# Dictionary Definition

variance

### Noun

1 an event that departs from expectations [syn:
discrepancy,
variant]

2 discord that splits a group [syn: division]

3 the second moment around the mean; the expected
value of the square of the deviations of a random variable from its
mean value

4 a difference between conflicting facts or
claims or opinions; "a growing divergence of opinion" [syn:
discrepancy,
disagreement,
divergence]

5 the quality of being subject to variation [syn:
variability,
variableness] [ant:
invariability,
invariability]

6 an activity that varies from a norm or
standard; "any variation in his routine was immediately reported"
[syn: variation]

# User Contributed Dictionary

## English

### Etymology

From variance.### Noun

- The act of varying or the state of being variable
- A difference between what is expected and what happens
- The state of differing or being in conflict
- A discrepancy, especially between two legal documents
- The second central moment in probability

#### Translations

The act of varying or the state of being
variable

A difference between what is expected and what
happens

The state of differing or being in conflict

- Finnish: eripuraisuus, erimielisyys

A discrepancy, especially between two legal
documents

## French

### Etymology

From variantia### Noun

fr-noun m# Extensive Definition

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.

## Definition

If random variable X has expected value (mean) μ = E(X), then the variance Var(X) of X is given by:- \operatorname(X) = \operatorname[ ( X - \mu ) ^ 2 ].\,

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.

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 .

### Discrete case

If the random variable is
discrete with probability
mass function x1↦p1, ..., xn↦pn, this is
equivalent to

- \sum_^n p_i (x_i - \mu)^2\,.

(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 variance of random variable X is typically designated as
Var(X), \scriptstyle\sigma_X^2, or simply σ2.

## Examples

### Exponential distribution

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:- f(x) = \lambda e^,\,

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

- \int_0^\infty f(x) (x - \mu)^2\,dx = \int_0^\infty \lambda e^ (x - \lambda^)^2\,dx = \lambda^.\,

So for an exponentially distributed random
variable σ2 = μ2.

### Fair die

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:- \sum_^6 \tfrac (i - 3.5)^2 = \tfrac\left((-2.5)^2(-1.5)^2(-0.5)^20.5^21.5^22.5^2\right) = \tfrac \cdot 17.50 \approx 2.92\,.

## Properties

Variance is non-negative because the squares are
positive or zero. The variance of a random variable is 0 if and
only if the variable is degenerate, that is, it takes on a constant
value with probability 1, 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:

- \operatorname(aX+b)=a^2\operatorname(X).

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

- Suppose that the observations can be partitioned into 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 or the law of total variance and plays an important role in the analysis of variance. 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.http://www.groupsrv.com/science/post-1990611.htmlThis 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 squares 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 mean 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.

## Properties, formal

8.a. Variance of the 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 random
variables is the sum of their variances:

- \operatorname\Big(\sum_^n X_i\Big) = \sum_^n \operatorname(X_i).

This statement 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

- \operatorname(\overline) = \operatorname\left(\frac\sum_^n X_i\right) = \frac n \sigma^2 = \frac .

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.

8.b. Variance of the sum of correlated
variables

In general, if the variables are correlated, then
the variance of their sum is the sum of their covariances:

- \operatorname\left(\sum_^n X_i\right) = \sum_^n \sum_^n \operatorname(X_i, X_j).

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

- \operatorname(\overline) = \frac + \frac \rho \sigma^2.

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

- \operatorname(\overline) = \frac + \frac \rho.

This formula is used in the
Spearman-Brown prediction formula of classical test theory.
This converges to ρ if n goes to infinity, provided that the
average correlation remains constant or converges too. So for the
variance of the mean of standardized variables with equal
correlations or converging average correlation we have

- \lim_ \operatorname(\overline) = \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 generally not converge to the population
mean, even though the Law
of large numbers states that the sample mean will converge for
independent variables.

8.c. Variance of a weighted sum of
variables

Properties 6 and 8, along with this property from
the covariance page:
Cov(aX, bY) = ab Cov(X, Y) jointly imply
that

- \operatorname(aX+bY) =a^2 \operatorname(X) + b^2 \operatorname(Y) + 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.

9. Decomposition of variance

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

- \operatorname(X) = \operatorname(\operatorname(X|Y))+ \operatorname(\operatorname(X|Y)).

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

- SS_ = SS_ + SS_.

It is also used in linear
regression analysis, where the corresponding formula is

- SS_ = SS_ + SS_.

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.

10. Computational formula for variance

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

- \operatorname(X)= \operatorname(X^2 - 2\,X\,\operatorname(X) + (\operatorname(X))^2),

- =\operatorname(X^2) - 2(\operatorname(X))^2 + (\operatorname(X))^2,

- =\operatorname(X^2) - (\operatorname(X))^2.

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

### Characteristic property

The second moment of a random variable attains the minimum value when taken around the mean of the random variable, i.e. \mathrm_m\,\mathrm((X - m)^2) = \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.## Approximating the variance of a function

The delta method uses second-order Taylor expansions to approximate the variance of a function of one or more random variables. For example, the approximate variance of a function of one variable is given by-
- \operatorname\left[f(X)\right]\approx \left(f'(\operatorname\left[X\right])\right)^2\operatorname\left[X\right]

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
of size N is given by

- \sigma^2 = \frac 1N \sum_^N

or if the population is an abstract
population with probability distribution Pr:

- \sigma^2 = \sum_^N

where \overline 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
(y_1,\dots,y_n) 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:

- s_n^2 = \frac 1n \sum_^n \left(y_i - \overline \right)^ 2 = \left(\frac \sum_^y_i^2\right) - \overline^2,

- s^2 = \frac \sum_^n\left(y_i - \overline \right)^ 2 = \frac\sum_^n y_i^2 - \frac \overline^2,

Both are referred to as sample variance. Most
advanced electronic calculators can calculate both s_n^2 and s^2at
the press of a button, in which case that button is usually labeled
\sigma^2 or \sigma_n^2 for s_n^2 and \sigma_^2 for s^2.

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 E[s^2] 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 s^2 of
the population variance, or to the variance \sigma^2 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 n, 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.
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.

### Distribution of the sample variance

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 y_i are independent observations from a normal distribution, Cochran's theorem shows that s^2 follows a scaled chi-square distribution:(n-1)\frac\sim\chi^2_.

As a direct consequence, it follows that
\operatorname(s^2)=\sigma^2.

However, even in the absence of the Normal
assumption, it is still possible to prove that s^2 is unbiased for
\sigma^2.

## Generalizations

If X is a vector-valued random variable, with values in \mathbb^n, and thought of as a column vector, then the natural generalization of variance is \operatorname((X - \mu)(X - \mu)^\operatorname), where \mu = \operatorname(X) and X^\operatorname is the transpose of X, and so is a row vector. This variance is a positive semi-definite square matrix, commonly referred to as the covariance matrix.If X is a complex-valued
random variable, with values in \mathbb, then its variance is
\operatorname((X - \mu)(X - \mu)^*), where X^* is the complex
conjugate of X. This variance is also a positive semi-definite
square matrix.

## History

The term variance was first introduced by Ronald Fisher in his 1918 paper The Correlation Between Relatives on the Supposition of Mendelian Inheritance: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 \theta_1 and \theta_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...

## Moment of inertia

The variance of a probability distribution is analogous to the moment of inertia in classical mechanics of a corresponding mass distribution along a line, with respect to rotation about its center of mass. It is because of this analogy that such things as the variance are called moments of probability distributions. (The covariance matrix is analogous to the moment of inertia tensor for multivariate distributions.)## See also

## References

## External links

variance in Arabic: تباين

variance in Catalan: Variància

variance in Czech: Rozptyl (statistika)

variance in Danish: Varians

variance in German: Varianz

variance in Estonian: Dispersioon

variance in Modern Greek (1453-):
Διακύμανση

variance in Spanish: Varianza

variance in Esperanto: Varianco

variance in Persian: واریانس

variance in French: Variance (statistiques et
probabilités)

variance in Galician: Varianza

variance in Korean: 분산

variance in Indonesian: Varians

variance in Italian: Varianza

variance in Hebrew: שונות

variance in Lithuanian: Dispersija

variance in Dutch: Variantie

variance in Japanese: 分散

variance in Norwegian: Varians

variance in Polish: Wariancja

variance in Portuguese: Variância

variance in Russian: Дисперсия случайной
величины

variance in Simple English: Variance

variance in Slovak: Rozptyl (štatistika)

variance in Slovenian: Varianca

variance in Sundanese: Varian

variance in Finnish: Varianssi

variance in Swedish: Varians

variance in Vietnamese: Phương sai

variance in Turkish: Varyans

variance in Ukrainian: Дисперсія випадкової
величини

variance in Urdu: تفاوت

variance in Chinese: 方差

# Synonyms, Antonyms and Related Words

agreement to disagree, alienation, antagonism, apostasy, argument, argumentation, at
variance, change,
clashing, conflict, contention, contradiction, contrariety, contrast, controversy,
counter-culture, cross-purposes, debate, departure, deviation, difference, difference of
opinion, difficulty,
disaccord, disaccordance, disagreement, disapprobation, disapproval, disconformity, discongruity, discord, discordance, discordancy, discrepancy, discreteness, disharmony, disparity, dispute, dissatisfaction,
dissension, dissent, dissentience, dissidence, dissimilarity, dissonance, distinction, distinctness, disunion, disunity, divergence, divergency, diversity, dividedness, division, dropping out,
faction, far cry,
fluctuation,
heterogeneity, in
disagreement, in dispute, inaccordance, incompatibility,
incongruity,
inconsistency,
inconsonance,
inequality, inharmoniousness,
inharmony, irreconcilability,
jarring, minority
opinion, misunderstanding,
mixture, negation, nonagreement, nonassent, nonconcurrence, nonconformity, nonconsent, odds, opposition, oppugnancy, otherness, polarization, quarrel, recusance, recusancy, rejection, repudiation, repugnance, rift, schism, secession, separateness, separation, severing, strife, unconformity, underground, unharmoniousness,
unlikeness, unorthodoxy, variation, variegation, variety, withdrawal