**Kurtosis**

➦ Even if we know about the measures of central tendency, dispersion, and skewness, we cannot fully comprehend a distribution.

➦ For a complete understanding of the shape of the distribution, we should also know another measure called Kurtosis.

➦ It is called the “convexity of a curve” by Prof. Karl Pearson. It measures the flatness of distributions.

➦ Kurtosis is another measure of the shape of a frequency curve. It is a Greek word, which means bulginess.

➦ While skewness signifies the extent of asymmetry, kurtosis measures the degree of peakedness of a frequency distribution.

➦ Kurtosis is a statistical measure of how much a distribution’s tails differ from the tails of a normal distribution.

➦ As such, kurtosis identifies whether a distribution features extreme values in the tails. Along with skewness, kurtosis is also used as a descriptive statistic to describe data distribution.

➦ However, the two concepts should not be confused with each other. Skewness is a measure of distribution symmetry, while kurtosis is a measure of tail heaviness.

➦ Financial risk is measured by kurtosis in finance. When the kurtosis is large, there is a high probability of extremely large and extremely small returns for an investment.

➦ A small kurtosis, on the other hand, indicates a low risk level because the probability of extreme returns is relatively low.

**Excess Kurtosis**

➦ An excess kurtosis metric compares a distribution’s kurtosis with the normal kurtosis. A normal distribution has a kurtosis of 3.

➦ So, it is easy to check whether there is excessive kurtosis by using the following formula:

**Excess Kurtosis = Kurtosis – 3**

➦ Excess kurtosis is a statistical measure that quantifies the degree to which a probability distribution deviates from the normal distribution in terms of its peakedness and tail behavior.

➦ In business statistics, excess kurtosis plays a crucial role in understanding the shape and characteristics of data distributions, particularly in financial analysis, risk management, and market research.

**Types of Kurtosis**

➦ The types of kurtosis are determined by the excess kurtosis of a particular distribution. The excess kurtosis can take positive or negative values, as well as values close to zero.

**1) Mesokurtic**

**[normal in shape]****When the kurtosis = 0**

➦ A Mesokurtic distribution will have a kurtosis of zero or close to zero. Therefore, if the data has a normal distribution, it also has a Mesokurtic distribution.

**2) Leptokurtic**

**[high and thin]****When the kurtosis > 0, there are high frequencies in only a small part of the curve (i.e, the curve is more peaked)**

➦ A positive excess kurtosis is indicated by leptokurtic. It is evident that there are large outliers on either side of the leptokurtic distribution.

➦ Leptokurtic distributions are prone to extreme values on either side of an investment return. A risky investment is one whose returns follow a leptokurtic distribution.

**3) Platykurtic**

**[flat and spread out]****When the kurtosis < 0, the frequencies throughout the curve are closer to be equal (i.e., the curve is more flat and wide)**

➦ There is a negative excess kurtosis with a Platykurtic distribution. According to the kurtosis, the distribution is flat.

➦ A flat tail indicates that an outlier has been found in the distribution. Investment returns are more likely to have a Platykurtic distribution in the financial context, since there is a small chance that the investment will experience extreme returns.

**References **

- Kenton, W. (2023, October 1).
*Kurtosis Definition, types, and importance*. Investopedia. https://www.investopedia.com/terms/k/kurtosis.asp#:~:text=There%20are%20three%20categories%20of,against%20a%20normal%20distribution%20curve. -
*Kurtosis: Definition, leptokurtic, platykurtic – Statistics How to*. (2024, January 19). Statistics How To. https://www.statisticshowto.com/probability-and-statistics/statistics-definitions/kurtosis-leptokurtic-platykurtic/

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It has been shown conclusively that kurtosis measures tailweight only, and nothing about the peak. For example, the beta(.5,1) distribution is infinitely peaked but has very low kurtosis. And the .0001Cauchy + .9999U(0,1) distribution appears perfectly flat over 99.99% of the observable data, but has infinite kurtosis. Have a look at more current references.