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In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, variance is the expectation of the squared deviation of a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
from its
population mean In statistics, a population is a set of similar items or events which is of interest for some question or experiment. A statistical population can be a group of existing objects (e.g. the set of all stars within the Milky Way galaxy) or a hypothe ...
or
sample mean The sample mean (or "empirical mean") and the sample covariance are statistics computed from a Sample (statistics), sample of data on one or more random variables. The sample mean is the average value (or mean, mean value) of a sample (statistic ...
. Variance is a measure of
dispersion Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns *Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variatio ...
, meaning it is a measure of how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include
descriptive statistics A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features from a collection of information, while descriptive statistics (in the mass noun sense) is the process of using and an ...
,
statistical inference Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical ...
,
hypothesis testing A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis. Hypothesis testing allows us to make probabilistic statements about population parameters. ...
,
goodness of fit The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measure ...
, and
Monte Carlo sampling Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determi ...
. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
, the second
central moment In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random ...
of a
distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations * Probability distribution, the probability of a particular value or value range of a vari ...
, and the
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
of the random variable with itself, and it is often represented by \sigma^2, s^2, \operatorname(X), V(X), or \mathbb(X). 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; 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. There are two distinct concepts that are both called "variance". One, as discussed above, is part of a theoretical
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
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 calculate an estimate of the population 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.


Etymology

The term ''variance'' was first introduced by
Ronald Fisher Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who a ...
in his 1918 paper ''
The Correlation Between Relatives on the Supposition of Mendelian Inheritance #REDIRECT The Correlation between Relatives on the Supposition of Mendelian Inheritance #REDIRECT The Correlation between Relatives on the Supposition of Mendelian Inheritance {{R from other capitalisation ...
{{R from other capitalisation ...
'':
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 In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
corresponding to the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
of the
mean square error In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between ...
. When there are two independent causes of variability capable of producing in an otherwise uniform population distributions with standard deviations \sigma_1 and \sigma_2, it is found that the distribution, when both causes act together, has a standard deviation \sqrt. 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...


Definition

The variance of a random variable X is the
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of the squared deviation from the mean of X, \mu = \operatorname /math>: : \operatorname(X) = \operatorname\left X - \mu)^2 \right This definition encompasses random variables that are generated by processes that are
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a ...
,
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
,
neither Neither is an English pronoun, adverb, and determiner signifying the absence of a choice in an either/or situation. Neither may also refer to: * ''Neither'' (opera), the only opera by Morton Feldman * "neither" (short story), a very short s ...
, or mixed. The variance can also be thought of as the
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
of a random variable with itself: : \operatorname(X) = \operatorname(X, X). The variance is also equivalent to the second
cumulant In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will ha ...
of a probability distribution that generates X. The variance is typically designated as \operatorname(X), or sometimes as V(X) or \mathbb(X), or symbolically as \sigma^2_X or simply \sigma^2 (pronounced "
sigma Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used as ...
squared"). The expression for the variance can be expanded as follows: :\begin \operatorname(X) &= \operatorname\left X_-_\operatorname[X^2\right.html"_;"title=".html"_;"title="X_-_\operatorname[X">X_-_\operatorname[X^2\right">.html"_;"title="X_-_\operatorname[X">X_-_\operatorname[X^2\right\\[4pt.html" ;"title="">X_-_\operatorname[X^2\right.html" ;"title=".html" ;"title="X - \operatorname[X">X - \operatorname[X^2\right">.html" ;"title="X - \operatorname[X">X - \operatorname[X^2\right\\[4pt">">X_-_\operatorname[X^2\right.html" ;"title=".html" ;"title="X - \operatorname[X">X - \operatorname[X^2\right">.html" ;"title="X - \operatorname[X">X - \operatorname[X^2\right\\[4pt&= \operatorname\left[X^2 - 2X\operatorname + \operatorname 2\right] \\ pt&= \operatorname\left ^2\right- 2\operatorname operatorname + \operatorname 2 \\ pt&= \operatorname\left[X^2 \right] - \operatorname 2 \end In other words, the variance of is equal to the mean of the square of minus the square of the mean of . This equation should not be used for computations using
floating point arithmetic In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be r ...
, because it suffers from
catastrophic cancellation In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers. For example, if there are two studs, one L_ ...
if the two components of the equation are similar in magnitude. For other numerically stable alternatives, see
Algorithms for calculating variance Algorithms for calculating variance play a major role in computational statistics. A key difficulty in the design of good algorithms for this problem is that formulas for the variance may involve sums of squares, which can lead to numerical instab ...
.


Discrete random variable

If the generator of random variable X is
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a ...
with
probability mass function In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
x_1 \mapsto p_1, x_2 \mapsto p_2, \ldots, x_n \mapsto p_n, then :\operatorname(X) = \sum_^n p_i\cdot(x_i - \mu)^2, where \mu is the expected value. That is, :\mu = \sum_^n p_i x_i . (When such a discrete weighted variance is specified by weights whose sum is not 1, then one divides by the sum of the weights.) The variance of a collection of n equally likely values can be written as : \operatorname(X) = \frac \sum_^n (x_i - \mu)^2 where \mu is the average value. That is, :\mu = \frac\sum_^n x_i . The variance of a set of n equally likely values can be equivalently expressed, without directly referring to the mean, in terms of squared deviations of all pairwise squared distances of points from each other: : \operatorname(X) = \frac \sum_^n \sum_^n \frac(x_i - x_j)^2 = \frac\sum_i \sum_ (x_i-x_j)^2.


Absolutely continuous random variable

If the random variable X has a
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
f(x), and F(x) is the corresponding
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...
, then :\begin \operatorname(X) = \sigma^2 &= \int_ (x-\mu)^2 f(x) \, dx \\ pt &= \int_ x^2f(x)\,dx -2\mu\int_ xf(x)\,dx + \mu^2\int_ f(x)\,dx \\ pt &= \int_ x^2 \,dF(x) - 2 \mu \int_ x \,dF(x) + \mu^2 \int_ \,dF(x) \\ pt &= \int_ x^2 \,dF(x) - 2 \mu \cdot \mu + \mu^2 \cdot 1 \\ pt &= \int_ x^2 \,dF(x) - \mu^2, \end or equivalently, :\operatorname(X) = \int_ x^2 f(x) \,dx - \mu^2 , where \mu is the expected value of X given by :\mu = \int_ x f(x) \, dx = \int_ x \, d F(x). In these formulas, the integrals with respect to dx and dF(x) are
Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
and Lebesgue–Stieltjes integrals, respectively. If the function x^2f(x) is
Riemann-integrable In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of G ...
on every finite interval ,bsubset\R, then :\operatorname(X) = \int^_ x^2 f(x) \, dx - \mu^2, where the integral is an
improper Riemann integral In mathematical analysis, an improper integral is the limit (mathematics), limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative infinity; or in some inst ...
.


Examples


Exponential distribution

The
exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
with parameter is a continuous distribution whose
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
is given by :f(x) = \lambda e^ on the interval . Its mean can be shown to be :\operatorname = \int_0^\infty \lambda xe^ \, dx = \frac. Using
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
and making use of the expected value already calculated, we have: :\begin \operatorname\left ^2\right&= \int_0^\infty \lambda x^2 e^ \, dx \\ &= \left -x^2 e^ \right0^\infty + \int_0^\infty 2xe^ \,dx \\ &= 0 + \frac\operatorname \\ &= \frac. \end Thus, the variance of is given by :\operatorname(X) = \operatorname\left ^2\right- \operatorname 2 = \frac - \left(\frac\right)^2 = \frac.


Fair die

A fair six-sided die can be modeled as a discrete random variable, , with outcomes 1 through 6, each with equal probability 1/6. The expected value of is (1 + 2 + 3 + 4 + 5 + 6)/6 = 7/2. Therefore, the variance of is :\begin \operatorname(X) &= \sum_^6 \frac\left(i - \frac\right)^2 \\ pt &= \frac\left((-5/2)^2 + (-3/2)^2 + (-1/2)^2 + (1/2)^2 + (3/2)^2 + (5/2)^2\right) \\ pt &= \frac \approx 2.92. \end The general formula for the variance of the outcome, , of an die is :\begin \operatorname(X) &= \operatorname\left(X^2\right) - (\operatorname(X))^2 \\ pt &= \frac\sum_^n i^2 - \left(\frac\sum_^n i\right)^2 \\ pt &= \frac - \left(\frac\right)^2 \\ pt &= \frac. \end


Commonly used probability distributions

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


Properties


Basic properties

Variance is non-negative because the squares are positive or zero: :\operatorname(X)\ge 0. The variance of a constant is zero. :\operatorname(a) = 0. Conversely, if the variance of a random variable is 0, then it is
almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
a constant. That is, it always has the same value: :\operatorname(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 The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
, 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 The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto ( ), is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actua ...
whose
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
k satisfies 1 < k \leq 2.


Decomposition

The general formula for variance decomposition or the
law of total variance In probability theory, the law of total variance or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law, states that if X and Y are random variables on the same probability space, and ...
is: If X and Y are two random variables, and the variance of X exists, then :\operatorname \operatorname(\operatorname \mid Y+\operatorname(\operatorname \mid Y. The
conditional expectation In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – give ...
\operatorname E(X\mid Y) of X given Y, and the
conditional variance In probability theory and statistics, a conditional variance is the variance of a random variable given the value(s) of one or more other variables. Particularly in econometrics, the conditional variance is also known as the scedastic function or ...
\operatorname(X\mid Y) may be understood as follows. Given any particular value ''y'' of the random variable ''Y'', there is a conditional expectation \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) = \operatorname E(X\mid Y=y). That same function evaluated at the random variable ''Y'' is the conditional expectation \operatorname E(X\mid Y) = g(Y). In particular, if Y is a discrete random variable assuming possible values y_1, y_2, y_3 \ldots with corresponding probabilities p_1, p_2, p_3 \ldots, , then in the formula for total variance, the first term on the right-hand side becomes :\operatorname(\operatorname \mid Y = \sum_i p_i \sigma^2_i, where \sigma^2_i = \operatorname \mid Y = y_i/math>. Similarly, the second term on the right-hand side becomes :\operatorname(\operatorname \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 \mu_i = \operatorname \mid Y = y_i/math> and \mu = \sum_i p_i \mu_i. Thus the total variance is given by :\operatorname = \sum_i p_i \sigma^2_i + \left( \sum_i p_i \mu_i^2 - \mu^2 \right). A similar formula is applied in
analysis of variance Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statisticia ...
, where the corresponding formula is :\mathit_\text = \mathit_\text + \mathit_\text; here \mathit refers to the Mean of the Squares. In
linear regression In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is call ...
analysis the corresponding formula is :\mathit_\text = \mathit_\text + \mathit_\text. 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, \mathit): :\mathit_\text = \mathit_\text + \mathit_\text, :\mathit_\text = \mathit_\text + \mathit_\text.


Calculation from the CDF

The population variance for a non-negative random variable can be expressed in terms of the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...
''F'' using :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 Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
, can be conveniently expressed.


Characteristic property

The second
moment Moment or Moments may refer to: * Present time Music * The Moments, American R&B vocal group Albums * ''Moment'' (Dark Tranquillity album), 2020 * ''Moment'' (Speed album), 1998 * ''Moments'' (Darude album) * ''Moments'' (Christine Guldbrand ...
of a random variable attains the minimum value when taken around the first moment (i.e., mean) of the random variable, i.e. \mathrm_m\,\mathrm\left(\left(X - m\right)^2\right) = \mathrm(X). Conversely, if a continuous function \varphi satisfies \mathrm_m\,\mathrm(\varphi(X - m)) = \mathrm(X) for all random variables ''X'', then it is necessarily of the form \varphi(x) = a x^2 + b, where . This also holds in the multidimensional case.


Units of measurement

Unlike the expected absolute deviation, the variance of a variable has units that are the square of the units of the variable itself. For example, a variable measured in meters will have a variance measured in meters squared. For this reason, describing data sets via their
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
or
root mean square deviation The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) is a frequently used measure of the differences between values (sample or population values) predicted by a model or an estimator and the values observed. The RMSD represents ...
is often preferred over using the variance. In the dice example the standard deviation is , 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 In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
, is used frequently in theoretical statistics; however the expected absolute deviation tends to be more
robust Robustness is the property of being strong and healthy in constitution. When it is transposed into a system, it refers to the ability of tolerating perturbations that might affect the system’s functional body. In the same line ''robustness'' ca ...
as it is less sensitive to
outlier In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are ...
s arising from measurement anomalies or an unduly
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. In many applications it is the right tail of the distrib ...
.


Propagation


Addition and multiplication by a constant

Variance is
invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
with respect to changes in a
location parameter In geography, location or place are used to denote a region (point, line, or area) on Earth's surface or elsewhere. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ...
. That is, if a constant is added to all values of the variable, the variance is unchanged: :\operatorname(X+a)=\operatorname(X). If all values are scaled by a constant, the variance is scaled by the square of that constant: :\operatorname(aX)=a^2\operatorname(X). The variance of a sum of two random variables is given by :\operatorname(aX+bY)=a^2\operatorname(X)+b^2\operatorname(Y)+2ab\, \operatorname(X,Y), :\operatorname(aX-bY)=a^2\operatorname(X)+b^2\operatorname(Y)-2ab\, \operatorname(X,Y), where \operatorname(X,Y) is the
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
.


Linear combinations

In general, for the sum of N random variables \, the variance becomes: :\operatorname\left(\sum_^N X_i\right)=\sum_^N\operatorname(X_i,X_j)=\sum_^N\operatorname(X_i)+\sum_\operatorname(X_i,X_j), see also general
Bienaymé's identity In probability theory, the general form of Bienaymé's identity states that :\operatorname\left( \sum_^n X_i \right)=\sum_^n \operatorname(X_i)+\sum_^n \operatorname(X_i,X_j)=\sum_^n\operatorname(X_i,X_j). This can be simplified if X_1, \ldots, X_n ...
. These results lead to the variance of a linear combination as: : \begin \operatorname\left( \sum_^N a_iX_i\right) &=\sum_^ a_ia_j\operatorname(X_i,X_j) \\ &=\sum_^N a_i^2\operatorname(X_i)+\sum_a_ia_j\operatorname(X_i,X_j)\\ & =\sum_^N a_i^2\operatorname(X_i)+2\sum_a_ia_j\operatorname(X_i,X_j). \end If the random variables X_1,\dots,X_N are such that :\operatorname(X_i,X_j)=0\ ,\ \forall\ (i\ne j) , then they are said to be
uncorrelated In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname Y- \operatorname \operatorname /math>, is zero. If two variables are uncorrelated, there ...
. It follows immediately from the expression given earlier that if the random variables X_1,\dots,X_N are uncorrelated, then the variance of their sum is equal to the sum of their variances, or, expressed symbolically: :\operatorname\left(\sum_^N X_i\right)=\sum_^N\operatorname(X_i). Since independent random variables are always uncorrelated (see ), the equation above holds in particular when the random variables 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 as a column vector of n random variables X_1, \ldots,X_n, and c as a column vector of n scalars c_1, \ldots,c_n. Therefore, c^\mathsf X is a linear combination of these random variables, where c^\mathsf denotes the
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
of c. Also let \Sigma be the
covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
of X. The variance of c^\mathsfX is then given by: :\operatorname\left(c^\mathsf X\right) = c^\mathsf \Sigma c . This implies that the variance of the mean can be written as (with a column vector of ones) :\operatorname\left(\bar\right) = \operatorname\left(\frac 1'X\right) = \frac 1'\Sigma 1.


Sum of variables


Sum of uncorrelated 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 In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname Y- \operatorname \operatorname /math>, is zero. If two variables are uncorrelated, there ...
random variables is the sum of their variances: :\operatorname\left(\sum_^n X_i\right) = \sum_^n \operatorname(X_i). This statement is called the Bienaymé formula and was discovered in 1853. It is often made with the stronger condition that the variables are
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independ ...
, but being uncorrelated suffices. So if all the variables have the same variance σ2, then, since division by ''n'' is a linear transformation, this formula immediately implies that the variance of their mean is : \operatorname\left(\overline\right) = \operatorname\left(\frac \sum_^n X_i\right) = \frac\sum_^n \operatorname\left(X_i\right) = \fracn\sigma^2 = \frac. 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 The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error ...
of the sample mean, which is used in the
central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselv ...
. To prove the initial statement, it suffices to show that :\operatorname(X + Y) = \operatorname(X) + \operatorname(Y). The general result then follows by induction. Starting with the definition, :\begin \operatorname(X + Y) &= \operatorname\left X + Y)^2\right- (\operatorname + Y^2 \\ pt &= \operatorname\left ^2 + 2XY + Y^2\right- (\operatorname + \operatorname ^2. \end Using the linearity of the
expectation operator In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
and the assumption of independence (or uncorrelatedness) of ''X'' and ''Y'', this further simplifies as follows: :\begin \operatorname(X + Y) &= \operatorname\left ^2\right+ 2\operatorname Y+ \operatorname\left ^2\right- \left(\operatorname 2 + 2\operatorname operatorname + \operatorname 2\right) \\ pt &= \operatorname\left ^2\right+ \operatorname\left ^2\right- \operatorname 2 - \operatorname 2 \\ pt &= \operatorname(X) + \operatorname(Y). \end


Sum of correlated variables


=Sum of correlated variables with fixed sample size

= In general, the variance of the sum of variables is the sum of their
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
s: :\operatorname\left(\sum_^n X_i\right) = \sum_^n \sum_^n \operatorname\left(X_i, X_j\right) = \sum_^n \operatorname\left(X_i\right) + 2\sum_\operatorname\left(X_i, X_j\right). (Note: The second equality comes from the fact that .) Here, \operatorname(\cdot,\cdot) is the
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
, 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 Cronbach's alpha (Cronbach's \alpha), also known as tau-equivalent reliability (\rho_T) or coefficient alpha (coefficient \alpha), is a reliability coefficient that provides a method of measuring internal consistency of tests and measures. Numero ...
in
classical test theory Classical test theory (CTT) is a body of related psychometric theory that predicts outcomes of psychological testing such as the difficulty of items or the ability of test-takers. It is a theory of testing based on the idea that a person's observe ...
. So if the variables have equal variance ''σ''2 and the average
correlation In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
of distinct variables is ''ρ'', then the variance of their mean is :\operatorname\left(\overline\right) = \frac + \frac\rho\sigma^2. This implies that the variance of the mean increases with the average of the correlations. In other words, additional correlated observations are not as effective as additional independent observations at reducing the uncertainty of the mean. Moreover, if the variables have unit variance, for example if they are standardized, then this simplifies to :\operatorname\left(\overline\right) = \frac + \frac\rho. This formula is used in the
Spearman–Brown prediction formula The Spearman–Brown prediction formula, also known as the Spearman–Brown prophecy formula, is a formula relating psychometric reliability to test length and used by psychometricians to predict the reliability of a test after changing the test len ...
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_ \operatorname\left(\overline\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 In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...
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 is a random variable whose variation adds to the variation of , such that, : \operatorname\left(\sum_^X_i\right)=\operatorname\left \rightoperatorname(X)+\operatorname(N)\operatorname\left \right2 which follows from the
law of total variance In probability theory, the law of total variance or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law, states that if X and Y are random variables on the same probability space, and ...
. If has a
Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
, then \operatorname \operatorname(N) with estimator = . So, the estimator of \operatorname\left(\sum_^X_i\right) becomes n^2+n\bar^2 giving \operatorname(\bar)=\sqrt


Weighted sum of variables

The scaling property and the Bienaymé formula, along with the property of the
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
jointly imply that :\operatorname(aX \pm bY) =a^2 \operatorname(X) + b^2 \operatorname(Y) \pm 2ab\, \operatorname(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: :\operatorname\left(\sum_^n a_iX_i\right) = \sum_^na_i^2 \operatorname(X_i) + 2\sum_\sum_a_ia_j\operatorname(X_i,X_j)


Product of variables


Product of independent variables

If two variables X and Y are
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independ ...
, the variance of their product is given by :\operatorname(XY) = operatorname(X)2 \operatorname(Y) + operatorname(Y)2 \operatorname(X) + \operatorname(X)\operatorname(Y). Equivalently, using the basic properties of expectation, it is given by :\operatorname(XY) = \operatorname\left(X^2\right) \operatorname\left(Y^2\right) - operatorname(X)2 operatorname(Y)2.


Product of statistically dependent variables

In general, if two variables are statistically dependent, then the variance of their product is given by: :\begin \operatorname(XY) = &\operatorname\left ^2 Y^2\right- operatorname(XY)2 \\ pt = &\operatorname\left(X^2, Y^2\right) + \operatorname(X^2)\operatorname\left(Y^2\right) - operatorname(XY)2 \\ pt = &\operatorname\left(X^2, Y^2\right) + \left(\operatorname(X) + operatorname(X)2\right)\left(\operatorname(Y) + operatorname(Y)2\right) \\ pt &- operatorname(X, Y) + \operatorname(X)\operatorname(Y)2 \end


Arbitrary functions

The
delta method In statistics, the delta method is a result concerning the approximate probability distribution for a function of an asymptotically normal statistical estimator from knowledge of the limiting variance of that estimator. History The delta method ...
uses second-order
Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
s to approximate the variance of a function of one or more random variables: see Taylor expansions for the moments of functions of random variables. For example, the approximate variance of a function of one variable is given by :\operatorname\left (X)\right\approx \left(f'(\operatorname\left \right\right)^2\operatorname\left \right/math> provided that ''f'' is twice differentiable and that the mean and variance of ''X'' are finite.


Population variance and sample variance

Real-world observations such as the measurements of yesterday's rain throughout the day typically cannot be complete sets of all possible observations that could be made. As such, the variance calculated from the finite set will in general not match the variance that would have been calculated from the full population of possible observations. This means that one
estimates {{otheruses, Estimate (disambiguation) In the Westminster system of government, the ''Estimates'' are an outline of government spending for the following fiscal year presented by the cabinet to parliament. The Estimates are drawn up by bureaucrat ...
the mean and variance from a limited set of observations by using an
estimator In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, the ...
equation. The estimator is a function of the
sample Sample or samples may refer to: Base meaning * Sample (statistics), a subset of a population – complete data set * Sample (signal), a digital discrete sample of a continuous analog signal * Sample (material), a specimen or small quantity of s ...
of ''n''
observations Observation is the active acquisition of information from a primary source. In living beings, observation employs the senses. In science, observation can also involve the perception and recording of data via the use of scientific instruments. The ...
drawn without observational bias from the whole
population Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using a ...
of potential observations. In this example that sample would be the set of actual measurements of yesterday's rainfall from available rain gauges within the geography of interest. The simplest estimators for population mean and population variance are simply the mean and variance of the sample, the sample mean and (uncorrected) sample variance – these are
consistent estimator In statistics, a consistent estimator or asymptotically consistent estimator is an estimator—a rule for computing estimates of a parameter ''θ''0—having the property that as the number of data points used increases indefinitely, the result ...
s (they converge to the correct value as the number of samples increases), but can be improved. Estimating the population variance by taking the sample's variance is close to optimal in general, but can be improved in two ways. Most simply, the sample variance is computed as an average of
squared deviations Squared deviations from the mean (SDM) result from squaring deviations. In probability theory and statistics, the definition of ''variance'' is either the expected value of the SDM (when considering a theoretical distribution) or its average valu ...
about the (sample) mean, by dividing by ''n.'' However, using values other than ''n'' improves the estimator in various ways. Four common values for the denominator are ''n,'' ''n'' − 1, ''n'' + 1, and ''n'' − 1.5: ''n'' is the simplest (population variance of the sample), ''n'' − 1 eliminates bias, ''n'' + 1 minimizes
mean squared error In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between ...
for the normal distribution, and ''n'' − 1.5 mostly eliminates bias in
unbiased estimation of standard deviation In statistics and in particular statistical theory, unbiased estimation of a standard deviation is the calculation from a statistical sample of an estimated value of the standard deviation (a measure of statistical dispersion) of a population of va ...
for the normal distribution. 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 In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In st ...
: it underestimates the variance by a factor of (''n'' − 1) / ''n''; correcting by this factor (dividing by ''n'' − 1 instead of ''n'') is called ''
Bessel's correction In statistics, Bessel's correction 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. This method corrects the bias in t ...
''. The resulting estimator is unbiased, and is called the (corrected) sample variance or unbiased sample variance. For example, when ''n'' = 1 the variance of a single observation about the sample mean (itself) is obviously zero regardless of the population variance. If the mean is determined in some other way than from the same samples used to estimate the variance then this bias does not arise and the variance can safely be estimated as that of the samples about the (independently known) mean. Secondly, the sample variance does not generally minimize
mean squared error In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between ...
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 In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosi ...
of the population (see mean squared error: variance), and introduces bias. This always consists of scaling down the unbiased estimator (dividing by a number larger than ''n'' − 1), and is a simple example of a
shrinkage estimator In statistics, shrinkage is the reduction in the effects of sampling variation. In regression analysis, a fitted relationship appears to perform less well on a new data set than on the data set used for fitting. In particular the value of the coeff ...
: one "shrinks" the unbiased estimator towards zero. For the normal distribution, dividing by ''n'' + 1 (instead of ''n'' − 1 or ''n'') minimizes mean squared error. The resulting estimator is biased, however, and is known as the biased sample variation.


Population variance

In general, the ''population variance'' of a ''finite''
population Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using a ...
of size ''N'' with values ''x''''i'' is given by :\begin \sigma^2 &= \frac \sum_^N \left(x_i - \mu\right)^2 = \frac \sum_^N \left(x_i^2 - 2\mu x_i + \mu^2 \right) \\ pt &= \left(\frac 1N \sum_^N x_i^2\right) - 2\mu \left(\frac \sum_^N x_i\right) + \mu^2 \\ pt &= \left(\frac \sum_^N x_i^2\right) - \mu^2 \end where the population mean is : \mu = \frac 1N \sum_^N x_i. The population variance can also be computed using :\sigma^2 = \frac \sum_\left( x_i-x_j \right)^2 = \frac \sum_^N\left( x_i-x_j \right)^2. This is true because : \begin &\frac \sum_^N\left( x_i - x_j \right)^2 \\ pt = &\frac \sum_^N\left( x_i^2 - 2x_i x_j + x_j^2 \right) \\ pt = &\frac \sum_^N\left(\frac \sum_^N x_i^2\right) - \left(\frac \sum_^N x_i\right)\left(\frac \sum_^N x_j\right) + \frac \sum_^N\left(\frac \sum_^N x_j^2\right) \\ pt = &\frac \left( \sigma^2 + \mu^2 \right) - \mu^2 + \frac \left( \sigma^2 + \mu^2 \right) \\ pt = &\sigma^2 \end 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


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 Sample or samples may refer to: Base meaning * Sample (statistics), a subset of a population – complete data set * Sample (signal), a digital discrete sample of a continuous analog signal * Sample (material), a specimen or small quantity of s ...
of the population. Sample variance can also be applied to the estimation of the variance of a continuous distribution from a sample of that distribution. We take a sample with replacement of ''n'' values ''Y''1, ..., ''Y''''n'' from the population, where ''n'' < ''N'', and estimate the variance on the basis of this sample. Directly taking the variance of the sample data gives the average of the
squared deviations Squared deviations from the mean (SDM) result from squaring deviations. In probability theory and statistics, the definition of ''variance'' is either the expected value of the SDM (when considering a theoretical distribution) or its average valu ...
: :\tilde_Y^2 = \frac \sum_^n \left(Y_i - \overline\right)^2 = \left(\frac 1n \sum_^n Y_i^2\right) - \overline^2 = \frac \sum_\left(Y_i - Y_j\right)^2. Here, \overline denotes the
sample mean The sample mean (or "empirical mean") and the sample covariance are statistics computed from a Sample (statistics), sample of data on one or more random variables. The sample mean is the average value (or mean, mean value) of a sample (statistic ...
: :\overline = \frac \sum_^n Y_i . Since the ''Y''''i'' are selected randomly, both \overline and \tilde_Y^2 are random variables. Their expected values can be evaluated by averaging over the ensemble of all possible samples of size ''n'' from the population. For \tilde_Y^2 this gives: :\begin \operatorname tilde_Y^2 &= \operatorname\left \frac \sum_^n \left(Y_i - \frac \sum_^n Y_j \right)^2 \right\\ pt &= \frac 1n \sum_^n \operatorname\left Y_i^2 - \frac Y_i \sum_^n Y_j + \frac \sum_^n Y_j \sum_^n Y_k \right\\ pt &= \frac 1n \sum_^n \left( \frac \operatorname\left _i^2\right- \frac \sum_ \operatorname\left _i Y_j\right+ \frac \sum_^n \sum_^n \operatorname\left _j Y_k\right+\frac \sum_^n \operatorname\left _j^2\right\right) \\ pt &= \frac 1n \sum_^n \left \frac \left(\sigma^2 + \mu^2\right) - \frac (n - 1)\mu^2 + \frac n(n - 1)\mu^2 + \frac \left(\sigma^2 + \mu^2\right) \right\\ pt &= \frac \sigma^2. \end Hence \tilde_Y^2 gives an estimate of the population variance that is biased by a factor of \frac. For this reason, \tilde_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: :S^2 = \frac \tilde_Y^2 = \frac \left \frac \sum_^n \left(Y_i - \overline\right)^2 \right= \frac \sum_^n \left(Y_i - \overline \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 In statistics, Bessel's correction 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. This method corrects the bias in t ...
, and it is also used in
sample covariance The sample mean (or "empirical mean") and the sample covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or mean value) of a sample of numbers taken from a larger po ...
and the
sample standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
(the square root of variance). The square root is a
concave function In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex. Definition A real-valued function f on an in ...
and thus introduces negative bias (by
Jensen's inequality In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier pr ...
), which depends on the distribution, and thus the corrected sample standard deviation (using Bessel's correction) is biased. The
unbiased estimation of standard deviation In statistics and in particular statistical theory, unbiased estimation of a standard deviation is the calculation from a statistical sample of an estimated value of the standard deviation (a measure of statistical dispersion) of a population of va ...
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 In statistical theory, a U-statistic is a class of statistics that is especially important in estimation theory; the letter "U" stands for unbiased. In elementary statistics, U-statistics arise naturally in producing minimum-variance unbiased est ...
for the function ''ƒ''(''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.


Distribution of the sample variance

Being a function of
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s, 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 In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
,
Cochran's theorem In statistics, Cochran's theorem, devised by William G. Cochran, is a theorem used to justify results relating to the probability distributions of statistics that are used in the analysis of variance. Statement Let ''U''1, ..., ''U'N'' be i.i. ...
shows that ''S''2 follows a scaled
chi-squared distribution In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squa ...
(see also: asymptotic properties): : (n - 1)\frac\sim\chi^2_. As a direct consequence, it follows that : \operatorname\left(S^2\right) = \operatorname\left(\frac \chi^2_\right) = \sigma^2 , and : \operatorname\left ^2\right= \operatorname\left(\frac \chi^2_\right) = \frac\operatorname\left(\chi^2_\right) = \frac. If the ''Y''''i'' are independent and identically distributed, but not necessarily normally distributed, then : \operatorname\left ^2\right= \sigma^2, \quad \operatorname\left ^2\right= \frac \left(\kappa - 1 + \frac \right) = \frac \left(\mu_4 - \frac\sigma^4\right), where ''κ'' is the
kurtosis In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosi ...
of the distribution and ''μ''4 is the fourth
central moment In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random ...
. If the conditions of the
law of large numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...
hold for the squared observations, ''S''2 is a
consistent estimator In statistics, a consistent estimator or asymptotically consistent estimator is an estimator—a rule for computing estimates of a parameter ''θ''0—having the property that as the number of data points used increases indefinitely, the result ...
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.).


Samuelson's inequality

Samuelson's inequality In statistics, Samuelson's inequality, named after the economist Paul Samuelson, also called the Laguerre–Samuelson inequality, after the mathematician Edmond Laguerre, states that every one of any collection ''x''1, ..., ''x'n'' ...
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. Values must lie within the limits \bar y \pm \sigma_Y (n-1)^.


Relations with the harmonic and arithmetic means

It has been shown that for a sample of positive real numbers, : \sigma_y^2 \le 2y_ (A - H), where ''y''max is the maximum of the sample, ''A'' is the arithmetic mean, ''H'' is the
harmonic mean In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired. The harmonic mean can be expressed as the recipro ...
of the sample and \sigma_y^2 is the (biased) variance of the sample. This bound has been improved, and it is known that variance is bounded by : \sigma_y^2 \le \frac, : \sigma_y^2 \ge \frac, where ''y''min is the minimum of the sample.


Tests of equality of variances

The
F-test of equality of variances In statistics, an ''F''-test of equality of variances is a test for the null hypothesis that two normal populations have the same variance. Notionally, any ''F''-test can be regarded as a comparison of two variances, but the specific case being ...
and the
chi square test Chi or CHI may refer to: Greek * Chi (letter), the Greek letter (uppercase Χ, lowercase χ); Chinese * ''Chi'' (length) (尺), a traditional unit of length, about ⅓ meter *Chi (mythology) (螭), a dragon *Chi (surname) (池, pinyin: ''chí' ...
s 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 A capon (from la, cāpō, genitive ''cāpōnis'') is a cockerel (rooster) that has been castrated or neutered, either physically or chemically, to improve the quality of its flesh for food, and, in some countries like Spain, fattened by forc ...
, Mood test, the Klotz test and the Sukhatme test. The Sukhatme test applies to two variances and requires that both
median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...
s 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 Lehmann is a German surname. Geographical distribution As of 2014, 75.3% of all bearers of the surname ''Lehmann'' were residents of Germany, 6.6% of the United States, 6.3% of Switzerland, 3.2% of France, 1.7% of Australia and 1.3% of Poland. In ...
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 A box (plural: boxes) is a container used for the storage or transportation of its contents. Most boxes have flat, parallel, rectangular sides. Boxes can be very small (like a matchbox) or very large (like a shipping box for furniture), and can ...
, the Box–Anderson test and the
Moses test Moses hbo, מֹשֶׁה, Mōše; also known as Moshe or Moshe Rabbeinu (Mishnaic Hebrew: מֹשֶׁה רַבֵּינוּ, ); syr, ܡܘܫܐ, Mūše; ar, موسى, Mūsā; grc, Mωϋσῆς, Mōÿsēs () is considered the most important Prop ...
. Resampling methods, which include the bootstrap and the jackknife, may be used to test the equality of variances.


Moment of inertia

The variance of a probability distribution is analogous to the
moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
in
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
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 or Moments may refer to: * Present time Music * The Moments, American R&B vocal group Albums * ''Moment'' (Dark Tranquillity album), 2020 * ''Moment'' (Speed album), 1998 * ''Moments'' (Darude album) * ''Moments'' (Christine Guldbrand ...
s'' of
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
s. The covariance matrix is related to the
moment of inertia tensor The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accelera ...
for multivariate distributions. The moment of inertia of a cloud of ''n'' points with a covariance matrix of \Sigma is given by :I = n\left(\mathbf_ \operatorname(\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 :\Sigma = \begin10 & 0 & 0 \\ 0 & 0.1 & 0 \\ 0 & 0 & 0.1\end. 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\begin0.2 & 0 & 0 \\ 0 & 10.1 & 0 \\ 0 & 0 & 10.1\end.


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:\text = \sum_(x_-\mu)^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. For inequalities associated with the semivariance, see .


Generalizations


For complex variables

If x is a scalar
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
-valued random variable, with values in \mathbb, then its variance is \operatorname\left x - \mu)(x - \mu)^*\right where x^* is the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of x. This variance is a real scalar.


For vector-valued random variables


As a matrix

If X is a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
-valued random variable, with values in \mathbb^n, and thought of as a column vector, then a natural generalization of variance is \operatorname\left X - \mu)(X - \mu)^\right where \mu = \operatorname(X) and X^ is the transpose of X, and so is a row vector. The result is a positive semi-definite square matrix, commonly referred to as the
variance-covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
(or simply as the ''covariance matrix''). If X is a vector- and complex-valued random variable, with values in \mathbb^n, then the covariance matrix is \operatorname\left X - \mu)(X - \mu)^\dagger\right where X^\dagger is the
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex con ...
of X. This matrix is also positive semi-definite and square.


As a scalar

Another generalization of variance for vector-valued random variables X, which results in a scalar value rather than in a matrix, is the
generalized variance The generalized variance is a scalar value which generalizes variance for multivariate random variables. It was introduced by Samuel S. Wilks. The generalized variance is defined as the determinant of the covariance matrix In probability ...
\det(C), the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the covariance matrix. The generalized variance can be shown to be related to the multidimensional scatter of points around their mean. A different generalization is obtained by considering the
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
between the random variable and its mean. This results in \operatorname\left X - \mu)^(X - \mu)\right= \operatorname(C), which is the
trace Trace may refer to: Arts and entertainment Music * Trace (Son Volt album), ''Trace'' (Son Volt album), 1995 * Trace (Died Pretty album), ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * The Trace (album), ''The ...
of the covariance matrix.


See also

* Bhatia–Davis inequality *
Coefficient of variation In probability theory and statistics, the coefficient of variation (CV), also known as relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is often expressed as ...
*
Homoscedasticity In statistics, a sequence (or a vector) of random variables is homoscedastic () if all its random variables have the same finite variance. This is also known as homogeneity of variance. The complementary notion is called heteroscedasticity. The s ...
*
Least-squares spectral analysis Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the most used spectral method in science, generally ...
for computing a
frequency spectrum The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
with spectral magnitudes in % of variance or in dB * Popoviciu's inequality on variances * Measures for statistical dispersion *
Variance-stabilizing transformation In applied statistics, a variance-stabilizing transformation is a data transformation that is specifically chosen either to simplify considerations in graphical exploratory data analysis or to allow the application of simple regression-based or anal ...


Types of variance

*
Correlation In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
*
Distance variance In statistics and in probability theory, distance correlation or distance covariance is a measure of Independence (probability theory), dependence between two paired random vectors of arbitrary, not necessarily equal, Euclidean vector, dimension. Th ...
*
Explained variance In statistics, explained variation measures the proportion to which a mathematical model accounts for the variation (dispersion) of a given data set. Often, variation is quantified as variance; then, the more specific term explained variance can be ...
*
Pooled variance In statistics, pooled variance (also known as combined variance, composite variance, or overall variance, and written \sigma^2) is a method for estimating variance of several different populations when the mean of each population may be different ...
* Pseudo-variance


References

{{Authority control Moment (mathematics) Statistical deviation and dispersion Articles containing proofs