Kurtosis tells you virtually nothing about the shape of the peak – its only unambiguous interpretation is in terms of tail extremity; i.e., either existing outliers (for the sample kurtosis) or propensity to produce outliers (for the kurtosis of a probability distribution).

Peter H. Westfall

This post is inspired by the nice blog of Eran Raviv, it tries to give a graphical illustration of the kurtosis formula which basically measures the outliers in a distribution. Indeed, the kurtosis measures the thickness or the thinness of a distribution’s tail.

I start with the first standardized moment:

In virtue of the expectation operator properties recalled in this post, we have:

Thus, the kurtosis is the fourth standardized moment:

Before moving to the graphical illustrations, I recall the formula for the sample kurtosis:

The kurtosis for a random variable that follows a Normal distribution is 3 (the ratio between the fourth moment and the square of the second moment, 3/(1^2)=3). We have a mesokurtic distribution. In figure 1, we have a distribution **with the usual properties for a normal distribution:**

The kurtosis for a random variable that follows a Uniform distribution is below 3 (the ratio between the fourth moment and the square of the second moment, 125/(25/3)^2=1.8). We have a platykurtic distribution. In figure 2, we have a distribution with **thinner tails than a normal distribution:**

The kurtosis for a random variable that follows a Laplace distribution is above 3 (the ratio between the fourth moment and the square of the second moment, 24/(2^2)=6). We have a leptokurtic distribution. In figure 3, we have a distribution with **thicker tails than a normal distribution:**

We can combine figures 1 to 3 to compare the kurtosis in figure 4:

We can superpose the three kernel density estimations in order to have a better view of the distributions respective thickness of the distribution’s tail:

The STATA code used to produce the graphs is reproduced below:

```
* Illustrate the Kurtosis with some graphs
*------------------------------------------------------------------------
version 15.1
set more off
cd "C:\Users\EconJamel\Latex\L1-STAT-I-CM\2019-2020\doc-supplementaires"
// Set the directory
capture log close
log using kurtosis.smcl, replace
// Apply the s2color scheme
set scheme s2color
// Generate random variables
set obs 10000
// Normal variable
capture gen normal = rnormal()
sum normal, detail
// Uniform variable
capture gen uniform = runiform(-5,5)
sum uniform, detail
// Laplace variable
capture gen laplace = rlaplace(0,1)
sum laplace, detail
/* histogram normal, frequency title("Normal, K=3") ///
subtitle(mesokurtic) kdensity
*/
histogram normal, title("Normal, K=3") ///
subtitle(mesokurtic) kdensity
capture graph rename normal, replace
capture graph export normal.png, replace
/* histogram uniform, frequency title("Uniform, K=1.8") ///
subtitle(platykurtic) kdensity
*/
histogram uniform, title("Uniform, K=1.8") ///
subtitle(platykurtic) kdensity
capture graph rename uniform, replace
capture graph export uniform.png, replace
/* histogram laplace, frequency title("Uniform, K=6") ///
subtitle(platikurtic) kdensity
*/
histogram laplace, title("Laplace, K=6") ///
subtitle(leptokurtic) kdensity
capture graph rename laplace, replace
capture graph export laplace.png, replace
graph combine normal uniform laplace, ycommon xcommon imargin(zero) ///
title(Kurtosis) ///
subtitle("Ilustration with Normal, Uniform and Laplace distribution")
capture graph rename kurtosis, replace
capture graph export kurtosis.png, replace
twoway (kdensity normal) (kdensity uniform) (kdensity laplace), ///
title(Kurtosis) ///
subtitle("Ilustration with Normal, Uniform and Laplace distr.")
capture graph rename kurtosis_gathered, replace
capture graph export kurtosis_gathered.png, replace
// Save the data
save ///
"C:\Users\EconJamel\Latex\L1-STAT-I-CM\2019-2020\doc-supplementaires\kurtosis.dta", ///
replace
log close
exit
Description
-----------
This file aims at illustrating the kurtosis formula with striking graphs.
```