Giving Kurtosis a Workout

Kurtosis is a statistical measure of a random process that is often used, but perhaps less widely understood. This blog mentions a couple of key issues and misunderstandings about kurtosis in a risk assessment model.

A high kurtosis figure is sometimes described as being associated with a distribution that has “fat tails”. However, by simply overlaying two Normal distributions with the same mean but different standard deviations (e.g. using @RISK to do so), it is visually clear that the distribution with the larger standard deviation has the “fatter tails”.  However, every Normal distribution has a kurtosis of 3 (sometimes “excess kurtosis” is referred to, whereby any base calculation has three subtracted from it; this is the case when using the Excel KURT function to calculate kurtosis, for example), so the kurtosis figure does not pick up the idea that one of the distributions has more weight in the tail.

In fact, kurtosis is a simultaneous measure of the “peakedness” of a distribution and the extent to which it has “fat tails”; the Normal distribution with the larger standard deviation will have fatter tails, but will also be less peaked, and in terms of how kurtosis is calculated, these effects balance out.  Kurtosis is then a bit like going for a workout, where you are required to push weights in a central direction whilst keeping your elbows up!

Another important aspect of kurtosis that is little appreciated is that the kurtosis of a binomial distribution (e.g. modelling an event risk that may or may not happen with a certain probability) increases as the probability of the event decreases. In this sense, distributions with high kurtosis figures are perhaps most easily understood as ones relating to events of low probability but high impact.

Such topics are very easy to explore with @RISK (risk analysis Monte Carlo software add-in to Excel), where the visual ability to view overlays, combined with the use of the RiskTheo functions to obtain in Excel the numerical values of statistics associated with distributions allows for a powerful environment to rapidly address such issues.

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