Kurtosis

by

Zhiyong Zhang, Wen Qu

• LAST REVIEWED: 24 February 2021
• LAST MODIFIED: 24 February 2021
• DOI: 10.1093/obo/9780199828340-0276

Introduction

In statistics, kurtosis is a measure of the probability distribution of a random variable or a vector of random variables. As mean measures the centrality and variance measures the spreadness of a probability distribution, kurtosis measures the tailedness of the distribution. Kurtosis for a univariate distribution was first introduced by Karl Pearson in 1905. Kurtosis, together with skewness, is widely used to quantify the non-normality—the deviation from a normal distribution—of a distribution. In psychology, kurtosis has often been studied in the field of quantitative psychology to evaluate its effects on psychometric models.

General Overviews

Although Mosteller and Tukey 1977 suggests that kurtosis should be considered as a vague concept like location and scale that can be formalized in many ways, as discussed by Balanda and MacGillivray 1988, the commonly used univariate kurtosis was defined in Pearson 1905, and the widely used multivariate kurtosis was defined in Mardia 1970. Pearson 1905 defines a statistic β₂ as the ratio of the fourth moments over the squared second moments of a distribution (p. 174). The statistic β₂ is the fourth standardized moment of a distribution and is now commonly referred to as kurtosis of a distribution. In the same paper, Pearson refers to β₂ − 3 as the degree of kurtosis (p. 181), and as later kurtosis (p. 195), now known as excess kurtosis. In the literature and software programs, “kurtosis” by itself has also been used to refer to the excess kurtosis. For a normal distribution, its kurtosis is 3 and its excess kurtosis is 0. However, a variable with skewness and excess kurtosis being zero is not necessarily normally distributed. If the kurtosis of a distribution is greater than 3, Pearson calls it platykurtic, and if the kurtosis is smaller than 3, leptokurtic. For a p-variate distribution, Mardia 1970 defines a multivariate kurtosis β_2,p. For a multivariate normal distribution, the Mardia’s kurtosis is p*(p+2). For a sample, different measures of sample kurtosis have been used for β₂ and they can have different magnitudes of bias (Joanes and Gill 1998). Yanagihara 2007 notes that Mardia’s kurtosis became more biased as the kurtosis itself increased and proposed a new way to estimate kurtosis.

• Balanda, K. P. and H. L. MacGillivray. 1988. Kurtosis: A critical review. The American Statistician 42.2: 111–119.

  DOI: 10.2307/2684482

  Critically reviews the concept of kurtosis and showed different formalizations of kurtosis.

• Joanes, D. N., and C. A. Gill. 1998. Comparing measures of sample skewness and kurtosis. Journal of the Royal Statistical Society: Series D (The Statistician) 47.1: 183–189.

  DOI: 10.1111/1467-9884.00122

  Compares the bias and mean-squared error of several widely used measures of sample kurtosis.

• Mardia, K. V. 1970. Measures of multivariate skewness and kurtosis with applications. Biometrika 57.3: 519–530.

  DOI: 10.2307/2334770

  Defines a multivariate skewness and a multivariate kurtosis for measuring multivariate non-normality.

• Mosteller, F., and J. W. Tukey. 1977. Data analysis and regression. Reading, MA: Addison-Wesley.

  A textbook on basic data analysis and regression analysis.

• Pearson, K. 1905. “Das Fehlergesetz und seine Verallgemeinerungen durch Fechner und Pearson. A Rejoinder” [The error law and its generalizations by Fechner and Pearson. A rejoinder]. Biometrika 4.1–2: 169–212.

  DOI: 10.2307/2331536

  The first article that defined kurtosis as a measure of a univariate distribution.

• Yanagihara H. 2007. A family of estimators for multivariate kurtosis in a nonnormal linear regression model. Journal of Multivariate Analysis 98.1: 1–29.

  DOI: 10.1016/j.jmva.2005.05.015

  Proposes a new multivariate kurtosis estimator that is less biased than the Mardia’s kurtosis under non-normal linear regression model.