Power law

On Boxing Day (December 26) 2004, a tsunami resulting from a 9.0+ magnitude earthquake killed about 250,000 people around the Indian Ocean. This was one of the deadliest natural disasters in recorded history. The Indian Ocean tsunami illustrated a major theme on this blog: the importance of catastrophe in human history, and in the history of life and the universe.

Earthquakes are one example of a phenomenon following a power law statistical distribution. The frequency of earthquakes drops off as an exponential function of their magnitude, so that on a logarithmic scale, the magnitude-frequency relationship looks linear. This is known as the Gutenberg-Ritter relation. (The deviation from linearity in the upper left part of the chart below may reflect measurement error, with a lot of tiny earthquakes not being detected.)

gutenberg-richter

Power law distributions are found in many other contexts, for example, in the frequency of wars versus their magnitude (as measured by the number of war deaths). A power law distribution is very different from the more familiar bell-curve Gaussian normal distribution: extreme “black swan” events that are astronomically unlikely under a normal distribution may happen at appreciable frequency under a power law distribution. Depending on the exponent, a power law distribution may not have a well-defined variance, or even a well-defined mean.

For a technical discussion of why small scale processes sometimes aggregate to generate normally distributed outcomes, and other times aggregate to produce power law distributions, here’s an article on The common patterns of nature. A take home lesson – not always covered in introductory treatments of statistics and probability theory – is that catastrophes and extreme outcomes can be an expectable part of the natural order.

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