Kurtosis

< Probability and statistics definitions < Kurtosis

What Is Kurtosis?

Kurtosis tells us about the amount of data in the tails of a probability distribution. According to some authors, it also tells us about the “peakedness” of a distribution.

• Positive kurtosis equals heavy tails (i.e., a lot of data in tails). These distributions tend to look flatter than the normal distribution.
• Negative kurtosis equals light tails  (i.e., little data in tails). These distributions often look more peaked than the normal distribution.

The standard normal distribution has a kurtosis of 3, so if your values are close to that then your data is nearly normal.

Kurtosis is the fourth moment in statistics.

Mesokurtic and platykurtic distributions

Mesokurtic distributions are defined as having a kurtosis of zero, although the distribution doesn’t have to be exactly zero — just close to it. The most common mesokurtic distributions are:

• The normal (Gaussian) distribution.
• Any distribution with a Gaussian (normal) shape and zero probability at other places on the real line.
• The binomial distribution is mesokurtic for some values (i.e. for p = 1/2±√(1/12).

Platykurtic distributions are far from ‘normal’, featuring a negative kurtosis and significantly thinner tails than the standard bell curve. In striking contrast, leptokuritc distributions have an excess of positive kurtosis with much fuller tails- clearly visible when compared to a normal distribution.

Excess Kurtosis

Excess kurtosis is usually defined as kurt – 3 (see note below). It is a measure of how the distribution’s tails compare to the normal [3].

• Excess for the normal distribution is 0 (i.e. 3 -3 = 0).
• Negative excess equals lighter tails than a normal distribution.
• Positive excess equals heavier tails than the normal.

If there is significant excess kurtosis — and there usually is — then your data is not normally distributed [4].

The following graph shows a variety of distributions and their excess kurtosis. Note how the tails are fatter or thinner than the normal (black):

Key:

• Red, kurt 3, Laplace (D)ouble exponential distribution;
• Orange, kurt 2, hyperbolic (S)ecant distribution;
• Green, kurt 1.2, (L)ogistic) distribution;
• Black, kurt 0, (N)ormal distribution;
• Cyan, kurt −0.593762…, raised (C)osine distribution;
• Blue, kurt −1, (W)igner semicircle distribution;
• Magenta, kurt −1.2, (U)niform distribution

Note: Which definition/equation you use is a matter of convention in your field, the particular software you’re working with, and sometimes the preference of the author. Therefore, it’s a good idea to check which formula you’re working with. This Cross Validated thread has a rundown of the different equations and which software uses which equation.

Why Does Kurtosis Matter?

Kurtosis is important because it tells us something about how our data behaves in different situations. For example, when calculating risk for investments or insurance policies, having an understanding of this concept can help us better assess potential losses and rewards associated with those investments or policies. Similarly, understanding the topic can provide insight into how errors propagate through systems like neural networks or machine learning algorithms when applied to real-world problems such as recognizing patterns in images or predicting future events based on past data points. This can provide crucial information needed for making informed decisions about investing money and developing these systems for optimal performance and accuracy.

Kurtosis also has real life applications in the world of economics. Fund managers usually focus on risks and returns, so it’s helpful to know if an investment is lepto- or platy-kurtic. According to stock trader and analyst Michael Harris, a leptokurtic return means that risks are coming from outlier events. This would be a stock for investors willing to take extreme risks. For example, real estate (with a kurt of 8.75) and High Yield US bonds (8.63) are high risk investments while Investment grade US bonds (1.06) and Small cap US stocks (1.08) would be considered safer investments.

The peakedness debate

The question of what kurtosis describes exactly — peakedness of a distribution or an indication of how much data is in the tails, is up for debate. Many articles have been published both advocating its interpretation as ’peakedness’ of a distributions and opposing it. See Crack [4] for a discussion of kurtosis that rebuts some recent research in the area and “gets to the root” of what the term means. In the author’s opinion, Kurtosis is both peakedness and fat tails relative to a normal distribution.

Statistician Peter Westfall [5] is firmly on the tail side:

“The incorrect notion that kurtosis somehow measures “peakedness” (flatness, pointiness or modality) of a distribution is remarkably persistent, despite attempts by statisticians to set the record straight. “

~ Peter Westfall.

Fiori & Zenga [6] offer a more neutral historical review.

References

[1] By Audriusa at the Lithuanian language Wikipedia, CC BY-SA 3.0.

[2] Leprokurtic vs. Playtykurtic. Jon Stakweather, Bayes (University of Northern Texas).

[3] Aldrich, E. (2014). Moments. Retrieved July 13, 2023 from: https://people.ucsc.edu/~ealdrich/Teaching/Econ114/LectureNotes/moments.html

[4] CRACK, T. (2022). Foundations for Scientific Investing: Capital Markets Intuition and Critical Thinking Skills. Timothy F. Crack, 11th ed.

[5] WESTFALL, P. H. (2014). Kurtosis as peakedness, 1905–2014. R.I.P. The American Statistician 68, 191–195.

[6] FIORI, A. M. & ZENGA, M. (2009). Karl Pearson and the origin of… International Statistical Review 77, 40–50