# The Significant Digit

Mike O'Dell mo at ccr.org
Sat May 8 11:44:27 CDT 2010

i read something on this somewhere else (i cannot remember where)
that suggests this is just another manifestation of Zipf's law
and the nature of 1/f distributions. i don't know enough to
hold an opinion based on anything other than opinion.

-mo

On 5/8/10 12:18 PM, Philip Miller Tate wrote:
> I'm sorry, I just don't buy Benford's "Law". I once saw a reference to
> it with the supporting statement, "Scientists and mathematicians have
> recognized this fact for many years, and designed the slide rule so that
> the low numbers were spaced farther apart to make access to them easier."
> Phil M1GWZ
>
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> On 7 May 2010, at 22:27, Andre Kesteloot wrote:
>> Friday, May 07, 2010
>>
>>
>>     *Benford's Law And A Theory of Everything*
>>
>> A new relationship between Benford's Law and the statistics of
>> fundamental physics may hint at a deeper theory of everything
>>
>> In 1938, the physicist Frank Benford made an extraordinary discovery
>> about numbers. He found that in many lists of numbers drawn from real
>> data, the leading digit is far more likely to be a 1 than a 9. In
>> fact, the distribution of first digits follows a logarithmic law. So
>> the first digit is likely to be 1 about 30 per cent of time while the
>> number 9 appears only five per cent of the time.
>>
>> That's an unsettling and counterintuitive discovery. Why aren't
>> numbers evenly distributed in such lists? One answer is that if
>> numbers have this type of distribution then it must be scale
>> invariant. So switching a data set measured in inches to one measured
>> in centimetres should not change the distribution. If that's the case,
>> then the only form such a distribution can take is logarithmic.
>>
>> But while this is a powerful argument, it does nothing to explan the
>> existence of the distribution in the first place.
>>
>> Then there is the fact that Benford Law seems to apply only to certain
>> types of data. Physicists have found that it crops up in an amazing
>> variety of data sets. Here are just a few: the areas of lakes, the
>> lengths of rivers, the physical constants, stock market indices, file
>> sizes in a personal computer and so on.
>>
>> However, there are many data sets that do not follow Benford's law,
>> such as lottery and telephone numbers.
>>
>> What's the difference between these data sets that makes Benford's law
>> apply or not? It's hard to escape the feeling that something deeper
>> must be going on.
>>
>> Today, Lijing Shao and Bo-Qiang Ma at Peking University in China
>> provide a new insight into the nature of Benford's law. They examine
>> how Benford's law applies to three kinds of statistical distributions
>> widely used in physics.
>>
>> These are: the Boltzmann-Gibbs distribution which is a probability
>> measure used to describe the distribution of the states of a system;
>> the Fermi-Dirac distribution which is a measure of the energies of
>> single particles that obey the Pauli exclusion principle (ie
>> fermions); and finally the Bose-Einstein distribution, a measure of
>> the energies of single particles that do not obey the Pauli exclusion
>> principle (ie bosons).
>>
>> Lijing and Bo-Qiang say that the Boltzmann-Gibbs and Fermi-Dirac
>> distributions distributions both fluctuate in a periodic manner around
>> the Benford distribution with respect to the temperature of the
>> system. The Bose Einstein distribution, on the other hand, conforms to
>> benford's Law exactly whatever the temperature is.
>>
>> What to make of this discovery? Lijing and Bo-Qiang say that
>> logarithmic distributions are a general feature of statistical physics
>> and so "might be a more fundamental principle behind the complexity of
>> the nature".
>>
>> That's an intriguing idea. Could it be that Benford's law hints at
>> some kind underlying theory that governs the nature of many physical
>> systems? Perhaps.
>>
>> But what then of data sets that do not conform to Benford's law? Any
>> decent explanation will need to explain why some data sets follow the
>> law and others don't and it seems that Lijing and Bo-Qiang are as far
>> as ever from this.
>>
>> Ref: arxiv.org/abs/1005.0660 <http://arxiv.org/abs/1005.0660>: The
>> Significant Digit Law In Statistical Physics <Benford's-law.jpg>
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