  Statistical laws describe regular patterns observed in diverse scientific
domains, ranging from the magnitude of earthquakes (Gutenberg-Richter law) and
metabolic rates in organisms (Kleiber's law), to the frequency distribution of
words in texts (Zipf's and Herdan-Heaps' laws), and productivity metrics of
cities (urban scaling laws). The origins of these laws, their empirical
validity, and the insights they provide into underlying systems have been
subjects of scientific inquiry for centuries. This monograph provides an
unifying approach to the study of statistical laws, critically evaluating their
role in the theoretical understanding of complex systems and the different
data-analysis methods used to evaluate them. Through a historical review and a
unified analysis, we uncover that the persistent controversies on the validity
of statistical laws are predominantly rooted not in novel empirical findings
but in the discordance among data-analysis techniques, mechanistic models, and
the interpretations of statistical laws. Starting with simple examples and
progressing to more advanced time-series and statistical methods, this
monograph and its accompanying repository provide comprehensive material for
researchers interested in analyzing data, testing and comparing different laws,
and interpreting results in both existing and new datasets.
