Kurtosis Inversely Related to Peakedness

Kurtosis Inversely Related to Peakedness Peter H. Westfall Texas Tech University (Emeritus) Abstract In 2014 I published the article, “Kurtosis as Peakedness, 1905 – 2014. R.I.P.,” in The American Statistician, soundly debunking once and for all the century-old notion that kurtosis measures peakedness and flatness. However, sources on the web and in peer-reviewed publications continue to present kurtosis, incorrectly, as a measure of peakedness and flatness. This article presents new results since 2014 that make the case even more strongly that kurtosis most emphatically does not measure peakedness/flatness, as families of distributions exist where increasing peakedness causes decreased kurtosis. Instead of peakedness, I show how kurtosis is precisely interpreted as a measure of tail weight, regardless of symmetry or modality. Finally, I recommend visualizing kurtosis using normal qq plots, as density/histogram plots do not show tail weight clearly. 1. Kurtosis Does Not Measure Peakedness or Flatness In 1905, Pearson started the incorrect interpretations, saying [departure from normality involves a] degree of flat-toppedness which is greater or less than that of the normal curve. Given two frequency distributions which have the same variability as measured by the standard deviation, they may be relatively more or less flat-topped than the normal curve. If more flat-topped I term them platykurtic, if less flat-topped leptokurtic, and if equally flat-topped mesokurtic. This interpretation appears essentially in several books and articles, including from many prestigious statisticians, up to this very day with more on the web, despite Westfall's (2014) debunking of it. For example, a (2025) article in Psychology and Marketing, "A Tutorial on What to Do With Skewness, Kurtosis, and Outliers," by Iacobucci, Roman, Moon, and Rouzies, claims that kurtosis measures peakedness/flatness, independent of outliers, and cites DeCarlo for this interpretation. Specifically, they state, incorrectly, It [excess kurtosis] reflects the extent to which the distribution appears peaked (leptokurtic, positive) or flat (platykurtic, negative) (DeCarlo 1997; see examples in the Appendix). DeCarlo (1997) makes the error in the very first sentence of the abstract of his paper in the journal Psyvhological Methods, stating incorrectly, For symmetric unimodal distributions, positive [excess] kurtosis indicates heavy tails and peakedness relative to the normal distribution, whereas negative [excess] kurtosis indicates light tails and flatness. This statement is clearly false. The current Wikipedia page (English version), gives two distributions, both symmetric and unimodal. The first has kurtosis <3 (“platykurtic”) but is infinitely peaked, and the second has near infinite kurtosis, but appears perfectly flat-topped. The first distribution is obtained by mixing a Beta(0.5,1.0) with its reflection about 0, then standardizing. The second distribution is obtained by mixing a U(-1,1) and T(4.0000001), with probabilities 0.999 and 0.001, then standardizing. But it gets worse for the “peakedness/flatness” interpretation. You can have entire infinite families of symmetric unimodal distributions, F q , indexed by a single parameter 0 < q < 1, all having mean zero and variance 1.0, wherein kurtosis ranges from 2.24 (“platykurtic”) to infinite, where the kurtosis decreases as peakedness increases, and where the distributions become more flat-topped as kurtosis increases. Here are graphs of four distributions within such a family. These distributions are defined as follows: The only logic ever given for the “peakedness” and “flatness” interpretations is that there exist families of distributions for which larger kurtosis implies peaked, and lower kurtosis implies flat. However, using that same logic, one could as easily state as follows, based on the above family of distributions: For symmetric unimodal distributions, positive [excess] kurtosis indicates heavy tails and flatness relative to the normal distribution, whereas negative [excess] kurtosis indicates light tails and peakedness. Like DeCarlo's incorrect interpretation, this interpretation is also wrong, because examples do not prove generalities. A bear is an example of a mammal, but not all mammals are bears. 2. Why Kurtosis Measures Tails I now give three mathematical presentations establishing that kurtosis measures tail weight/leverage. 2.1 The extremes completely determine large kurtosis. In 2014, Westfall proved that large kurtosis is determined by the tail, specifically mass that is > b standard deviations from the mean, where b is arbitrarily large. Specifically, he proved the following: In particular, the arrangement of the density within b standard deviations of the mean is irrelevant for large kurtosis. 2.2 The arrangement of the density within 1 standard deviation has little import Since kurtosis is E(Z4), it can be expressed as the integral of z4 f(z), or the area under the graph of the function z4 f(z). Taken from the crossvalidated stackexchange site, the following figure shows graphs of z4 f(z) for various distributions. z It is clear that the character of the distribution f(z) within one standard deviation (z between -1 and 1), has little effect on kurtosis, since it is “dampened” by z4. Westfall (2014) proved that the area under these curves between -1 and 1 is always <1, and <0.5 for typical classes of distributions. 2.3 Kurtosis is precisely interpreted as “tail leverage” for all types of distributions Given two distributions (or data sets), one having larger kurtosis than the other, what can you say? There is always a precise comparison involving tail weight, and it goes as follows. Suppose the random variables are X and Y, and let their standardized versions be V and W, respectively. These may be discrete, so include the case of sample data. Suppose the kurtosis of Y is higher, so that E(W4 ) > E(V 4 ). Now, draw the graph of the distribution of V 4 . Since the mean is the point of balance, the graph balances at the kurtosis of X, k(X). Now, draw a graph of the the distribution of W 4 , it balances at k(Y). So if you place a fulcrum at k(X) under the distribution of W 4 , the graph will “fall to the right.” Now, what makes it “fall to the right”? Is it greater peakedness or concentration to the mean in the distribution of Y? No, those aspects would make it “fall to the left.” It is instead the extreme values in the distribution of Y that are responsible. See the following graphs. The final graph illustrates the “falling to the right” when the fulcrum is placed at the kurtosis of the data set with smaller kurtosis. 3. Higher Kurtosis Does Not Mean “More in the Tails” or “More in the Center” Kurtosis measures tails, as shown by the “point of balance” graphs in the previous section. However, sources on the web have described higher kurtosis as meaning “more probability in the tails.” This interpretation is also false, as tail weight shown in the graphs is a kind of leverage, a combination of both mass and extension. Less mass, coupled with higher extension, can just as easily cause higher kurtosis, as shown by the following example. In this family, the tail probability, q , tends to zero while the kurtosis simultaneously increases to infinity. Another misinterpretation of kurtosis, related to the peakedness misinterpretation, is that higher kurtosis implies more probability in the range within a standard deviation. This is also false, as the following example shows. 4. Visualizing Kurtosis The density or histogram graphs that you find online to illustrate kurtosis differences are typically very bad. People think that heaviness of tails should be visible in these graphs, and they grossly exaggerate the tail thickness so much that the distribution actually looks light-tailed, like a uniform distribution. Or they show graphs that simply differ in variance. Or they show graphs where the density is uniformly larger for higher kurtosis, which is impossible because then the areas cannot be 1.0. Or they show differences in peakedness, perpetuating the myth that kurtosis measures peakedness. Tails, even when "heavy," are still very close to zero, and thus are not clearly shown in histogram/density plots. The point of heavy tails is that are relatively far from zero compared to the normal distribution tails. For example, 10−3 is relatively huge compared to 10−100, but both are very close to zero in an absolute sense, and cannot be distinguished in a density graph. The solution is simple: Stop using density/histogram plots to illustrate kurtosis. Instead, use normal q-q plots. Tail behavior is clearly visible in q-q plots, and there is a direct mathematical connection between the appearance of these plots and the kurtosis statistic, described as follows: 5. Conclusion It has been more than a decade since I buried the “kurtosis as peakedness” interpretation. This article simply adds more nails to the coffin. References DeCarlo, L. T. (1997). On the meaning and use of kurtosis. Psychological Methods, 2(3), 292–307. Iacobucci, Roman, Moon, and Rouzies, (2025). A Tutorial on What to Do With Skewness, Kurtosis, and Outliers, Psychology & Marketing. Pearson K. (1905). Das Fehlergesetz und seine Verallgemeinerungen durch Fechner und Pearson. A Rejoinder. Biometrika. 1905;4:169–212. Westfall, P. H. (2014). Kurtosis as Peakedness, 1905 - 2014. R.I.P. The American Statistician, 68(3), 191–195.