Kurtosis

Kurtosis (from Greek: κυρτός (kyrtos or kurtos), meaning 'curved, arching') refers to the degree of tailedness in the probability distribution of a real-valued, random variable in probability theory and statistics. Similar to skewness, kurtosis provides insight into specific characteristics of a distribution. Various methods exist for quantifying kurtosis in theoretical distributions, and corresponding techniques allow estimation based on sample data from a population. Different measures of kurtosis can yield varying interpretations. The standard measure of a distribution's kurtosis, originating with Karl Pearson, is a scaled version of the fourth moment of the distribution. This number is related to the tails of the distribution, not its peak; hence, the sometimes-seen characterization of kurtosis as peakedness is incorrect. For this measure, higher kurtosis corresponds to greater extremity of deviations (or outliers), and not the configuration of data near the mean. The widespread misunderstanding of kurtosis may derive from Pearson's own description of kurtosis, as quoted by Westfall, as a departure from normality, a : 'degree of flat-toppedness which is greater or less than that of the normal curve'. Excess kurtosis, typically compared to a value of 0, characterizes the tailedness of a distribution. A univariate normal distribution has an excess kurtosis of 0. Negative excess kurtosis indicates a platykurtic distribution, which does not necessarily have a flat top but produces fewer or less extreme outliers than the normal distribution. For instance, the uniform distribution (i.e., one that is uniformly finite over some bound and zero elsewhere) is platykurtic. On the other hand, positive excess kurtosis signifies a leptokurtic distribution. The Laplace distribution for example, has tails that decay more slowly than a normal one, resulting in more outliers. To simplify comparison with the normal distribution, excess kurtosis is calculated as Pearson's kurtosis minus 3. Some authors and software packages use kurtosis to...