Asymmetric distribution - Probability How To

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An asymmetric distribution is any distribution that isn’t symmetric. In other words, the left and right portions of the data are not mirror images.

*The symmetric distribution (blue) has two halves that are mirror images. In comparison, the asymmetric distribution (orange) does not have two mirrored halves.*

Asymmetric distribution vs. skewed distribution

The terms asymmetric and skewed are often used interchangeably, but they are not the same thing. Asymmetrical distributions include any distribution that lacks symmetry around the mean. A skewed distribution is a specific type of asymmetric distribution with a long tail that “pulls” the mean away from the center.

*The asymmetric distribution on the left is not skewed because it does not have a “tail”. The distribution on the right is a classic skewed distribution with a long tail on the right.*

A bimodal distribution is an example of a probability distribution that is often asymmetrical but not skewed. It consists of two peaks, or modes, indicating the presence of two distinct groups of data points. Unlike a symmetric distribution, the mean of a bimodal distribution is located between the two peaks.

an example of an asymmetric distribution — *This bimodal distribution is asymmetric, but it lacks a tail and is therefore not called a skewed distribution.*

One major difference between a bimodal distribution and a skewed distribution is that the mean of a bimodal distribution is not influenced by a long tail. Other types of distributions that are usually asymmetric — but not skewed include trimodal distributions (three peaks) and multimodal distributions.

Describing the asymmetric distribution

Skewness helps describe asymmetry in a distribution.

central tendency and skewed distribution graph — *Skewed distributions include negative (left) skewed and positive (right) skewed [1].*

The distribution shown on the above right is an example of right skewness, also called positive skewness. Although most values are concentrated in the left part of the distribution, it is the long tail on the right side with extremely large values that determines the overall skewness. These extreme values can significantly influence the mean, while the median is usually more anchored in the center. This characteristic makes the median a resistant measure of location; it is resistant because the median is the value that lies in the center of the data when arranged in ascending order and doesn’t change when extreme values are present in the data.

Similarly, the distribution shown on the left shows left skewness. The presence of extreme values in the lower part of the distribution pulls the mean toward the left, while the median is less affected [2].

We can also calculate a numerical value for skewness with tools such as the skewness test and excess kurtosis test. A distribution with a skewness of zero is symmetrical; any other number for skewness means that the distribution is asymmetric.

Limitations of asymmetric distributions

While we can use skewness to describe asymmetric distributions, there isn’t a rigorous mathematical foundation to establish what is meant by kurtosis of an asymmetric distribution and what is needed to measure it properly [3]. When modeling stock market volatility, Gabaix et al. [4] consider distributions with large values of moment based skewness and kurtosis and goes as far as stating that “The use of [moment] kurtosis should be banished from use with fat-tailed distributions.”

References

[1] Rodolfo Hermans (Godot) at en.wikipedia., CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons

[2] Day, R. MAT 312: Probability and Statistics for Middle School Teachers

[3] Eberl, A. & Bernhard, K. Centre-free kurtosis orderings for asymmetric distributions. Retrieved July 13, 2023 from: https://publikationen.bibliothek.kit.edu/1000156220

[4] GABAIX, X., GOPIKRISHNAN, P., PLEROU, V. & STANLEY, H. (2006). Institutional investors and stock market volatility. The Quarterly Journal of Economics 121, 461–504.