quality:corruption, structural:summarization, stylistic:style_transfer, structural:expansion
In probability theory and statistics, kurtosis refers to the degree of “tailedness” in the probability distribution of a real-valued random variable. 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. It’s important to note that 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, and not the configuration of data near the mean. 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 doesn’t necessarily have a flat top but produces fewer or less extreme outliers than the normal distribution. For instance, the uniform distribution 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 Gaussian, 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 refer specifically to excess kurtosis, but this article distinguishes between the two for clarity. Alternative measures of kurtosis are: the L-kurtosis, which is a scaled version of the fourth L-moment; measures based on four population or sample quantiles. These are analogous to the alternative measures of skewness that are not based on ordinary moments.