Severe problems with kurtosis

Kurtosis is routinely taught in introductory statistics courses, where it is sometimes still wrongly interpreted as "peakedness" even though the statistical profession has dismissed that notion. Furthermore, the internet abounds with wrong illustrations of kurtosis. Moreover, our literature search found only two actual published practical applications of the kurtosis statistic. To assess the merits of kurtosis, or lack thereof, we review the definition of kurtosis and present simulations that demonstrated extreme instability of kurtosis estimates, especially when viewed jointly with skewness. The simulations were conducted with 100,000 draws of samples of size 30, 100 and 1000 from standard normal, standard uniform, lognormal, arcsine and standard triangular distributions. Results were presented graphically for the marginal and joint distributions of the skewness and kurtosis estimates. With the lognormal distribution, which is highly skewed, the vast majority of skewness-kurtosis pairs fell very far from the theoretical population values even in samples of size 1000. At the same time, high asymmetry of sampling distribution of kurtosis when sampling from a symmetric distribution was observed for the arcsine and the standard triangular distribution. In addition, we made two sets of didactic simulations with data from standard normal distribution to illustrate the huge sampling variability of kurtosis estimates as compared to estimates of lower moments. Hence, we believe that kurtosis should be avoided in non-specialist introductory statistics courses, it should not be routinely calculated as part of numerical data description (whereby its standard error is particularly misleading), and it should not serve routinely as criterion for assessing appropriateness of using the normal distribution as the model for an empirical dataset.