Kurtosis

Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. That is, data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak. A uniform distribution would be the extreme case. For univariate data Y₁, Y₂,......., Y_N, the formula for kurtosis is:

where is the mean, is the standard deviation, and N is the number of data points. The kurtosis for a standard normal distribution is three. For this reason, some sources use the following definition of kurtosis (often referred to as excess kurtosis):

This definition is used so that the standard normal distribution has a kurtosis of zero. In addition, with the second definition positive kurtosis indicates a peaked distribution and negative kurtosis indicates a flat distribution.

Which definition of kurtosis is used is a matter of convention (this handbook uses the original definition). When using software to compute the sample kurtosis, you need to be aware of which convention is being followed. Many sources use the term kurtosis when they are actually computing excess kurtosis, so it may not always be clear.