Characterizing Distribution Shape in Terms of Skewness and Kurtosis

1 Characterizing Distribution Shape in Terms of Skewness and Kurtosis 

Indices of distribution central tendency and spread are not reviewed here (see Vose 2000, Section 3.2.1). The concept of skewness of a distribution relates to deviations from symmetry of the pdf. The normal distribution has a skewness of zero (the distribution is symmetric, with the familiar bell-shaped pdf). For a distribution with positive skewness the right tail of the distribution is more extended than the left tail a distribution with the left tail more extended has negative skewness. In many cases, it seems that skewness is associated with a constraint on the permissible values of a variable (Vose 2000, Section 6.7). The idea is that the distribution tail can be more extended in the direction opposite to a bound than in the direction of the bound. 

Some literature has defined kurtosis in terms of pdfs that are relatively flat versus relatively peaked at the mode. A tendency in more recent literature is to emphasize the idea of tail weight. Leptokurtic distributions have relatively heavier pdf tails, while platykurtic distributions have relatively lighter tails (Balanda and MacGillivray 1998). 

Leptokurtic distributions are more outlier-prone. When fitting distributions to data, it may sometimes be difficult to decide whether one should assume a leptokurtic distribution (say, a Student t distribution with relatively few degrees of freedom) or assume the presence of a few outliers. 

Environmental concentrations and other environmental variables tend to have positive skewness. Therefore, environmental statistics texts often focus on positive skew distributions such as the log-normal, gamma, and Weibull. Discussions of distributions with nonnormal kurtosis are somewhat more scarce. 

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