Kurtosis is a parameter that describes the shape of a random variable’s probability density function (PDF). Consider the two PDFs in Exhibit 1:
Which would you say has the greater standard deviation? It is impossible to say. The PDF on the right is more peaked at the center, which might lead us to believe that it has a lower standard deviation. It has fatter tails, which might lead us to believe that it has a higher standard deviation. If the effect of the peakedness exactly offsets that of the fat tails, the two PDFs will have the same standard deviation. The different shapes of the two PDFs illustrate kurtosis. The PDF on the right has a greater kurtosis than the one on the left. The kurtosis of a random variable X is denoted
where
A normal random variable has a kurtosis of 3 irrespective of its mean or standard deviation. If a random variable’s kurtosis is greater than 3, it is said to be leptokurtic. If its kurtosis is less than 3, it is said to be platykurtic. Leptokurtosis is associated with PDFs that are simultaneously “peaked” and have “fat tails.” Platykurtosis is associated with PDFs that are simultaneously less peaked and have thinner tails. They are said to have "shoulders." In Exhibit 1, the PDF on the left is platykurtic. The one on the right is leptokurtic.
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or kurt(X). It is defined as
and
are the
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