Regency EF
Visualizing Kurtosis (304008)
*Peter Westfall, Texas Tech University
Keywords: graphics, kurtosis, normal quantile-quantile plot
Along with the mean, standard deviation, and skewness, kurtosis is a statistic that describes the distribution of data. However, unlike the other statistics, kurtosis is difficult to visualize in a histogram or density plot. The reason is that kurtosis is primarily a measure of heaviness of tails, but even when tails are heavy, they are still close to zero, and thus difficult to distinguish in the density or histogram. Because of this difficulty, soundly debunked interpretations of kurtosis as a measure of "peakedness," "flatness," "concentration toward mean," and "height of distribution" persist to this day.
A better way to visualize kurtosis is via the normal quantile-quantile plot. I show how the excess kurtosis statistic arises precisely as a weighted average of distances between the squared axes of the plot, where the weights are Euclidean distances from the origin. I also show that the kurtosis statistic is precisely the least squares slope of a simple transformation of the axes. Finally, I show that the kurtosis statistic is precisely the center of balance of the sample distribution of the z^4-values, which clearly explains why kurtosis is a measure of tail leverage.