Benford's Law of Leading Digits
A Statistical and Mathematical Analysis of First-Digits
Benford's law is an interesting examination of the phenomenon wherein the leading digits of all numerical data seems to follow an illogical statistical pattern.
There are aspects of math which are truly important and lead to outstanding breakthroughs in science, engineering or other areas of thought. Then there are the aspects of math which are simply interesting, even if they might not be important in the long run.
Benford’s Law does not fall easily into either category. It is not exactly a pointless exercise, and it is not exactly an entirely useful, groundbreaking achievement. It is interesting, though, and while it may be difficult to find real world applications for it, they certainly do exist.
Formulation of the Law
Benford’s law, discovered by American physicist Frank Benford in 1938 (though the phenomenon in question had been noticed as early as 1881 by Astronomer Simon Newcomb) begins with an examination of the frequency of certain numbers and comes to some rather odd conclusions. Think about it this way:
Imagine spending an entire day cataloguing every available number which has some sort of real world representation (meaning that they can’t just be random, made-up numbers – they actually have to have some sort of meaning). This means street addresses, phone numbers, social security numbers, heights of various trees, lengths of rivers, numbers of gumballs in a jar, populations of various towns and cities, mathematical constants etc… the possible source of numbers is practically limitless.
Now, take all of these numbers and sort them by their leading digit (that is, the number that comes first).
According to standard probability theory (the basics of which are really not so difficult to understand), each of the nine digits (excluding zero, which is generally never used as a leading digit) should occur with relatively equivalent frequency. That is, they should each appear roughly eleven percent of the time. The results would probably vary slightly, but the larger sample one takes, the more one would expect the numbers to arrive at their most probable locations in the distribution of probabilities. However, the crux of Benford’s law is the fact that this is not at all what happens, no matter how large the sample size.
In actuality (and after having analyzed quite the volume of data), in situations such as this, there is no even distribution of leading digits. In fact, Benford’s law states that the number 1 occurs more frequently than any other number – making up nearly a third of all leading digits! Number 2 is far behind in second place, occurring only 17.6% of the time, then number 3 with 12.5% and so on, all the way to number 9, occurring only 4.6% of the time.
The frequency of leading digits inexplicably proceeds in perfect numerical order from 1 to 9, despite the logic that would insist that there should be some sort of even distribution.
Further Examination
To make matters even stranger, mathematicians studying this phenomenon since the time of Benford have discovered that this pattern continues to hold true even if the units of measurement are changed (feet changed to yards, yards changed to kilometers, pounds changed to ounces, etc…) or even if the bases are changed entirely (a base is the fundamental unit of counting or measurement – the decimal system uses base 10 while the standard system used in America generally uses a base of 12 – other bases such as 6 and 16 also exist).
Mathematicians have even discovered equations which describe this phenomenon using logarithms (a calculus tool in which certain base numbers are raised exponentially). While explanation behind this phenomenon is not exactly a simple one, but it does exist, and to mathematicians it even makes some sense.
Benford’s Law Applied
So what can Benford’s law do for humanity? On the surface it doesn’t seem like much, but practical uses do exist, though such applications have the potential to be somewhat controversial.
For example: Imagine that a national election is held. The ballot box returns from each state are analyzed and fed into a computer, which then analyzes the distribution of votes in each precinct of each state to find that, indeed, the vast majority of locations very clearly can be defined by Benford’s law – with 1 being the first digit roughly 30.1% of the time.
However, what if there was one state where the returns do not resemble this distribution at all, but are rather spread out in a much more even and random manner? To the eye of a trained mathematician, this discrepancy in such a situation might lead one to assume that there may have been a bit of election fraud occurring in the area being analyzed, as when humans falsify data it usually ends up being much more random than it would normally be in any natural flow of information.
So this could very well be a possible stepping stone toward a full investigation of election fraud, which can also potentially be used by insurance companies and in analyzing demographic statistics from around the world (third world countries have been known to falsify their data to make themselves seem less needy than they really are).
It is somewhat obvious why such methods might be a little controversial, as they are relying on mathematical methods in which there is always a margin of error. When dealing with numbers seemingly at random, isn’t there always a chance that a discrepancy might occur here and there which would change the probability distribution? Of course there is. But at least by analyzing groups of numbers by way of Benford’s law, mathematicians have another tool at their disposal both for discovering certain types of fraud and for understanding a bit more about the behavior of numbers in general.
In any case, no matter what the uses for Benford’s law might be, it is at the very least an interesting phenomenon – one of a great many. Numbers are very interesting things and every time it seems as if mathematicians have understood their behaviors, new surprises surely await.
References:
Brown, Malcolm W. “Following Benford’s Law, or Looking Out for No. 1.”
“Benford’s Law.” Math Pages.
