Kurtosis is a measure of whether the data in a data set are heavy-tailed or light-tailed relative to a normal distribution. That is, data sets with high kurtosis tend to have heavy tails, or outliers. Data sets with low kurtosis tend to have light tails, or lack of outliers.
In computing kurtosis the formula used is μ4/σ4 where μ4 is Pearson’s fourth moment about the mean and sigma is the standard deviation.
The normal distribution (Gaussian) is found to have a kurtosis of three. The formula μ4/σ4 - 3 is the formula for excess kurtosis. We could then classify a distribution from its excess kurtosis:
- Mesokurtic distributions have excess kurtosis of zero.
- Platykurtic distributions (light tails) have negative excess kurtosis.
- Leptokurtic distributions (heavy tails) have positive excess kurtosis.
The mathematical formulas used in google/excel spreadsheet statistical functions that are used in wiki statistical spreadsheets:
- Google Definition of kurtosis
- Gummy's Tutorial on Kurtosis (includes skew), from Gummy stuff (hosted by Financial Wisdom)
- James X. Xiong, CFA, and Thomas M. Idzorek, CFA, The Impact of Skewness and Fat Tails on the Asset Allocation Decision. Ibbotson, (March 23, 2011)