Dist. Skewness & Kurtosis

Source: Notion | Last edited: 2023-06-14 | ID: a8b5ed0d-580...

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Table of contents

Use graphics to help me illustrate the different ranges of values in skewness an…

❓ Prompt

Use graphics to help me illustrate the different ranges of values in skewness and kurtosis

💬 Answer

Here are histograms of four different types of distributions: Normal, Exponential, Uniform, and Cauchy. Each histogram is labeled with the skewness and kurtosis of the distribution.

https://www.wolframcloud.com/obj/3e9c7922-f021-46da-b2ba-feb06a46e5d4

Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. Positive skewness indicates a distribution with an asymmetric tail extending towards more positive values. Negative skewness indicates a distribution with an asymmetric tail extending towards more negative values.

Kurtosis is a measure of the “tailedness” of the probability distribution of a real-valued random variable. Higher kurtosis indicates a distribution with more extreme values (either extremely high or extremely low, or both) compared to a normal distribution. Lower kurtosis indicates a distribution with less extreme values.

As you can see from the histograms:

• The Normal Distribution has skewness close to 0 and kurtosis close to 3, which is typical for a normal distribution.
• The Exponential Distribution has positive skewness, indicating a longer tail on the right, and higher kurtosis, indicating more extreme values.
• The Uniform Distribution has skewness close to 0 and lower kurtosis, indicating less extreme values.
• The Cauchy Distribution has undefined skewness and kurtosis due to its heavy tails.

Chart 2