Kurtosis Definition, Types, and Importance

By
WILL KENTON

Updated December 03, 2022

Reviewed by CHARLES POTTERS

Fact checked by
TIMOTHY LI

What Is Kurtosis?
Kurtosis is a statistical measure used to describe a characteristic of a dataset. When normally
distributed data is plotted on a graph, it generally takes the form of an upsidedown bell. This
is called the bell curve. The plotted data that are furthest from the mean of the data usually
form the tails on each side of the curve. Kurtosis indicates how much data resides in the
tails.

Distributions with a large kurtosis have more tail data than normally distributed data, which
appears to bring the tails in toward the mean. Distributions with low kurtosis have fewer tail
data, which appears to push the tails of the bell curve away from the mean.

For investors, high kurtosis of the return distribution curve implies that there have been
many price fluctuations in the past (positive or negative) away from the average returns for
the investment. So, an investor might experience extreme price fluctuations with an
investment with high kurtosis. This phenomenon is known as kurtosis risk.

KEY TAKEAWAYS

 Kurtosis describes the "fatness" of the tails found in probability distributions.

 There are three kurtosis categories—mesokurtic (normal), platykurtic (less than
normal), and leptokurtic (more than normal).
 Kurtosis risk is a measurement of how often an investment's price moves
dramatically.
 A curve's kurtosis characteristic tells you how much kurtosis risk the investment
you're evaluating has.

Kurtosis

Understanding Kurtosis
Kurtosis is a measure of the combined weight of a distribution's tails relative to the center of
the distribution curve (the mean). For example, when a set of approximately normal data is
graphed via a histogram, it shows a bell peak, with most of the data residing within three
standard deviations (plus or minus) of the mean. However, when high kurtosis is present,
the tails extend farther than the three standard deviations of the normal bell-curved
distribution.

Kurtosis is sometimes confused with a measure of the peakedness of a distribution.

However, kurtosis is a measure that describes the shape of a distribution's tails in relation
to its overall shape. A distribution can be sharply peaked with low kurtosis, and a
distribution can have a lower peak with high kurtosis. Thus, kurtosis measures "tailedness,"
not "peakedness."
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Formula and Calculation of Kurtosis
Calculating With Spreadsheets

There are several different methods for calculating kurtosis. The simplest way is to use the
Excel or Google Sheets formula. For instance, assume you have the following sample data:
4, 5, 6, 3, 4, 5, 6, 7, 5, and 8 residing in cells A1 through A10 on your spreadsheet. The
spreadsheets use this formula for calculating kurtosis:1

{[n(n+1)/(n-1)(n-2)(n- - -{[3(n-1)2]/[(n-2)(n-3
)]}
However, we'll use the following formula in Google Sheets, which calculates it for us,
assuming the data resides in cells A1 through A10:2

=KURT(A1:A10)
The result is a kurtosis of -0.1518, indicating the curve has lighter tails and is platykurtic.

Calculating By Hand

Calculating kurtosis by hand is a lengthy endeavor, and takes several steps to get to the
results. We'll use new data points and limit their number to simplify the calculation. The new
data points are 27, 13, 17, 57, 113, and 25.

It's important to note that a sample size should be much larger than this; we are using six
numbers to reduce the calculation steps. A good rule of thumb is to use 30% of your data
for populations under 1,000. For larger populations, you can use 10%.3

First, you need to calculate the mean. Add up the numbers and divide by six to get 42.
Next, use the following formulas to calculate two sums, s2 (the square of the deviation from
the mean) and s4 (the square of the deviation from the mean squared). Note—these
numbers do not represent standard deviation; they represent the variance of each data
point.45

 s2 = Σ ( yi - ȳ )2
 s4 = Σ ( yi - ȳ )4

Where:

 yi = the ith variable of the sample

 ȳ = the mean

To get s2, use each variable, subtract the mean, and then square the result. Add all of the
results together:

 (27 - 42)2 = (-15)2 = 225

 (13 - 42)2 = (-29)2 = 841
 (17 - 42)2 = (-25)2 = 625
 (57 - 42)2 = (15)2 = 225
 (113 - 42)2 = (71)2 = 5041
 (25 - 42)2 = (-17)2 = 289
 225 + 841 + 625+ 225 + 5,041 + 289 = 7,246

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To get s4, use each variable, subtract the mean, and raise the result to the fourth power.
Add all of the results together:

 (27 - 42)4 = (-15)4 = 50,625

 (13 - 42)4 = (-29)4 = 707,281
 (17 - 42)4 = (-25)4 = 390,625
 (57 - 42)4 = (15)4 = 50,625
 (113 - 42)4 = (71)4 = 25,411,681
 (25 - 42)4 = (-17)4 = 83,521
 50,625+707,281+390,625+50,625+25,411,681+83,521 = 26,694,358

So, our sums are:

 s2 = 7,246
 s4 = 26,694,358

Now, calculate m2 and m4, the second and fourth moments of the kurtosis formula:

 m2 = s2 / n, or 7,246 / 6 = 1,207.67
 m4 = s4 / n, or 26,694,358 / 6 = 4,449,059.67

We can now calculate kurtosis using a formula found in many statistics textbooks that
assumes a perfectly normal distribution with kurtosis of zero:

k = ( m4 / m22 ) - 3
Where:

 k = kurtosis
 m4 = fourth moment
 m2 = second moment
 4,449,059.67 / 1,458,466.83 = 3.05

So, the kurtosis for the sample variables is 3.05 - 3, or .05.

Types of Kurtosis
There are three categories of kurtosis that a set of data can display—mesokurtic,
leptokurtic, and platykurtic. All measures of kurtosis are compared against a normal
distribution curve.

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Kurtosis.
Investopedia

Mesokurtic (kurtosis = 3.0)

The first category of kurtosis is mesokurtic distribution. This distribution has a kurtosis similar
to that of the normal distribution, meaning the extreme value characteristic of the
distribution is similar to that of a normal distribution. Therefore, a stock with a mesokurtic
distribution generally depicts a moderate level of risk.

Leptokurtic (kurtosis > 3.0)

The second category is leptokurtic distribution. Any distribution that is leptokurtic displays
greater kurtosis than a mesokurtic distribution. This distribution appears as a curve one with
long tails (outliers.) The "skinniness" of a leptokurtic distribution is a consequence of the
outliers, which stretch the horizontal axis of the histogram graph, making the bulk of the
data appear in a narrow ("skinny") vertical range.

A stock with a leptokurtic distribution generally depicts a high level of risk but the possibility
of higher returns because the stock has typically demonstrated large price movements.

While a leptokurtic distribution may be "skinny" in the center, it also features "fat tails".

Platykurtic (kurtosis < 3.0)

The final type of distribution is platykurtic distribution. These types of distributions have short
tails (fewer outliers.). Platykurtic distributions have demonstrated more stability than other
curves because extreme price movements rarely occurred in the past. This translates into a
less-than-moderate level of risk.

Using Kurtosis

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Kurtosis is used in financial analysis to measure an investment's risk of price volatility.
Kurtosis risk differs from more commonly used measurements such as alpha, beta, r-
squared, or the Sharpe ratio. Alpha measures excess return relative to a benchmark index,
and beta measures the volatility a stock has compared to the broader market.

R-squared measures the percent of movement a portfolio or fund has that can be explained
by a benchmark, and the Sharpe ratio compares return to risk. Kurtosis measures the
amount of volatility an investment's price has experienced regularly.

For example, imagine a stock had an average price of $25.85 per share. If the stock's price
swung widely and often enough, the bell curve would have heavy tails (high kurtosis). This
means that there is a lot of variation in the stock price—an investor should anticipate wide
price swings often.

If the stock had light tails (low kurtosis), the investor might anticipate that the stock price
would swing widely only occasionally.

Why Is Kurtosis Important?

Kurtosis explains how often observations in some data sets fall in the tails vs. the center of
a probability distribution. In finance and investing, excess kurtosis is interpreted as a type of
risk known as "tail risk," or the chance of a loss occurring due to a rare event, as predicted
by a probability distribution. If such events are more common than predicted by a
distribution, the tails are said to be "fat."

What Is Excess Kurtosis?

Excess kurtosis compares the kurtosis coefficient with that of a normal distribution. Most
normal distributions are assumed to have a kurtosis of three, so excess kurtosis would be
more or less than three; however, some models assume a normal distribution has a
kurtosis of zero, so excess kurtosis would be more or less than zero.5

Is Kurtosis the Same As Skewness?

No. Kurtosis measures how much of the data in a probability distribution are centered
around the middle (mean) vs. the tails. Skewness instead measures the relative symmetry
of a distribution around the mean.

The Bottom Line

Kurtosis describes how much of a probability distribution falls in the tails instead of its
center. In a normal distribution, the kurtosis is equal to three (or zero in some models).
Positive or negative excess kurtosis will then change the shape of the distribution
accordingly. For investors, kurtosis is important in understanding tail risk, or how frequently
"infrequent" events occur, given one's assumption about the distribution of price returns