8/2/2019 Benford's Law 14oct10

http://slidepdf.com/reader/full/benfords-law-14oct10 1/2

New ScientistCurious mathematical law is rife in nature

* 14 October 2010 by Rachel Courtland* Magazine issue 2782.

WHAT do earthquakes, spinning stellar remnants, bright space objects and ahost of other natural phenomena have in common? Some of their propertiesconform to a curious and little known mathematical law, which could now findnew uses.

A subject of fascination to mathematicians, Benford's law states that for manysets of numbers, the first or "leading" digit of each number is not random.Instead, there is a 30.1 per cent chance that a number's leading digit is a 1.Progressively higher leading digits get increasingly unlikely, and a number has

 just a 4.6 per cent chance of beginning with a 9 (see diagram).

 The law is named after physicist Frank Benford, who in 1938 showed that the

trend appears in many number sets, from the surface area of rivers to baseballstatistics to figures picked randomly from a newspaper. It later emerged thatsuch distributions are "scale-invariant": if you convert the units of the numbersin the set, from metres to yards, say, the set will still conform to Benford's law.

Not all sets of numbers obey this law, but it crops up surprisingly often. As aresult, mathematicians have put it to work, using deviations from it to detectcases of tax fraud, voter fraud and even digital image manipulation.

Now Malcolm Sambridge of the Australian National University in Canberra andcolleagues have extended the list of natural phenomena with properties thatfollow Benford's law. Their new list includes the depths of almost 250,000earthquakes that occurred worldwide between 1989 and 2009, the brightnessof gamma rays that reach Earth as recorded by the Fermi space telescope, therotation rates of spinning star remnants known as pulsars, and 987 infectiousdisease numbers reported to the World Health Organization in 2007(Geophysical Research Letters, DOI: 10.1029/2010GL044830).It applies to quakes, the brightness of gamma rays reaching Earth and therotations of dead stars

 That Benford's law pops up in so many natural phenomena won't surprisemathematicians but may shock some scientists. When Sambridge's teampresented Benford's law findings at a 2009 geoscience conference, onedubious attendee "thought we were having a laugh", he recalls.

 Yet geoscience is ripe for new uses of the law, he says. As well as measuringearthquake depths, Sambridge's team also looked at the verticaldisplacements of the ground in Peru as the tsunami-triggering Sumatra-Andaman earthquake of 2004 progressed. A set of ground shifts before theearthquake proper, when small movements were due to "background noise",did not follow Benford's law, but shifts that occurred during the quake itself did.

8/2/2019 Benford's Law 14oct10

http://slidepdf.com/reader/full/benfords-law-14oct10 2/2

 The team also examined seismic data recorded at the same time by a stationin Canberra. The overall patterns in the shifts persisted but the exact extent of the adherence to the law varied differently over time than in the Peruvianmeasurements. The team then looked more closely at Canberra seismogramsand found that they were consistent with a minor, local earthquake occurringat the same time, which could be the source of the discrepancy between the

two measurements.

"That's the first time I know of where something physical like that was actuallydiscovered using Benford's law," says Theodore Hill, a mathematician at theGeorgia Institute of Technology in Atlanta, not involved with the work.

As well as using Benford's law to detect mild earthquakes, Sambridge says itcould find other uses. "I'm hoping people will check it out in their data. It couldsignal something strange and something to investigate, perhaps somethingthat you might not have spotted in another way." And checking if propertiesthat adhere to Benford's law in nature also do so in computer simulations couldbe a way to check and improve misbehaving models.

 Just how widespread the law is in nature is not known. When the team lookedat the masses of 400 extrasolar planets, there was an anomalous bump innumbers starting with 6. This may be an artefact of a small sample, a problemwith the measurement technique or a sign that exoplanet masses do not fitBenford's law.