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EntityQ175199· pop 66· linked from 1,253 articles

variance

thumb|400px|right|Example of samples from two populations with the same mean but different variances. The red population has mean and variance (), while the blue population has mean and variance ().

AI overview

Variance measures how spread out data points are from their average—a small variance means numbers cluster close together, while a large variance means they're scattered far apart. It matters because two groups can have the same average but tell very different stories; for example, one group's values might be tightly bunched while another's are wildly scattered, which affects how predictable or stable that group is.

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Stack Exchange tag: stackoverflow.com/tags/variance
Commons category: Variance

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~45 min read

Article

50 sections

Contents

Definition
Discrete random variable
Absolutely continuous random variable
Examples
Exponential distribution
Fair die
Commonly used probability distributions
Properties
Basic properties
Issues of finiteness
Decomposition
Calculation from the CDF
Characteristic property
Units of measurement
Propagation
Addition and multiplication by a constant
Linear combinations
Matrix notation for the variance of a linear combination
Sum of variables
Sum of uncorrelated variables
Sum of correlated variables
Sum of correlated variables with fixed sample size
Sum of uncorrelated variables with random sample size
Weighted sum of variables
Product of variables
Product of independent variables
Product of statistically dependent variables
Arbitrary functions
Population variance and sample variance
Population variance
Sample variance
Biased sample variance
Unbiased sample variance
Example
Distribution of the sample variance
Samuelson's inequality
Effect of adding one observation on variance
Relations with the harmonic and arithmetic means
Tests of equality of variances
Moment of inertia
Semivariance
Etymology
Generalizations
For complex variables
For vector-valued random variables
As a matrix
As a scalar
See also
Types of variance
References

thumb|400px|right|Example of samples from two populations with the same mean but different variances. The red population has mean and variance (), while the blue population has mean and variance ().

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers are spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by , , {{tmath|\operatorname{Var}(X)}}, , or {{tmath|\mathbb{V}(X)}}.