sampling Cluster Stratified Opinion poll Questionnaire Standard error Controlled experiments Blocking Factorial experiment Interaction Random assignment Randomized controlled trial Randomized experiment Scientific control Adaptive designs Adaptive clinical trial Stochastic approximation Up-and-down designs Observational studies Cohort study Cross-sectional study Natural experiment Quasi-experiment Statistical inference Statistical theory Population Statistic Probability distribution Sampling distribution Order statistic Empirical distribution Density estimation Statistical model Model specification Lp space Parameter location scale shape Parametric family Likelihood (monotone) Location–scale family Exponential family Completeness Sufficiency Statistical functional Bootstrap U V Optimal decision loss function Efficiency Statistical distance divergence Asymptotics Robustness Frequentist inference Point estimation Estimating equations Maximum likelihood Method of moments M-estimator Minimum distance Unbiased estimators Mean-unbiased minimum-variance Rao–Blackwellization Lehmann–Scheffé theorem Median unbiased Plug-in Interval estimation Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling Bootstrap Jackknife Testing hypotheses 1- & 2-tails Power Uniformly most powerful test Permutation test Randomization test Multiple comparisons Parametric tests Likelihood-ratio Score/Lagrange multiplier Wald Specific tests Z-test (normal) Student's t-test F-test Goodness of fit Chi-squared G-test Kolmogorov–Smirnov Anderson–Darling Lilliefors Jarque–Bera Normality (Shapiro–Wilk) Likelihood-ratio test Model selection Cross validation AIC BIC Rank statistics Sign Sample median Signed rank (Wilcoxon) Hodges–Lehmann estimator Rank sum (Mann–Whitney) Nonparametric anova 1-way (Kruskal–Wallis) 2-way (Friedman) Ordered alternative (Jonckheere–Terpstra) Van der Waerden test Bayesian inference Bayesian probability prior posterior Credible interval Bayes factor Bayesian estimator Maximum posterior estimator Correlation Regression analysis Correlation Pearson product-moment Partial correlation Confounding variable Coefficient of determination Regression analysis Errors and residuals Regression validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS) Template:Least squares and regression analysis Linear regression Simple linear regression Ordinary least squares General linear model Bayesian regression Non-standard predictors Nonlinear regression Nonparametric Semiparametric Isotonic Robust Homoscedasticity and Heteroscedasticity Generalized linear model Exponential families Logistic (Bernoulli) / Binomial / Poisson regressions Partition of variance Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom Categorical / multivariate / time-series / survival analysis Categorical Cohen's kappa Contingency table Graphical model Log-linear model McNemar's test Cochran–Mantel–Haenszel statistics Multivariate Regression Manova Principal components Canonical correlation Discriminant analysis Cluster analysis Classification Structural equation model Factor analysis Multivariate distributions Elliptical distributions Normal Time-series General Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality Specific tests Dickey–Fuller Johansen Q-statistic (Ljung–Box) Durbin–Watson Breusch–Godfrey Time domain Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model (Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR) (Autoregressive model (AR)) Frequency domain Spectral density estimation Fourier analysis Least-squares spectral analysis Wavelet Whittle likelihood Survival Survival function Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time Hazard function Nelson–Aalen estimator Test Log-rank test Applications Biostatistics Bioinformatics Clinical trials / studies Epidemiology Medical statistics Engineering statistics Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification Social statistics Actuarial science Census Crime statistics Demography 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Adapted from the Wikipedia article [Variance](https://en.wikipedia.org/wiki/Variance) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Variance?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.