# SQL Server population kurtosis function

KURTOSIS_P

Updated: 24 May 2013

Use the aggregate function KURTOSIS_P to calculate the kurtosis for an entire population. The KURTOSIS_P function is an implementation of the kurtosis “excess” as formulated by Abramowitz and Stegun in Handboook of Mathematical Functions, 26.1.16.

Syntax
Arguments
@Known_x
the values to be used in the calculation. @Known_x must be of a type float or of a type that implicitly converts to float.
Return Types
float
Remarks
·         If you want to measure the kurtosis for a sample, then use the KURTOSIS_S function.
·         To calculate the population skewness use the SKEWNESS_P function.
·         To calculate the sample skewness use the SKEWNESS_S function.
Examples
SELECT wct.KURTOSIS_P(x) as KURTOSIS_P
FROM (
SELECT 30000.0000216303 UNION ALL
SELECT 30000.0000565854 UNION ALL
SELECT 30000.000038137 UNION ALL
SELECT 30000.0000495983 UNION ALL
SELECT 30000.0000185861 UNION ALL
SELECT 30000.0000863479 UNION ALL
SELECT 30000.0000776366 UNION ALL
SELECT 30000.0000637985 UNION ALL
SELECT 30000.0000939786 UNION ALL
SELECT 30000.000031191 UNION ALL
SELECT 30000.0000550457 UNION ALL
SELECT 30000.0000207558 UNION ALL
SELECT 30000.0000805531 UNION ALL
SELECT 30000.0000241287
)n(x)
This produces the following result

KURTOSIS_P
----------------------
-1.34100844093842

In this example, we generate 100 random numbers form the standard normal distribution using the
wct.SeriesFloat function and calculate the population kurtosis.
SELECT wct.KURTOSIS_P(k.SeriesValue) as KURTOSIS_P
FROM wctMath.wct.SeriesFloat(0,1,NULL,100,'N')k
This produces the following result (your results will be different).
KURTOSIS_P
----------------------
-0.0599061758404065