Benford's Law: F(d) = log[(1 +1/d)]

In an attempt to escape the general nausea that gripped me everytime I began to think about blogging on whatever inane and asine political porridge was being slopped up this week, be it the latest crap our Attorney General "Gonzo All Lies" was asking the American Public and the Senate Congressional Hearing to swallow, where said crap was being presented as high cuisine to be savored ...  the futility of the Dem's latest run-up-the-hill, Jack & Jill Veto Show now improved with Half Now, Half Later Easy Payment Plan... I grew weary and sick at heart to even attempt to put on a Sarcastic Smile, to once again write with great verve and wit how fundamentally fucked up this Facade of a Government is. 

So, perhaps, one can forgive me for straying from that milieu. Avoid, if even for a scant moment, this arduous and thankless task of bringing our so-called leaders to... task.  Hey, let's just have some quality time with science, shall we?  As I was perusing some articles in the Science Section of Some News Portal I came across a curious Law.  Benford's Law.

An Astronomer, Simon Newcomb, noticed a rather strange thing.  The Book of Logarithms in his University's Library was dirtier towards the front of the book than the back.  Why?  Well, a book of logs is just a big tables of logarithms.  Page after page of numbers in columnar fashion.  So?  Well, it turns out that the science geeks of the time that were using that book of logarithmic numbers were finding their answers much more so towards the front end of the book than the last, hence the dirtier entries at the front.  See?  Hmm... so?  So it meant that the pages towards the front, which are based around the first digit of a number being 1 had a higher probability than those that were at the end of the book, the 9's.  In other words, the equations that involved looking up logs of numbers were not uniformly distributed.  And that includes LOTS of equations.  What is going on that would shake the probability tree so hard that 1 was more likely than 2 which is more likely than 3, and so on and so forth?  It didn't matter what the equations were, they were in fact random in that sense, this Law held sway.  Newcomb wrote a paper in 1881 on this curious property of numbers and it was promptly forgotten, more or less.

Fast forward to 1935.  Enter Frank Benford, from whom the Law get's it's name.    Working as a physicist at General Electric (he also invented the light pointer) he also noted, independently, that the Log Book was dirtier at the front of the book.  He took it a step further and began to ask his colleagues what equations they were using.  As I indicated earlier, the results ran a gamut of scientific and statistical formula.  No rhyme or reason could be found for this proclivity.  The studies ranged from lengths or areas of the rivers around the world, statistics of the baseball league, a set of numbers extracted from magazines, the house numbers of the 342 people awarded Scientists of the Year, the set of physical and chemical constants, and series generated by some mathematical functions. Furthermore, the prevalence of the number 1 was in the same proportion in all these cases, and for other digits the results were also similar when all the groups of data were compared. He compiled a large amount of information to check this tendency before publishing his discovery. He presented a study on 20,229 data of 20 different types. He also rediscovered the logarithmic form of this first digit law. This time the discovery became notorious and in spite of the recovery of Newcomb’s previous publication, nowadays this logarithmic law is known as Benford’s Law or the Law of Anomalous Numbers as he called it in his paper.

* in the table above it is obvious that in all these non-related cases that the number 1 is much more likely to be the first digit (approximating to F(d) = log[(1+1/d)] = log(1 + 1/1) = 30.1%).

It's rather easy to understand why some equations work out with 1 more often 'chosen' as the first digit.   Take the fact that there's a higher probability that house addresses begin with a 1 or 2 than 9.  Obviously the 90's and such don't occur as often as 10's and 20's.  on a street.  However, while many natural law equations amazingly fit the Benford Curve, not all do. 

Practically any group of data obtained carrying out ‘measurements’ satisfied the law, provided the numbers were not arbitrarily assigned and without restrictions (telephone numbers, identity cards or passport numbers, dates, etc), and neither random uniform nor normal distributions (lottery, weight and/or height of adult people, etc). The really astonishing thing was the variety of different data obeying the law. Benford’s law was revealed to be a statistical law, and therefore the greater the number of elements in a
numerical group, the better it would fit to this law.

It was demonstrated that series of numbers generated by some mathematical functions such as the square root x1/2, the inverse 1/x or the square x2 do not satisfy Benford’s law, whereas some other functions fulfil the law in an exact way: prime numbers, exponential function a^x, factorial x!, Fibonacci’s succession, Tartaglia’s triangle, or Newton’s binomial coefficients. The law was generalized to be applied to the digits following the first one, also using Benford’s formula. For the second significant digit, the prevalence of number 1 was also stated, and for the third digit too, but each time with less importance because the frequency of appearance of each digit started becoming equal. 

What is surprising, at least to this layman, is that there are so many Natural Laws that will conform to this statistical property!  Equations that cannot have any 'Man Made' skew show this property, such as alpha decay half lives.  And yet other mathematical models of Nature do not exhibit this behavior. For instance, atomic ionization energies—do not obey Benford’s law.

In 1961, Pinkham discovered the first general relevant result, demonstrating that Benford’s law is scale invariant and is also the only law referring to digits which can have this scale invariance. That is to say, as the length of the rivers of the world in kilometres fulfil Benford’s law, it is certain that these same data expressed in miles, light years, microns or in any other length units will also fulfil it. Scientific interest in the law increased as a consequence of this very attractive unicity theorem. However, when Raimi in 1969 published a review in which he summarized the science behind Benford’s law, it was still something disconcerting and the theoretical studies undertaken seemed fruitless, for example, studies of the possible relationship between Benford’s law and fractals (because of the scale invariance that both phenomena share) were not successful.

Hmm.  I would have bet that fractals had this property.  Perhaps it's some fractal law that they not obey other laws, he he.

Now that I've laid the framework for this Odd Law, what applications might there be?

Well, Fraud Protection for one.  It seems that people that are committing fraud will attempt to hide their number juggling with a preference toward 5's and 6's (hide in the middle).  Auditors can apply the Benford Law to show when numbers are being tainted with fraudulent entries.  This has also been applied to Voter Results.

Some studies based on Benford’s law have shown some anomalies in the voting results of the controversial presidential election in the USA, concretely in the Florida state, or in the presidential referendum in Venezuela in 2004. Recently, the last Mexican presidential election in July 2006, whose closed results and accusations of fraud between different candidates have generated serious problems, has also been analyzed with the help of Benford’s law. Mansilla, from the Institute of Physics of the Universidad Aut´onoma de Mexico (UNAM), concluded in his work: ‘It is very difficult to explain why the real data of the voting are how they are, if you compare them with the theoretical results ( . . . ) and every divergence of Benford’s law must be looked with suspicion’.

Let me state that first part again: Some studies based on Benford’s law have shown some anomalies in the voting results of the controversial presidential election in the USA, concretely in the Florida state.

Oh my hanging chads!  We have here mathematical and statistical proof that the reason I feel like throwing up whenever I feel the need to speak out against this Administration is because this Administration is Fraudulent. It all makes sense now.  Sweet, mathematically pure sense.  Mountain Spring pure, even.  Next time you feel a bit nauseous over some incredibly stupid thing Dumbya and his League of Satanic Minions are shoving down your throat just remember this: F(d) = log[(1 + 1/d)] ... and try not to swallow.

Zenny, Math Wiz and Dirty Book of Logs Lover

Sources:

European Journal of Physics: How do numbers begin? The first digit law) - Free Signup required.

Physorg.com : Numbers follow a surprising law of digits, and scientists can't explain why

Wikipedia: Benford's Law

Wolfram MathWorld: Benford's Law

Neato Flash Graphic demonstrating Benford's Law