The significant digit law: a paradigm of statistical scale symmetries

In many different topics, the most significant digits of data series display a non-uniform distribution which points to an equiprobability of logarithms. This surprising ubiquitous property, known as the significant digit law, is shown here to follow from two similar, albeit different, scale symmetries: the scale-invariance and the scale-ratio invariance. After having legitimized these symmetries in the present context, the corresponding symmetric distributions are determined by implementing a covariance criterion. The logarithmic distribution is identified as the only distribution satisfying both symmetries. Attraction of other distributions to this most symmetric distribution by dilation, stretching and merging is investigated and clarified. The natures of both the scale-invariance and the scale-ratio invariance are further analyzed by determining the structure of the sets composed by the corresponding symmetric distributions. Altogether, these results provide new insights into the meaning and the role of scale symmetries in statistics. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2006