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

Copyright 1997 by the American PsychologicalAssociation,Inc. 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 andflatness. 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; ingeneralized tests and estimators, as well as limitations of and alternatives to the kurtosis measure [32, are discussed.

It is t y p i c a l l y noted in i n t r o d u c t o r y 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 aspect of shape, which is kurtosis, is either not discussed or, worse yet, is often described or illustrated incorrectly. Kurtosis is also frequently not reported in research 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 aspectsof 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. O n the M e a n i n g o f K u r t o s i s Kurtosis can be formally defined as the standardized fourth population moment about the mean, E (X - IX)4 132 = ( E ( X - IX)2)2 IX4
0.4'

standard deviation. The normal distribution has a kurtosis of3, and 132 - 3 is often used so that the reference 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 adefinition 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 has heavier tails and a higher peak than the normal, whereas the right panel shows that adistribution with negative kurtosis has lighter tails and is flatter. Kurtosis and Well-Known Distributions Although a stylized figure such as Figure 1 is useful for illustrating kurtosis, a comparison of well-known distributions to the normal is also informative. The t distribution, which is discussed in introductory textbooks, provides a useful example. Figure 2 shows the t distribution with 5 df which has a positive kurtosis of [32 - 3 = 6, and the normal distribution, for which 132 - 3 = 0. Note that the t distribution with 5 d f has a variance of 5/3, and the normal distribution shown in the figure is scaled to also have a variance of 5/3. The figure shows that the t 5 distribution has heavier

where E is the expectation operator, IX is the mean, 1,1,4 is the fourth moment aboutthe 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 addressed to Lawrence T. DeCarlo, Department of Psychology, Fordham University, Bronx, New York 10458. Electronic mail may be sent via Internet to decarlo@ murray.fordham.edu.

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