Manuals/calci/KURT

KURT()

• Parameters are any values to calculate kurtosis.
  • KURT(), returns the kurtosis of a data set.

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	${\frac {6}{n-4}}$

ZOS

• The syntax is to calculate KURTOSIS in ZOS is $KURT()$
  • Parameters are any values to calculate kurtosis.
• For e.g., KURT([-1..-10,20..25..0.5])

KURTOSIS

Examples

Spreadsheet

	A	B	C	D	E
1	14	11	23	54	38
2	6	7	8	9	10
3	1898	1987	1786	1947
4	26	16	12

1. =KURT(A1:E1) = -0.8704870491886512
2. =KURT(A2:E2) = -1.199999999
3. =KURT(A3:D3) = 0.8709011137293157
4. =KURT(A4:C4) = NAN

Related Videos

Kurtosis

See Also

• SKEW
• STDEV
• STDEVP

References

Kurtosis

• List of Main Z Functions

• Z3 home