Kurtosis: Definition, Leptokurtic, Platykurtic

Contents:

1. What is Kurtosis?

• A positive value tells you that you have heavy-tails (i.e. a lot of data in your tails).
• A negative value means that you have light-tails (i.e. little data in your tails).

This heaviness or lightness in the tails usually means that your data looks flatter (or less flat) compared to the normal distribution. The standard normal distribution has a kurtosis of 3, so if your values are close to that then your graph’s tails are nearly normal. These distributions are called mesokurtic.

Kurtosis is the fourth moment in statistics.

The distribution on the left has a very negative kurtosis (no tails); the one on the right has positive kurtosis (heavier tails compared to the normal distribution).

2. Mesokurtic

Mesokurtic distributions are technically defined as having a kurtosis of zero, although the distribution doesn’t have to be exactly zero in order for it to be classified as mesokurtic. The most common mesokurtic distributions are:

• The normal 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).

3. What is Excess kurtosis?

Excess kurtosis is usually defined as kurt – 3 (see Important note about equations). It is a measure of how the distribution’s tails compare to the normal (Aldrich, E, 2014).

• Excess kurt 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.

The following graph shows a variety of distributions. 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.

4. Calculating Kurtosis.

Important Note about formulas: There’s no real consensus for exactly what the correct equation is for calculating kurt. 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 an excellent rundown of the different equations and which software uses which equation.

For Minitab and SPSS, you can find the option in the “Descriptive Statistics” tab.

Kurtosis in Excel 2013

Note: The “KURT” reported by Excel is actually the excess kurtosis. See the note on formulas above.

Watch the video or read the steps below:

Negative kurt(left) and positive kurt (right)

There are two options in Excel for finding kurtosis: the KURT Function and the Data Analysis Toolpak (How to load the Data Analysis Toolpak).

Kurtosis Excel 2013: KURT function

Step 1: Type your data into columns in an Excel worksheet.
Step 2: Click a blank cell.
Step 3: Type “=KURT(A1:A99)” where A1:99 is the cell locations for your data.

Kurtosis Excel 2013: Data Analysis

Step 1: Click the “Data” tab and then click “Data Analysis.”
Step 2: Click “Descriptive Statistics” and then click “OK.”
Step 3: Click the Input Range box and then type the location for your data. For example, if you typed your data into cells A1 to A10, type “A1:A10” into that box
Step 4: Click the radio button for Rows or Columns, depending on how your data is laid out.
Step 5: Click the “Labels in first row” box if your data has column headers.
Step 6: Click the “Descriptive Statistics” check box.
Step 7: Select a location for your output. For example, click the “New Worksheet” radio button.
Step 8: Click “OK.”

Platykurtic

Platykurtic distributions have negative kurtosis. The tails are very thin compared to the normal distribution, or — as in the case of the uniform distribution— non-existent.

Platykurtic (left) and leptokurtic (right).

An example of a very platykurtic distribution is the uniform distribution.

A uniform distribution.

Leptokurtic

A leptokurtic distribution has excess positive kurtosis, where the kurtosis is greater than 3. The tails are fatter than the normal distribution. The following illustration1 shows a leptokurtic distribution along with a normal distribution (dotted line).

The Leptokurtic T-Test

The T distribution is an example of a leptokurtic distribution. It has fatter tails than the normal (you can also look at the first image above to see the fatter tails). Therefore, the critical values in a Student’s t-test will be larger than the critical values from a z-test.

The t-distribution.

Financial Markets

Kurtosis isn’t just a theory confined to mathematical textbooks; it has real life applications, especially in the world of economics. Fund managers usually focus on risks and returns, kurtosis (in particular 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.

References

Aldrich, E. (2014). Moments. Retrieved December 9, 2017 from: https://people.ucsc.edu/~ealdrich/Teaching/Econ114/LectureNotes/moments.html
DeCarlo, L. 1997. On the Meaning and Use of Kurtosis Psychological Methods. Vol 2. No.3, 292-307.
Wuensch, K. Undated. Skewness, Kurtosis, and the Normal Curve.

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