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# Variance

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Calculates the variance of a given set of data.
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Contents

C++

## Variance

 template doublevariance( int n T* data bool total = false )
Consider a discrete random variable $\inline&space;X$. The variance $\inline&space;&space;X$ of is defined as
$var[X]=E[(X-E[X])^2]$

Note that $\inline&space;(X-E[X])^2$ is a new random variable (it's a function of $\inline&space;X$ ). The variance is also denoted as $\inline&space;\sigma^2$. A useful formula that follows inmediately from the definition is that
$var[X]=E[X^2]-E[X]^2$

In words, the variance of $\inline&space;&space;X$ is the second moment of $\inline&space;X$ minus the first moment squared. The variance of a random variable determines a level of variation of the possible values of $\inline&space;X$ around its mean. However, as this measure is squared, the standard deviation is used instead when one wants to talk about how much a random variable varies around its expected value.

If we cannot analyze a whole population but we have to take a sample, we define its variance (denoted as $\inline&space;s^2$) with the formula:

$s^2=\frac{1}{N-1}\sum_{i=1}^N(x_i-\overline{x})^2$

where $\inline&space;&space;\overline{x}$ is the aritmetic mean . The value for $\inline&space;s^2$ is an estimator for $\inline&space;\sigma$. \[]

If the value of the boolean argument <em> total </em> is true, then the variance is computed using the following formula:
$Var=\frac{1}{N}\sum_{i=1}^N(x_i-\overline{x})^2$

## References:

PlanetMath, http:planetmath.org/encyclopedia/Variance.html

### Example 1

#include <codecogs/statistics/moments/variance.h>
#include <iostream>
int main()
{
int x[5] = {3 , 1 , 5 , 6 , 9};
double var = Statistics::Moments::variance<int>(5, x);
std::cout << "The population variance is: " << var << std::endl;
return 0;
}
Output:
The population variance is: 9.2

### Parameters

 n the size of the population data the actual population data given as an array total Default value = false

### Returns

return value the variance of the given set of data

### Authors

Anca Filibiu (August 2005)
##### Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

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