Kurtosis

Kurtosis refers to the degree of presence of outliers (extreme values) in the distribution.

Kurtosis is a statistical measure, whether the data is heavy-tailed or light-tailed in a normal distribution.

Types of excess kurtosis

Excess Kurtosis

The excess kurtosis is used in statistics and probability theory to compare the kurtosis coefficient with that normal distribution. Excess kurtosis can be positive (Leptokurtic distribution), negative (Platykurtic distribution), or near to zero (Mesokurtic distribution).

Since normal distributions have a kurtosis of 3, excess kurtosis is calculating by subtracting kurtosis by 3.

Excess kurtosis  =  Kurt – 3

1. Mesokurtic (kurtosis same as the normal distribution).
2. Leptokurtic or heavy-tailed distribution (kurtosis more than normal distribution).
3. Platykurtic or short-tailed distribution (kurtosis less than normal distribution).

1. Mesokurtic (kurtosis = 3)

Mesokurtic is the same as the normal distribution, which means kurtosis is near to 0. In Mesokurtic, distributions are moderate in breadth, and curves are a medium peaked height.

Excess kurtosis in Mesokurtic = 3-3 = 0

2. Leptokurtic (kurtosis > 3)

Leptokurtic is having very long and skinny tails, which means there are more chances of outliers. Positive values of kurtosis indicate that distribution is peaked and possesses thick tails. An extreme positive kurtosis indicates a distribution where more of the numbers are located in the tails of the distribution instead of around the mean.

3. Platykurtic (kurtosis < 3)

Platykurtic having a lower tail and stretched around center tails means most of the data points are present in high proximity with mean. A platykurtic distribution is flatter (less peaked) when compared with the normal distribution.

Summary Table

	Kurtosis	Excess Kurtosis
Mesokurtic	=3	=0
Leptokurtic	>3	>0
Platykurtic	<3	<0

Video

In Hindi: Kurtosis Video

Code (Python)

# if scipy library is not installed
pip install scipy

x = [55, 78, 65, 98, 97, 60, 67, 65, 83, 65]


from scipy.stats import kurtosis

print(kurtosis(x, fisher=False))

OUTPUT
2.0453729382893178

Note: We set the fisher=False in the above code, so it calculate Pearson’s definition of kurtosis where the kurtosis value for normal distribution = 3.

The output of kurtosis is around 2.05 which is less than 3 and the value of excess kurtosis is around -0.95. So we can conclude the distribution is Platykurtic, i.e. a distribution with thicker tails and flatter than the normal distribution.