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# Kurtosis

A student asked me if the kurtosis of the normal distribution was 0 or 3. It seems that I’d said both at different times. Strangely, the answer is that kurtosis of a normal distribution is sometimes 0 and sometimes 3. If you look at Wikipedia you see this old-timer equation for Kurtosis, where μ_{4} is the fourth moment about the mean and σ is the standard deviation. As Wikipedia explains, this is sometimes used as the definition of kurtosis in older works.

## Old Timer Kurtosis Equation

\beta_{2}=\frac{\mu_{4}}{\sigma_{4}}

Wikipedia goes on to say kurtosis is more commonly defined as the fourth cumulant divided by the square of the second cumulant, which is equal to the fourth moment around the mean divided by the square of the variance of the probability distribution *minus 3*, which is also known as **excess kurtosis**. The “minus 3” at the end of this formula is often explained as a correction to make the kurtosis of the normal distribution equal to zero.

## Modern Equation for Kurtosis

\gamma{2}=\frac{k_{4}}{k^{2}_{2}}=\frac{\mu_{4}}{\sigma_{4}}-3

Regardless of whether you use 0 or 3 as the number representing the kurtosis of the normal distribution, kurtosis is a measure of the peakedness or flatness of the distribution. A high kurtosis distribution has a sharper *peak *and longer, fatter *tails*, while a low kurtosis distribution has a more rounded peak and shorter thinner tails. Distributions that are more peaked than the normal are called leptokurtic, while those that are flatter than the normal are called platykurtic. To help you remember the difference, think of the platypus, a flat critter if ever there was one. Platykurtic. The kangaroo, which loves to *leap to *(groan) strikes a much more peaked pose!

As for my answer to the student, it seems obvious (at least to me) that I used 0 back in the old days, and 3 now. Makes sense, right? Either that, or I made a mistake.

## One response to “Kurtosis”

Actually, kurtosis *only* measures tail heaviness (or more specifically, leverage), and nothing about pointiness or flatness of the peak. You can have an infinitely pointy distribution with negative excess kurtosis (beta(.5,1), eg), and you can have a distribution that is perfectly flat over 99.99% of the observable data that has infinite kurtosis (.9999U(0,1) .0001Cauchy, eg.)

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