Kurtosis as Peakedness, 1905–2014. R.I.P.

The incorrect notion that kurtosis somehow measures “peakedness” (flatness, pointiness, or modality) of a distribution is remarkably persistent, despite attempts by statisticians to set the record straight. This article puts the notion to rest once and for all. Kurtosis tells you virtually nothing about the shape of the peak—its only unambiguous interpretation is in terms of tail extremity, that is, either existing outliers (for the sample kurtosis) or propensity to produce outliers (for the kurtosis of a probability distribution). To clarify this point, relevant literature is reviewed, counterexample distributions are given, and it is shown that the proportion of the kurtosis that is determined by the central μ ± σ range is usually quite small.

[1]  P. Westfall,et al.  Understanding Advanced Statistical Methods , 2013 .

[2]  S. Kotz,et al.  Visualizing Peak and Tails to Introduce Kurtosis , 2008 .

[3]  E. Agerbo.  Basic Concepts in Statistics and Epidemiology , 2008, Journal of Epidemiology & Community Health.

[4]  B. G. R. Fowler,et al.  A Book Review: Anfara, V. A., & Mertz, N. T. (Eds.). (2006). Theoretical frameworks in qualitative research. Thousand Oaks, CA: Sage , 2008 .

[5]  S. Ejaz Ahmed,et al.  Improving the performance of kurtosis estimator , 2008, Comput. Stat. Data Anal..

[6]  Basic Psychological Measurement, Research Designs, and Statistics Without Math , 2007 .

[7]  J. Livesey,et al.  Kurtosis provides a good omnibus test for outliers in small samples. , 2007, Clinical biochemistry.

[8]  John C. Reinard.  Communication research statistics , 2006 .

[9]  Hal S. Stern,et al.  CHAPMAN & HALL/CRC Texts in Statistical Science Series , 2002 .

[10]  Chunsheng Ma,et al.  On peakedness of distributions of convex combinations , 1998 .

[11]  Takuji Nishimura,et al.  Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator , 1998, TOMC.

[12]  L. T. DeCarlo.  On the meaning and use of kurtosis. , 1997 .

[13]  Barry H. Cohen.  Explaining Psychological Statistics , 2013 .

[14]  Cheng-Few Lee,et al.  Statistics For Business And Financial Economics , 1993 .

[15]  K. Balanda,et al.  Kurtosis: A Critical Review , 1988 .

[16]  J. Moors,et al.  A quantile alternative for kurtosis , 1988 .

[17]  D. Ruppert.  What is Kurtosis? An Influence Function Approach , 1987 .

[18]  Richard J. Beckman,et al.  A New Family of Probability Distributions with Applications to Monte Carlo Studies , 1980 .

[19]  Robert J. Beaver,et al.  An Introduction to Probability Theory and Mathematical Statistics , 1977 .

[20]  Mukhtar M. Ali.  Stochastic Ordering and Kurtosis Measure , 1974 .

[21]  K. Mardia.  Measures of multivariate skewness and kurtosis with applications , 1970 .

[22]  Brad S. Chissom,et al.  Interpretation of the Kurtosis Statistic , 1970 .

[23]  Richard B. Darlington,et al.  Is Kurtosis Really “Peakedness?” , 1970 .

[24]  P. A. P. Moran,et al.  An introduction to probability theory , 1968 .

[25]  Irving Kaplansky,et al.  A Common Error concerning Kurtosis , 1945 .

[26]  Karl Pearson.  “DAS FEHLERGESETZ UND SEINE VERALLGEMEINER-UNGEN DURCH FECHNER UND PEARSON.” A REJOINDER , 1905 .