In probability theory and statistics, the variance of a random variable, probability distribution, or sample is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value (mean). Whereas the mean is a way to describe the location of a distribution, the variance is a way to capture its scale or degree of being spread out. The unit of variance is the square of the unit of the original variable. The positive square root of the variance, called the standard deviation, has the same units as the original variable and can be easier to interpret for this reason.
The variance of a real-valued random variable is its second central moment, and it also happens to be its second cumulant. Just as some distributions do not have a mean, some do not have a variance. The mean exists whenever the variance exists, but not vice versa.
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If random variable X has expected value (mean) μ = E(X), then the variance Var(X) of X is given by:
This definition encompasses random variables that are discrete, continuous, or neither. Of all the points about which squared deviations could have been calculated, the mean produces the minimum value for the averaged sum of squared deviations.
The variance of random variable X is typically designated as Var(X), , or simply σ2. If a distribution does not have an expected value, as is the case for the Cauchy distribution, it does not have a variance either. Many other distributions for which the expected value does exist do not have a finite variance because the relevant integral diverges. An example is a Pareto distribution whose Pareto index k satisfies 1 < k ≤ 2.
If the random variable X is continuous with probability density function p(x),
where
and where the integrals are definite integrals taken for x ranging over the range of X.
If the random variable X is discrete with probability mass function x1 ↦ p1, ..., xn ↦ pn,
(Note: this variance should be divided by the sum of weights in the case of a discrete weighted variance.) That is, it is the expected value of the square of the deviation of X from its own mean. In plain language, it can be expressed as "The average of the square of the distance of each data point from the mean". It is thus the mean squared deviation.
The exponential distribution with parameter λ is a continuous distribution whose support is the semi-infinite interval [0,∞). Its probability density function is given by:
and it has expected value μ = λ−1. Therefore the variance is equal to:
So for an exponentially distributed random variable σ2 = μ2.
A six-sided fair die can be modelled with a discrete random variable with outcomes 1 through 6, each with equal probability 1/6. The expected value is (1+2+3+4+5+6)/6 = 3.5. Therefore the variance can be computed to be:
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 is 0 if and only if all entries have the same value.
Variance is invariant with respect to changes in a location parameter. That is, if a constant is added to all values of the variable, the variance is unchanged. 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 that for uncorrelated variables covariance is zero.
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.[2]
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.
One reason for the use of the variance in preference to other measures of dispersion is that the variance of the sum (or the difference) of uncorrelated random variables is the sum of their variances:
This statement is called the Bienaymé formula.[1] and was discovered in 1853. It is often made with the stronger condition that the variables are independent, but uncorrelatedness suffices. So if 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 with n. This fact is used in the definition of the standard error of the sample mean, which is used in the central limit theorem.
In general, if the variables are correlated, then the variance of their sum is the sum of their covariances:
(Note: This by definition includes the variance of each variable, since Cov(X,X)=Var(X).)
Here Cov is the covariance, which is zero for independent random variables (if it exists). The formula states that the variance of a sum is equal to the sum of all elements in the covariance matrix of the components. This formula is used in the theory of Cronbach's alpha in classical test theory.
So if the variables have equal variance σ2 and the average correlation of distinct variables is ρ, then the variance of their mean is
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 states that the sample mean will converge for independent variables.
Properties 6 and 8, along with this property from the covariance page: Cov(aX, bY) = ab Cov(X, Y) 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 general formula for variance decomposition or the law of total variance is: If X and Y are two random variables and the variance of X exists, then
Here, E(X|Y) is the conditional expectation of X given Y, and Var(X|Y) is the conditional variance 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, where the corresponding formula is
It is also used in linear regression analysis, where the corresponding formula is
This can also be derived from the additivity of variances (property 8), since the total (observed) score is the sum of the predicted score and the error score, where the latter two are uncorrelated.
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.
The second moment of a random variable attains the minimum value when taken around the first moment (i.e., mean) of the random variable, i.e. . Conversely, if a continuous function
satisfies
for all random variables X, then it is necessarily of the form
, where a > 0. This also holds in the multidimensional case.[2]
The delta method uses second-order Taylor expansions to approximate the variance of a function of one or more random variables. 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.
In general, the population variance of a finite population of size N is given by
or if the population is an abstract population with probability distribution Pr:
where is the population mean. This is merely a special case of the general definition of variance introduced above, but restricted to finite populations.
In many practical situations, the true variance of a population is not known a priori and must be computed somehow. When dealing with infinite populations, this is generally impossible.
A common method is estimating the variance of large (finite or infinite) populations from a sample. We take a sample of n values from the population, and estimate the variance on the basis of this sample. There are several good estimators. Two of them are well known:
and
Both are referred to as sample variance. Most advanced electronic calculators can calculate both sn2 and s2 at the press of a button, in which case that button is usually labeled σ2 or σn2 for sn2 and σn-12 for s2.
The two estimators only differ slightly as we see, and for larger values of the sample size n the difference is negligible. The second one is an unbiased estimator of the population variance, meaning that its expected value is equal to the true variance of the sampled random variable. The first one may be seen as the variance of the sample considered as a population.
Common sense would suggest to apply the population formula to the sample as well. The reason that it is biased is that the sample mean is generally somewhat closer to the observations in the sample than the population mean is to these observations. This is so because the sample mean is by definition in the middle of the sample, while the population mean may even lie outside the sample. So the deviations to the sample mean will often be smaller than the deviations to the population mean, and so, if the same formula is applied to both, then this variance estimate will on average be somewhat smaller in the sample than in the population.
One common source of confusion is that the term sample variance may refer to either the unbiased estimator of the population variance, or to the variance
of the sample viewed as a finite population. Both can be used to estimate the true population variance. Apart from theoretical considerations, it doesn't really matter which one is used, as for small sample sizes both are inaccurate and for large values of n they are practically the same. Naively computing the variance by dividing by n instead of n-1 systematically underestimates the population variance. Moreover, in practical applications most people report the standard deviation rather than the sample variance, and the standard deviation that is obtained from the unbiased n-1 version of the sample variance has a slight negative bias (though for normally distributed samples a theoretically interesting but rarely used slight correction exists to eliminate this bias). Nevertheless, in applied statistics it is a convention to use the n-1 version if the variance or the standard deviation is computed from a sample.
In practice, for large , the distinction is often a minor one. In the course of statistical measurements, sample sizes so small as to warrant the use of the unbiased variance virtually never occur. In this context Press et al.[3] commented that if the difference between n and n−1 ever matters to you, then you are probably up to no good anyway - e.g., trying to substantiate a questionable hypothesis with marginal data.
Being a function of random variables, the sample variance is itself a random variable, and it is natural to study its distribution. In the case that are independent observations from a normal distribution, Cochran's theorem shows that
follows a scaled chi-square distribution:
As a direct consequence, it follows that
However, even in the absence of the Normal assumption, it is still possible to prove that is unbiased for
.
If is a vector-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.
If is a complex-valued random variable, with values in
, then its variance is
, where
is the complex conjugate of
. This variance is also a positive semi-definite square matrix.
If one's (real) random variables are defined on an n-dimensional continuum x, the cross-covariance of variables A[x] and B[x] as a function of n-dimensional vector displacement (or lag) Δx may be defined as σAB[Δx] ≡ 〈(A[x+Δx]-μA)(B[x]-μB)〉x. Here the population (as distinct from sample) average over x is denoted by angle brackets 〈 〉x or the Greek letter μ.
This quantity, called a second-moment correlation measure because it's a generalization of the second-moment statistic variance, is sometimes put into dimensionless form by normalizing with the population standard deviations of A and B (e.g. σA≡Sqrt[σAA[0]]). This results in a correlation coefficient ρAB[Δx] ≡ σAB[Δx]/(σAσB) that takes on values between plus and minus one. When A is the same as B, the foregoing expressions yield values for autocovariance, a quantity also known in scattering theory as the pair-correlation (or Patterson) function.
If one defines sample bias coefficient ρ as an average of the autocorrelation-coefficient ρAA[Δx] over all point pairs in a set of M sample points[4], an unbiased estimate for expected error in the mean of A is the square root of: sample variance (taken as a population) times (1+(M-1)ρ)/((M-1)(1-ρ)). When ρ is much greater than 1/(M-1), this reduces to the square root of: sample variance (taken as a population) times ρ/(1-ρ). When |ρ| is much less than 1/(M-1) this yields the more familiar expression for standard error, namely the square root of: sample variance (taken as a population) over (M-1).
The term variance was first introduced by Ronald Fisher in his 1918 paper The Correlation Between Relatives on the Supposition of Mendelian Inheritance[5]:
The great body of available statistics show us that the deviations of a human measurement from its mean follow very closely the Normal Law of Errors, and, therefore, that the variability may be uniformly measured by the standard deviation corresponding to the square root of the mean square error. When there are two independent causes of variability capable of producing in an otherwise uniform population distributions with standard deviations
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...
The variance of a probability distribution is analogous to the moment of inertia in classical mechanics of a corresponding mass distribution along a line, with respect to rotation about its center of mass. It is because of this analogy that such things as the variance are called moments of probability distributions. The covariance matrix is related to the moment of inertia tensor for multivariate distributions. The moment of inertia of a cloud of n points with a covariance matrix of 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 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
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