Manuals/calci/KURT

KURT(n1,n2,…)

$n1$ , $n2$ are values to calculate kurtosis.

Description

This function gives the value of Kurtosis of a given set.
Kurtosis is the peak or flatness of a frequency distribution graph especially with respect to the concentration of values near the Mean as compared with the Normal Distribution.
A normal distribution has a Kurtosis of 3.
Distributions having higher Kurtosis have flatter tails or more extreme values that phenomenon called 'leptokurtosis' also it is the positive excess Kurtosis , and those with lower Kurtosis have fatter middles or fewer extreme value that phenomenon called 'Platykurtosis' also it is the negative excess Kurtosis.
Example for positive Kurtosis(leptokurtosis) is Exponential distribution, Poisson distribution, Laplace Distribution.
Example for Negative Kurtosis(platykurtosis) is Bernoulli distribution, Uniform distribution.
Kurtosis has no units.
Kurtosis is defined by:
Kurtosis=:

${\frac {n(n+1)}{(n-1)(n-2)(n-3)}}{\frac {\sum (x_{i}-{\bar {x}})^{4}}{s}}-{\frac {3(n-1)^{2}}{(n-2)(n-3)}}$ , where $s$ is the Sample Standard Deviation. ${\bar {x}}$ is the Arithmetic Mean.

In this function arguments may be any type like numbers,names,arrays or references that contain numbers.
We can give logical values and text references also directly.
Suppose the referred argument contains any null cells, logical values like that values are not considered.
This function will return the result as error when

1.Any one of the argument is non-numeric.
2.Suppose the number of data points are less than four or the standard deviation of the sample is zero
3.The referred arguments could not convert
  in to numbers.

When calculating kurtosis, a result of +3.00 indicates the absence of kurtosis (distribution is mesokurtic).
For simplicity in its interpretation, some statisticians adjust this result to zero (i.e. kurtosis minus 3 equals zero), and then any reading other than zero is referred to as excess kurtosis.
Negative numbers indicate a platykurtic distribution and positive numbers indicate a leptokurtic distribution.
The below table is listing the Kurtosis excess for the number of common distributions:

Spreadsheet
Distribution	Kurtosis excess
Bernoulli distribution	${\frac {1}{1-p}}+{\frac {1}{p}}-6$
Beta distribution	${\frac {6[a^{3}+a^{2}(1-2b)+b^{2}(1+b)-2ab(2+b)]}{ab(2+a+b)(3+a+b)}}$
Binomial distribution	${\frac {6p^{2}-6p+1}{np(1-p)}}$
Chi squared distribution	${\frac {12}{r}}$
Exponential distribution	6
Gamma distribution	${\frac {6}{a}}$
Log normal distribution	$e^{4S^{2}}+2e^{3S^{2}}+3e^{2S^{2}}-6$
Negative binomial distribution	${\frac {6-p(6-p)}{r(1-p)}}$
Normal distribution	0
Poisson distribution	${\frac {1}{v}}$
Student's t distribution	${\frac {6}{n-4}}$