Kurtosis

Explained: kurtosis leptokurtosis platykurtosis

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 or kurt(X). It is defined as

where and are the mean and standard deviation of X.

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.

Related Internal Links
chi-squared distribution If you square a normal random variable, the result is a chi-squared random variable. Cornish-Fisher expansion A formula for approximating quintiles of a random variable based only on its first few cumulants. expected value A parameter describing the "center of gravity" of a PDF. joint normal distribution A multivariate distribution with normal marginal distributions. lognormal distribution A random variable is lognormal if its logarithm is normal. normal distribution A normal random variable has a "bell shaped" PDF. quantile A notion from probability that can be used as a parameter. skewness A parameter that describes the lack of symmetry of a PDF. stable Paretian distribution A non-normal stable distribution. standard deviation A parameter describing the dispersion of a PDF. uniform distribution A random variable is uniform if it has constant probability on a finite interval.
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