What is kurtosis?

Kurtosis is a statistical measure that shows the differences between the tails of a distribution and those of a Normal Distribution. It enables us to understand the relative concentration of data points in the tails of a distribution compared to its center. Kurtosis is a dimensionless statistic, meaning it does not depend on the units of measurement, making it a versatile tool for analyzing various datasets.

Where,

$n$ is the number of data points or samples.
$x_i$ is a single data point or sample.
$\bar{x}$ is the mean of the data points.
$s$ is the standard deviation of the data points.

Interpretation

The interpretation of kurtosis values lies in understanding the relationship between the kurtosis coefficient and the characteristics of the distribution. Positive kurtosis indicates a distribution with heavier tails and a sharper peak compared to the normal distribution (mesokurtic). Conversely, negative kurtosis suggests lighter tails and a flatter peak.

Types of kurtosis

Kurtosis can be categorized into three primary types:

Mesokurtic: In a mesokurtic distribution, the tails are moderate in thickness, and the peak is moderate in height and width. The data exhibits relative balance, and the occurrence of extreme values or outliers aligns with what one would expect in a normal distribution.
Leptokurtic: It indicates a distribution with heavier tails and a sharper peak compared to the normal distribution. In a leptokurtic distribution, the tails extend farther from the center, suggesting a higher probability of extreme values or outliers. The peak of the distribution is relatively narrow and pronounced, indicating a greater concentration of data around the mean.
Platykurtic: It signifies a distribution with lighter tails and a flatter peak compared to the normal distribution. In a platykurtic distribution, the tails are shorter, indicating a lower probability of extreme values or outliers. The distribution is relatively spread out, with data points dispersed more evenly across the range.

Understanding kurtosis types

If the value of kurtosis is 3 on a given set of data points, they form a mesokurtic distribution.
If the value of kurtosis is greater than 3 on a given set of data points, they form a leptokurtic distribution.
If the value of kurtosis is less than 3 on a given set of data points, they form a platykurtic distribution.

Applications

As a measure of data distribution, kurtosis finds practical applications in various fields. Some of the key areas where kurtosis plays a significant role are as follows:

Finance and risk management: Kurtosis helps quantify the shape of return distributions, providing insights into the potential occurrence of extreme events or outliers. Financial analysts can make informed decisions by analyzing kurtosis and assessing the risk associated with different investment options.
Image processing and anomaly detection: By calculating the kurtosis of pixel intensities in a digital image, it becomes possible to identify and flag outliers or abnormalities that may indicate image defects, noise, or tampering. This application is particularly valuable in surveillance, forensics, and medical imaging fields.
Social sciences and psychology: By analyzing the kurtosis of survey data, psychologists can gain insights into the distribution of responses and identify unusual patterns. This information helps researchers better understand human behavior, attitudes, and perceptions.

Conclusion

Understanding kurtosis allows data analysts, researchers, and professionals from various domains to make informed decisions based on the characteristics of their datasets.