Skewness-Kurtosis: Small samples and power-law behavior

Skewness and kurtosis are fundamental statistical moments commonly used to quantify asymmetry and tail heaviness and peakedness of probability distributions. Despite their widespread application in statistical mechanics, condensed matter physics, and complex systems, important aspects of their empirical behavior remain unclear — particularly in small samples and in relation to their hypothesized power-law scaling. In this work, we address both issues using a combination of empirical and synthetic data. First, we establish a lower bound for sample kurtosis as a function of sample size and skewness. In doing this, a remarkable deltoid-shaped domain appears for n = 4, which in case of discrete distributions exhibits fractal properties reflecting the underlying geometrical constraints of skewness–kurtosis space. Second, we examine the conditions under which the 4/3 power-law relationship between kurtosis and skewness emerges, effectively extending Taylor’s power-law to higher-order moments. Our results show that this scaling predominantly occurs in data sampled from heavy-tailed distributions and medium/large sample sizes, highlighting the interplay between tail behavior and sample size in shaping the empirical behavior of higher-order moments.