Earthquake size: An example of a statistical distribution that lacks a well-defined mean

Power-law distributions are observed to describe many physical phenomena with remarkable accuracy. In some cases, the distribution gives no indication of a cutoff in the tail, which poses interesting theoretical problems, because its average is then infinite. It is also known that the averages of samples of such data do not approach a normal distribution, even if the sample size increases. These problems have previously been studied in the context of random walks. Here, we present another example in which the sample average increases with the sample size. In the Gutenberg-Richter law for earthquakes, we show that the cumulative energy released by earthquakes grows faster than linearly with time. Here, increasing the time span of observation corresponds to increasing the sample size. While the mean of released energy is not well defined, its distribution obeys a non-trivial scaling law. (C) 2022 Published under an exclusive license by American Association of Physics Teachers.