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# Kurtosis

Calculates the kurtosis of a given set of data.
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Contents

C++

## Kurtosis

 template doublekurtosis( int n T* data )
A fundamental task in many statistical analyses is to characterize the location and variability of a data set. A further characterization of the data includes skewness and kurtosis.

Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. That is, data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails.

Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak. A uniform distribution would be the extreme case.

The kurtosis for a standard normal distribution is three. For this reason, excess kurtosis is defined as
$\eta_2&space;=\frac{\sum_{i=1}^n&space;(x_i-\overline{x})^4}&space;{(N-1)\sigma^4}$
where x is the actual population and $\inline&space;\sigma$ is the standard deviation. This way the standard normal distribution has a kurtosis of zero. Positive kurtosis indicates a peaked distribution and negative kurtosis indicates a flat distribution.

## References:

NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm

### Example 1

#include <codecogs/statistics/moments/kurtosis.h>
#include <iostream>

int main()
{
float x[5] = {3.4 , 7.1 , 1.5 , 8.6 , 4.9};
double kurt = Stats::Moments::kurtosis<float>(5, x);
std::cout << "The population kurtosis is: " << kurt << std::endl;
return 0;
}
Output:
The population kurtosis is: -0.928457

### Parameters

 n the size of the population data the actual population data given as an array

### Returns

the kurtosis 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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