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# Management Notes

Reference Notes for Management

# Kurtosis – Types of Kurtosis | Business Statistics

## 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

• 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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### 1 thought on “Kurtosis – Types of Kurtosis | Business Statistics”

1. 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.