Variance

:''This article is about mathematics. Alternate meaning: variance (land use).'' In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. The variance of a real number-valued random variable is its second moment about the mean, and also its second cumulant (cumulants differ from central moments only at and above degree 4). ==Definition== If μ = E(''X'') is the expected value (mean) of the random variable ''X'', then the variance is :

\operatorname{var}(X)=\operatorname{E}((X-\mu)^2).

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

\operatorname{var}(X)

\sigma_X^2

, or simply

\sigma^2

. Note that the above definition can be used for both discrete random variable and continuous random variable random variables. Many distributions, such as the Cauchy distribution, do not have a variance because the relevant integral diverges. In particular, if a distribution does not have expected value, it does not have variance either. The opposite is not true: there are distributions for which expected value exists, but variance does not. == Properties == If the variance is defined, we can conclude that it is never negative because the squares are positive or zero. The unit of variance is the square of the unit of observation. For example, the variance of a set of heights measured in centimeters will be given in square centimeters. This fact is inconvenient and has motivated many statisticians to instead use the square root of the variance, known as the standard deviation, as a summary of dispersion. It can be proven easily from the definition that the variance does not depend on the mean value

\mu

. That is, if the variable is "displaced" an amount ''b'' by taking ''X''+''b'', the variance of the resulting random variable is left untouched. By contrast, if the variable is multiplied by a scaling factor ''a'', the variance is multiplied by ''a²''. More formally, if ''a'' and ''b'' are real constants and ''X'' is a random variable whose variance is defined, :

\operatorname{var}(aX+b)=a^2\operatorname{var}(X)

Another formula for the variance that follows in a straightforward manner from the above definition is: :

\operatorname{var}(X)=\operatorname{E}(X^2) - (\operatorname{E}(X))^2.

This is often used to calculate the variance in practice. One reason for the use of the variance in preference to other measures of dispersion is that the variance of the sum (or difference) of statistical independence random variables is the sum of their variances. A weaker condition than independence, called uncorrelatedness also suffices. In general, :

\operatorname{var}(aX+bY) =a^2 \operatorname{var}(X) + b^2 \operatorname{var}(Y)
 + 2ab \operatorname{cov}(X, Y).

Here

\operatorname{cov}

is the covariance, which is zero for uncorrelated random variables. == Population variance and sample variance == In statistics, the concept of variance can also be used to describe a set of data. When the set of data is a statistical population, it is called the ''population variance''. If the set is a statistical sample, we call it the ''sample variance''. The population variance of a finite population ''y_i'' where ''i'' = 1, 2, ..., ''N'' is given by :

\sigma^2 = \frac{1}{N} \sum_{i=1}^N
 \left( y_i - \mu \right) ^ 2,

where

\mu

is the population mean. In practice, when dealing with large populations, it is almost never possible to find the exact value of the population variance, due to time, cost, and other resource constraints. A common method of estimating the population variance is sampling (statistics). When estimating the population variance using ''n'' random samples ''x_i'' where ''i'' = 1, 2, …, ''n'', the following formula is an unbiased estimator: :

s^2 = \frac{1}{n-1} \sum_{i=1}^n
 \left( x_i - \overline{x} \right) ^ 2,

where

\overline{x}

is the sample mean. Note that the ''n'' − 1 in the denominator above contrasts with the equation for

\sigma^2

. One common source of confusion is that the term ''sample variance'' may refer to either the unbiased estimator

s^2

of the population variance given above, or to what is strictly speaking the variance of the sample, computed by using ''n'' instead of ''n'' − 1. Intuitively, computing the variance by dividing by ''n'' instead of ''n'' − 1 gives an underestimate of the population variance. This is because we are using the sample mean

\overline{x}

as an estimate of the population mean

\mu

, which we do not know. In practice, for large ''n'', the distinction is often a minor one. === An unbiased estimator === We will demonstrate why

s^2

is an unbiased estimator of the population variance. An estimator

\hat{\theta}

for a parameter

\theta

is unbiased if

\operatorname{E}\{ \hat{\theta}\} = \theta

. Therefore, to prove that

s^2

is unbiased, we will show that

\operatorname{E}\{ s^2\} = \sigma^2

. As an assumption, the population which the

x_i

are drawn from has mean

\mu

and variance

\sigma^2

. :

\operatorname{E} \{ s^2 \}

= \operatorname{E} \left\{ \frac{1}{n-1} \sum_{i=1}^n  \left( x_i - \overline{x} \right) ^ 2 \right\}

= \frac{1}{n-1} \sum_{i=1}^n  \operatorname{E} \left\{ \left( x_i - \overline{x} \right) ^ 2 \right\}

= \frac{1}{n-1} \sum_{i=1}^n  \operatorname{E} \left\{ \left( (x_i - \mu) - (\overline{x} - \mu) \right) ^ 2 \right\}

= \frac{1}{n-1} \sum_{i=1}^n  \operatorname{E} \left\{ (x_i - \mu)^2 \right\}

- 2 \operatorname{E} \left\{ (x_i - \mu) (\overline{x} - \mu) \right\} 

+ \operatorname{E} \left\{ (\overline{x} - \mu)  ^ 2 \right\}

= \frac{1}{n-1} \sum_{i=1}^n  \sigma^2

- 2 \left( \frac{1}{n} \sum_{j=1}^n \operatorname{E} \left\{ (x_i - \mu) (x_j - \mu) \right\} \right)

+ \frac{1}{n^2} \sum_{j=1}^n \sum_{k=1}^n \operatorname{E} \left\{ (x_j - \mu) (x_k - \mu) \right\}

= \frac{1}{n-1} \sum_{i=1}^n  \sigma^2

- \frac{2 \sigma^2}{n}

+ \frac{\sigma^2}{n}

= \frac{1}{n-1} \sum_{i=1}^n \frac{(n-1)\sigma^2}{n}

= \frac{(n-1)\sigma^2}{n-1} = \sigma^2

''See also algorithms for calculating variance.'' == Generalizations == If ''X'' is a vector (spatial)-valued random variable, with values in ''R''^''n'', and thought of as a column vector, then the natural generalization of variance is E[(''X'' − μ)(''X'' − μ)^T], where μ = E(''X'') and ''X''^T is the transpose of ''X'', and so is a row vector. This variance is a nonnegative-definite square matrix, commonly referred to as the covariance matrix. If ''X'' is a complex-valued random variable, then its variance is E[(''X'' − μ)(''X'' − μ)^*], where ''X''^* is the complex conjugate of ''X''. This variance is a nonnegative real number. == History == The term ''variance'' was first introduced by Ronald Fisher in 1918 paper ''The Correlation Between Relatives on the Supposition of Mendelian Inheritance''. ==Moment of inertia== The variance of a probability distribution is equal to the moment of inertia in classical mechanics of a corresponding linear mass distribution, with respect to rotation about its center of mass. It is because of this analogy that such things as the variance are called ''moment (mathematics)'' of probability distributions. ==See also== * expected value * standard deviation * skewness * kurtosis * statistical dispersion * an inequality on location and scale parameters * law of total variance Probability theory Statistics su:Varian

Variance

Wow. This (and Mean, and Standard Deviation) are horrible. Undecipherable. Tons of variables being used without explanation. The grammar is awful. It reads like it was ripped from one of those 100 page math books from the turn of the 19th century--absolutely useless without the lectures. :I was concerned when I read the words above, since I dislike bad grammar and overly complicated verbiage (see my recent editing of counterexample). But then I looked at the page, and it looks as if any intelligent undergraduate would be able to follow it without much effort. No "variables" are unexplained (and I have often been upset to find Wikipedia articles in which mathematical notation is unexplained; I'm a stickler about such things). But do go ahead and improve it if you can. And this article is quite light on the use of "variables"; I don't understand why you say "tons". (I have not looked at mean and standard deviation today.) User:Michael Hardy 18:06, 5 Apr 2004 (UTC) == Variance as analogous to moment of inertia!?!? == I removed this aside someone had put at the bottom, as it was just plain silly. : ... and I've put it back, since it OBVIOUSLY makes sense. The mathematical analogy between the two is clear. Whether it is in some way fruitful is not clear to me at this moment; maybe someone can add something. User:Michael Hardy 22:48, 15 Apr 2005 (UTC) ::I clarified the sentence.--User:Patrick 00:19, 16 Apr 2005 (UTC)

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Variance
Variance
Variance/Algorithm
Variance_(accounting)
Variance_(land_use)
Variance_analysis
Variance_analysis_(accounting)
Variance_swap