Variance

# Variance

Overview
In probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

, describing how far the numbers lie from the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

(expected value). In particular, the variance is one of the moments
Moment (mathematics)
In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...

of a distribution. In that context, it forms part of a systematic approach to distinguishing between probability distributions. While other such approaches have been developed, those based on moments
Moment (mathematics)
In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...

are advantageous in terms of mathematical and computational simplicity.

The variance is a parameter describing in part either the actual probability distribution of an observed population of numbers, or the theoretical probability distribution of a not-fully-observed population of numbers.
Discussion

Recent Discussions
Encyclopedia
In probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

, describing how far the numbers lie from the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

(expected value). In particular, the variance is one of the moments
Moment (mathematics)
In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...

of a distribution. In that context, it forms part of a systematic approach to distinguishing between probability distributions. While other such approaches have been developed, those based on moments
Moment (mathematics)
In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...

are advantageous in terms of mathematical and computational simplicity.

The variance is a parameter describing in part either the actual probability distribution of an observed population of numbers, or the theoretical probability distribution of a not-fully-observed population of numbers. In the latter case a sample of data from such a distribution can be used to construct an estimate of its variance: in the simplest cases this estimate can be the sample variance, defined below.

### Examples

The variance of a random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

or distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

is the expectation
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

, or mean, of the squared deviation
Absolute deviation
In statistics, the absolute deviation of an element of a data set is the absolute difference between that element and a given point. Typically the point from which the deviation is measured is a measure of central tendency, most often the median or sometimes the mean of the data set.D_i = |x_i-m|...

of that variable from its expected value or mean. Thus the variance is a measure of the amount of variation of the values of that variable, taking account of all possible values and their probabilities or weightings (not just the extremes which give the range).

For example, a perfect , when thrown, has expected value of

Its expected absolute deviation
Absolute deviation
In statistics, the absolute deviation of an element of a data set is the absolute difference between that element and a given point. Typically the point from which the deviation is measured is a measure of central tendency, most often the median or sometimes the mean of the data set.D_i = |x_i-m|...

- the mean of the equally likely absolute deviations from the mean - is

But its expected squared deviation - its variance (the mean of the equally likely squared deviations) - is

As another example, if a coin is tossed twice, the number of heads is: 0 with probability 0.25, 1 with probability 0.5 and 2 with probability 0.25. Thus the mean of the number of heads is 0.25 × 0 + 0.5 × 1 + 0.25 × 2 = 1, and the variance is 0.25 × (0 − 1)2 + 0.5 × (1 − 1)2 + 0.25 × (2 − 1)2 = 0.25 + 0 + 0.25 = 0.5.

### 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 inches will have a variance measured in square inches. For this reason, describing data sets via their standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

or root mean square deviation
Root mean square deviation
The root-mean-square deviation is the measure of the average distance between the atoms of superimposed proteins...

is often preferred over using the variance. In the dice example the standard deviation is √(17.5/6) ≈ 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
Covariance
In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...

, is used frequently in theoretical statistics; however the expected absolute deviation tends to be more robust
Robust statistics
Robust statistics provides an alternative approach to classical statistical methods. The motivation is to produce estimators that are not unduly affected by small departures from model assumptions.- Introduction :...

as it is less sensitive to outlier
Outlier
In statistics, an outlier is an observation that is numerically distant from the rest of the data. Grubbs defined an outlier as: An outlying observation, or outlier, is one that appears to deviate markedly from other members of the sample in which it occurs....

s arising from measurement anomalies or an unduly heavy-tailed distribution
Heavy-tailed distribution
In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution...

.

### Estimating the variance

Real-world distributions such as the distribution of yesterday's rain throughout the day are typically not fully known, unlike the behavior of perfect dice or an ideal distribution such as the normal distribution, because it is impractical to account for every raindrop. Instead one estimates
Estimation theory
Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the...

the mean and variance of the whole distribution as the computed mean and variance of a sample
Sample (statistics)
In statistics, a sample is a subset of a population. Typically, the population is very large, making a census or a complete enumeration of all the values in the population impractical or impossible. The sample represents a subset of manageable size...

of n observations drawn suitably randomly from the whole sample space, in this example the set of all measurements of yesterday's rainfall in all available rain gauges.

This method of estimation is close to optimal, with the caveat that it underestimates the variance by a factor of (n−1) / n. (For example, when n = 1 the variance of a single observation is obviously zero regardless of the true variance). This gives a bias
Unbiased estimation of standard deviation
The question of unbiased estimation of a standard deviation arises in statistics mainly as question in statistical theory. Except in some important situations, outlined later, the task has little relevance to applications of statistics since its need is avoided by standard procedures, such as the...

which should be corrected for when n is small by multiplying by n / (n−1). 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.

To illustrate the relation between the population variance and the sample variance, suppose that in the (not entirely observed) population of numerical values, the value 1 occurs 1/3 of the time, the value 2 occurs 1/3 of the time, and the value 4 occurs 1/3 of the time. The population mean is (1/3)[1+2+4] = 7/3. The equally likely deviations from the population mean are 1 – 7/3, 2 – 7/3, and 4 – 7/3. The population variance — the expected squared deviation from the mean 7/3 — is (1/3)[(–4/3)2 + (–1/3)2 + (5/3)2] = 14/9. Now suppose for the sake of a simple example that we take a very small sample of n=2 observations, and consider the nine equally likely possibilities for the set of numbers within that sample: (1, 1), (1, 2), (1,4), (2, 1), (2,2), (2, 4), (4,1), (4, 2), and (4, 4). For these nine possible samples, the sample variance of the two numbers is respectively 0, 1/4, 9/4, 1/4, 0, 4/4, 9/4, 4/4, and 0. With our plan to observe two values, we could end up computing any of these sample variances (and indeed if we hypothetically could observe a pair of numbers many times, we would compute each of these sample variances 1/9 of the time). So the expected value, over all possible samples that might be drawn from the population, of the computed sample variance is (1/9)[0 + 1/4 + 9/4 + 1/4 + 0 + 4/4 + 9/4 + 4/4 + 0] = 7/9. This value of 7/9 for the expected value of our sample variance computation is a substantial underestimate of the true population variance, which we computed as 14/9, because our sample size of just two observations was so small. But if we adjust for this downward bias by multiplying our computed sample variance, whichever it may be, by n/(n – 1) = 2/(2 – 1) = 2, then our estimate of the population variance would be any one of 0, 1/2, 9/2, 1/2, 0, 4/2, 9/2, 4/2, and 0. The average of these is indeed the correct population variance of 14/9, so on average over all possible samples we would have the correct estimate of the population variance.

### Related concepts

The variance of a real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

-valued random variable is its second central moment
Central moment
In probability theory and statistics, central moments form one set of values by which the properties of a probability distribution can be usefully characterised...

, and it also happens to be its second cumulant
Cumulant
In probability theory and statistics, the cumulants κn of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. The moments determine the cumulants in the sense that any two probability distributions whose moments are identical will have...

. Just as some distributions do not have a mean, some do not have a variance. The mean exists whenever the variance exists, but the converse is not necessarily true.

## Definition

If a random variable X has the expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

(mean) , then the variance of X is given by:

That is, the variance is the expected value of the squared difference between the variable's realization and the variable's mean. This definition encompasses random variables that are discrete, continuous, or neither (or mixed). It can be expanded as follows:

A mnemonic for the above expression is "mean of square minus square of mean".
The variance of random variable X is typically designated as Var(X), , or simply σ2 ' onMouseout='HidePop("34869")' href="http://www.absoluteastronomy.com/topics/Sigma">sigma
Sigma
Sigma is the eighteenth letter of the Greek alphabet, and carries the 'S' sound. In the system of Greek numerals it has a value of 200. When used at the end of a word, and the word is not all upper case, the final form is used, e.g...

squared").

### Continuous random variable

If the random variable X is continuous with probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

f(x),

where is the expected value, i.e.
and where the integrals are definite integrals taken for x ranging over the range of X.

If a continuous distribution does not have an expected value, as is the case for the Cauchy distribution
Cauchy distribution
The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous probability distribution. As a probability distribution, it is known as the Cauchy distribution, while among physicists, it is known as the Lorentz distribution, Lorentz function, or Breit–Wigner...

, 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
Pareto index
In economics the Pareto index, named after the Italian economist and sociologist Vilfredo Pareto, is a measure of the breadth of income or wealth distribution. It is one of the parameters specifying a Pareto distribution and embodies the Pareto principle...

k satisfies .

### Discrete random variable

If the random variable X is discrete with probability mass function
Probability mass function
In probability theory and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value...

x1 ↦ p1, ..., xn ↦ pn, then

where is the expected value, i.e. .

(When such a discrete weighted variance is specified by weights whose sum is not 1, then one divides by the sum of the weights.) That is, it is the expected value of the square of the deviation
Squared deviations
In probability theory and statistics, the definition of variance is either the expected value , or average value , of squared deviations from the mean. Computations for analysis of variance involve the partitioning of a sum of squared deviations...

of X from its own mean. In plain language, it can be expressed as "The mean of the squares of the deviations of the data points from the average". It is thus the mean squared deviation.

### Exponential distribution

The exponential distribution
Exponential distribution
In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...

with parameter λ is a continuous distribution whose support is the semi-infinite interval [0,∞). Its probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

is given by:

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

So for an exponentially distributed random variable σ2 = μ2.

### Fair dice

A six-sided fair die
Dice
A die is a small throwable object with multiple resting positions, used for generating random numbers...

can be modelled with a discrete random variable with outcomes 1 through 6, each with equal probability . The expected value is (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5. Therefore the variance can be computed to be:

## Properties

Variance is non-negative because the squares are positive or zero. The variance of a constant random variable is zero, and the variance of a variable in a data set
Data set
A data set is a collection of data, usually presented in tabular form. Each column represents a particular variable. Each row corresponds to a given member of the data set in question. Its values for each of the variables, such as height and weight of an object or values of random numbers. Each...

is 0 if and only if all entries have the same value.

Variance is invariant
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...

with respect to changes in a location parameter
Location parameter
In statistics, a location family is a class of probability distributions that is parametrized by a scalar- or vector-valued parameter μ, which determines the "location" or shift of the distribution...

. That is, if a constant is added to all values of the variable, the variance is unchanged. If all values are scaled by a constant, the variance is scaled by the square of that constant. These two properties can be expressed in the following formula:

The variance of a finite sum of uncorrelated random variables is equal to the sum of their variances. This stems from the identity

and from the fact that for uncorrelated variables the covariance
Covariance
In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...

is zero.

In general, for the sum of variables: , we have:
or

Suppose that the observations can be partitioned into equal-sized subgroups according to some second variable. Then the variance of the total group is equal to the mean of the variances of the subgroups plus the variance of the means of the subgroups. This property is known as variance decomposition
Variance decomposition
Variance decomposition or forecast error variance decomposition indicates the amount of information each variable contributes to the other variables in a vector autoregression models...

or the law of total variance
Law of total variance
In probability theory, the law of total variance or variance decomposition formula states that if X and Y are random variables on the same probability space, and the variance of Y is finite, then...

and plays an important role in the analysis of variance
Analysis of variance
In statistics, analysis of variance is a collection of statistical models, and their associated procedures, in which the observed variance in a particular variable is partitioned into components attributable to different sources of variation...

. For example, suppose that a group consists of a subgroup of men and an equally large subgroup of women. Suppose that the men have a mean body length of 180 and that the variance of their lengths is 100. Suppose that the women have a mean length of 160 and that the variance of their lengths is 50. Then the mean of the variances is (100 + 50) / 2 = 75; the variance of the means is the variance of 180, 160 which is 100. Then, for the total group of men and women combined, the variance of the body lengths will be 75 + 100 = 175. Note that this uses N for the denominator instead of N − 1.
In a more general case, if the subgroups have unequal sizes, then they must be weighted proportionally to their size in the computations of the means and variances. The formula is also valid with more than two groups, and even if the grouping variable is continuous.
This formula implies that the variance of the total group cannot be smaller than the mean of the variances of the subgroups. Note, however, that the total variance is not necessarily larger than the variances of the subgroups. In the above example, when the subgroups are analyzed separately, the variance is influenced only by the man-man differences and the woman-woman differences. If the two groups are combined, however, then the men-women differences enter into the variance also.

Many computational formulas for the variance are based on this equality: The variance is equal to the mean of the square minus the square of the mean:

For example, if we consider the numbers 1, 2, 3, 4 then the mean of the squares is (1 × 1 + 2 × 2 + 3 × 3 + 4 × 4) / 4 = 7.5. The regular mean of all four numbers is 2.5, so the square of the mean is 6.25. Therefore the variance is 7.5 − 6.25 = 1.25, which is indeed the same result obtained earlier with the definition formulas. Many pocket calculators use an algorithm that is based on this formula and that allows them to compute the variance while the data are entered, without storing all values in memory. The algorithm is to adjust only three variables when a new data value is entered: The number of data entered so far (n), the sum of the values so far (S), and the sum of the squared values so far (SS). For example, if the data are 1, 2, 3, 4, then after entering the first value, the algorithm would have n = 1, S = 1 and SS = 1. After entering the second value (2), it would have n = 2, S = 3 and SS = 5. When all data are entered, it would have n = 4, S = 10 and SS = 30. Next, the mean is computed as M = S / n, and finally the variance is computed as SS / n − M × M. In this example the outcome would be 30 / 4 − 2.5 × 2.5 = 7.5 − 6.25 = 1.25. If the unbiased sample estimate is to be computed, the outcome will be multiplied by n / (n − 1), which yields 1.667 in this example.

### 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
Uncorrelated
In probability theory and statistics, two real-valued random variables are said to be uncorrelated if their covariance is zero. Uncorrelatedness is by definition pairwise; i.e...

random variables is the sum of their variances:

This statement is called the Bienaymé
Irénée-Jules Bienaymé
Irénée-Jules Bienaymé , was a French statistician. He built on the legacy of Laplace generalizing his least squares method. He contributed to the fields and probability, and statistics and to their application to finance, demography and social sciences...

formula. and was discovered in 1853. It is often made with the stronger condition that the variables are independent
Statistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...

, but uncorrelatedness suffices. So if all the variables have the same variance σ2, then, since division by n is a linear transformation, this formula immediately implies that the variance of their mean is

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
Standard error (statistics)
The standard error is the standard deviation of the sampling distribution of a statistic. The term may also be used to refer to an estimate of that standard deviation, derived from a particular sample used to compute the estimate....

of the sample mean, which is used in the central limit theorem
Central limit theorem
In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...

.

### Product of independent variables

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

### Sum of correlated variables

In general, if the variables are correlated, then the variance of their sum is the sum of their covariance
Covariance
In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...

s:

(Note: This by definition includes the variance of each variable, since Cov(Xi,Xi) = Var(Xi).)

Here Cov is the covariance, which is zero for independent random variables (if it exists). The formula states that the variance of a sum is equal to the sum of all elements in the covariance matrix of the components. This formula is used in the theory of Cronbach's alpha
Cronbach's alpha
Cronbach's \alpha is a coefficient of reliability. It is commonly used as a measure of the internal consistency or reliability of a psychometric test score for a sample of examinees. It was first named alpha by Lee Cronbach in 1951, as he had intended to continue with further coefficients...

in classical test theory
Classical test theory
Classical test theory is a body of related psychometric theory that predict outcomes of psychological testing such as the difficulty of items or the ability of test-takers. Generally speaking, the aim of classical test theory is to understand and improve the reliability of psychological...

.

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

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

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

Therefore, the variance of the mean of a large number of standardized variables is approximately equal to their average correlation. This makes clear that the sample mean of correlated variables does generally not converge to the population mean, even though the Law of large numbers
Law of large numbers
In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times...

states that the sample mean will converge for independent variables.

### Weighted sum of variables

The scaling property and the Bienaymé formula, along with this property from the covariance
Covariance
In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...

page: Cov(aXbY) = ab Cov(XY) jointly imply that

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:

### Decomposition

The general formula for variance decomposition or the law of total variance
Law of total variance
In probability theory, the law of total variance or variance decomposition formula states that if X and Y are random variables on the same probability space, and the variance of Y is finite, then...

is: If X and Y are two random variables and the variance of X exists, then

Here, E(X|Y) is the conditional expectation
Conditional expectation
In probability theory, a conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution....

of X given Y, and Var(X|Y) is the conditional variance
Conditional variance
In probability theory and statistics, a conditional variance is the variance of a conditional probability distribution. Particularly in econometrics, the conditional variance is also known as the scedastic function or skedastic function...

of X given Y. (A more intuitive explanation is that given a particular value of Y, then X follows a distribution with mean E(X|Y) and variance Var(X|Y). The above formula tells how to find Var(X) based on the distributions of these two quantities when Y is allowed to vary.) This formula is often applied in analysis of variance
Analysis of variance
In statistics, analysis of variance is a collection of statistical models, and their associated procedures, in which the observed variance in a particular variable is partitioned into components attributable to different sources of variation...

, where the corresponding formula is

here MS refers to the Mean of the Squares. It is also used in linear regression
Linear regression
In statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple regression...

analysis, where the corresponding formula is

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, SS):

### Computational formula

The computational formula for the variance follows in a straightforward manner from the linearity of expected values and the above definition:

This is often used to calculate the variance in practice, although it suffers from catastrophic cancellation if the two components of the equation are similar in magnitude.

### Characteristic property

The second moment
Moment (mathematics)
In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...

of a random variable attains the minimum value when taken around the first moment (i.e., mean) of the random variable, i.e. . Conversely, if a continuous function satisfies for all random variables X, then it is necessarily of the form , where . This also holds in the multidimensional case.

### Calculation from the CDF

The population variance for a non-negative random variable can be expressed in terms of the cumulative distribution function
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

F using

where H(u) = 1 − F(u) is the right tail function. This expression can be used to calculate the variance in situations where the CDF, but not the density
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

, can be conveniently expressed.

## Approximating the variance of a function

The delta method
Delta method
In statistics, the delta method is a method for deriving an approximate probability distribution for a function of an asymptotically normal statistical estimator from knowledge of the limiting variance of that estimator...

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
Taylor expansions for the moments of functions of random variables
In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite...

. For example, the approximate variance of a function of one variable is given by

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

## Population variance and sample variance

In general, the population variance of a finite population
Statistical population
A statistical population is a set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population. For example, if we were interested in generalizations about crows, then we would describe the set of crows that is of interest...

of size N is given by

where

is the population mean.

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.

A common task is to estimate the variance of a population from a sample
Sampling (statistics)
In statistics and survey methodology, sampling is concerned with the selection of a subset of individuals from within a population to estimate characteristics of the whole population....

. 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. There are several good estimators. Two of them are well known:
and

The first estimator, also known as the second central moment
Central moment
In probability theory and statistics, central moments form one set of values by which the properties of a probability distribution can be usefully characterised...

, is called the biased sample variance. The second estimator is called the unbiased sample variance. Either estimator may be simply referred to as the sample variance when the version can be determined by context. Here, denotes the sample mean:
The two estimators only differ slightly as can be seen, and for larger values of the sample size
Sample size
Sample size determination is the act of choosing the number of observations to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample...

n the difference is negligible. While the first one may be seen as the variance of the sample considered as a population, the second one is the unbiased estimator of the population variance, meaning that its expected value E[s2] is equal to the true variance of the sampled random variable; the use of the term n − 1 is called Bessel's correction
Bessel's correction
In statistics, Bessel's correction, named after Friedrich Bessel, is the use of n − 1 instead of n in the formula for the sample variance and sample standard deviation, where n is the number of observations in a sample: it corrects the bias in the estimation of the population variance,...

. The unbiased sample variance is a U-statistic
U-statistic
In statistical theory, a U-statistic is a class of statistics that is especially important in estimation theory. In elementary statistics, U-statistics arise naturally in producing minimum-variance unbiased estimators...

for the function ƒ(x1x2) = (x1 − x2)2/2, meaning that it is obtained by averaging a 2-sample statistic over 2-element subsets of the population.

In contrast,

### Distribution of the sample variance

Being a function of random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

s, 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
Cochran's theorem
In statistics, Cochran's theorem, devised by William G. Cochran, is a theorem used in to justify results relating to the probability distributions of statistics that are used in the analysis of variance.- Statement :...

shows that s2 follows a scaled chi-squared distribution:

As a direct consequence, it follows that E(s2)  = σ2.

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

where κ is the kurtosis
Kurtosis
In probability theory and statistics, kurtosis is any measure of the "peakedness" of the probability distribution of a real-valued random variable...

of the distribution. If the conditions of the law of large numbers
Law of large numbers
In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times...

hold, s2 is a consistent estimator
Consistent estimator
In statistics, a sequence of estimators for parameter θ0 is said to be consistent if this sequence converges in probability to θ0...

of σ2.

## Generalizations

If is a vector
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

-valued random variable, with values in , and thought of as a column vector, then the natural generalization of variance is , where and is the transpose of , and so is a row vector. This variance is a positive semi-definite square matrix, commonly referred to as the covariance matrix
Covariance matrix
In probability theory and statistics, a 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...

.

If is a complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

-valued random variable, with values in , then its variance is , where is the conjugate transpose
Conjugate transpose
In mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry...

of . This variance is also a positive semi-definite square matrix.

## History

The term variance was first introduced by Ronald Fisher
Ronald Fisher
Sir Ronald Aylmer Fisher FRS was an English statistician, evolutionary biologist, eugenicist and geneticist. Among other things, Fisher is well known for his contributions to statistics by creating Fisher's exact test and Fisher's equation...

in his 1918 paper The Correlation Between Relatives on the Supposition of Mendelian Inheritance
The Correlation Between Relatives on the Supposition of Mendelian Inheritance
The Correlation Between Relatives on the Supposition of Mendelian Inheritance is a scientific paper by R.A. Fisher which was published in the Philosophical Transactions of the Royal Society of Edinburgh in 1918,...

:

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
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

corresponding to the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

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 and , it is found that the distribution, when both causes act together, has a standard deviation . It is therefore desirable in analysing the causes of variability to deal with the square of the standard deviation as the measure of variability. We shall term this quantity the Variance...

## Moment of Inertia

The variance of a probability distribution is analogous to the moment of inertia
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...

in classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

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 moment
Moment (mathematics)
In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...

s
of probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

s. 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 is given by
This difference between moment of inertia in physics and in statistics is clear for points that are gathered along a line. Suppose many points are close to the x axis and distributed along it. The covariance matrix might look like
That is, there is the most variance in the x direction. However, physicists would consider this to have a low moment about the x axis so the moment-of-inertia tensor is