kurtosis-2 - Psychological Methods 1997 Vol.2 No 3,292-307...

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Psychological Methods Copyright 1997 by the American PsychologicalAssociation,Inc. 1997, Vol. 2, No. 3,292-307 1082-989X/97/$3.00 On the Meaning and Use of Kurtosis Lawrence T. DeCarlo Fordham University For symmetric unimodal distributions, positive kurtosis indicates heavy tails and peakedness relative to the normal distribution, whereas negative kurtosis indicates light tails and flatness. Many textbooks, however, describe or illustrate kurtosis incompletely or incorrectly. In this article, kurtosis is illustrated with well-known distributions, and aspects of its interpretation and misinterpretation are discussed. The role of kurtosis in testing univariate and multivariate normality; as a measure of departures from normality; in issues of robustness, outliers, and bimodality; in generalized tests and estimators, as well as limitations of and alternatives to the kurtosis measure [32, are discussed. It is typically noted in introductory statistics courses that distributions can be characterized in terms of central tendency, variability, and shape. With respect to shape, virtually every textbook defines and illustrates skewness. On the other hand, another as- pect of shape, which is kurtosis, is either not discussed or, worse yet, is often described or illustrated incor- rectly. Kurtosis is also frequently not reported in re- search articles, in spite of the fact that virtually every statistical package provides a measure of kurtosis. This occurs most likely because kurtosis is not well understood and because the role of kurtosis in various aspects of statistical analysis is not widely recognized. The purpose of this article is to clarify the meaning of kurtosis and to show why and how it is useful. On the Meaning of Kurtosis Kurtosis can be formally defined as the standard- ized fourth population moment about the mean, E (X - IX)4 IX4 132 = (E (X- IX)2)2 0.4' where E is the expectation operator, IX is the mean, 1,1,4 is the fourth moment about the mean, and 0. is the I thank Barry H. Cohen for motivating me to write the article and Richard B. Darlington and Donald T. Searls for helpful comments. Correspondence concerning this article should be ad- dressed to Lawrence T. DeCarlo, Department of Psy- chology, Fordham University, Bronx, New York 10458. Electronic mail may be sent via Internet to [email protected] murray.fordham.edu. standard deviation. The normal distribution has a kur- tosis of 3, and 132 - 3 is often used so that the refer- ence normal distribution has a kurtosis of zero (132 - 3 is sometimes denoted as Y2)- A sample counterpart to 132 can be obtained by replacing the population moments with the sample moments, which gives ~(X i -- S)4/n b2 (•(X i - ~')2/n)2' where b 2 is the sample kurtosis, X bar is the sample mean, and n is the number of observations. Given a definition of kurtosis, what information does it give about the shape of a distribution? The left and right panels of Figure 1 illustrate distributions with positive kurtosis (leptokurtic), 132 - 3 > 0, and negative kurtosis (platykurtic), [32 - 3 < 0. The left panel shows that a distribution with positive kurtosis
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