Patterns. They are all around us. From preschool playground “investigations” on leaves and flowers to senior year Calculus classes, we have been taught to observe and analyze patterns. When I think of the word pattern, I often think of beautiful geometric figures that create intricate artwork. Or I might think of patterns in my favorite team’s stats. To a child, patterns could mean a series of numbers or shapes. However simple patterns from children’s TV shows like Sesame Street may be, there are certain patterns that remain hidden but govern our world.
What do the NBA, tax evasion, cheating on a test, and the COVID-19 pandemic all have in common? Although that might sound like the beginnings of a joke, the answer is actually the mathematical pattern that connects us all: Benford’s Law.
The name might not live up to the intense build up I gave it, but the idea of one pattern being found in so many places does seem crazy, right? Well, in order to evaluate the validity of Benford’s Law, we need to understand what this pattern is. Basically, Benford’s Law challenges the idea that all random outcomes are equally random. Let’s take a random newspaper, say the New York Times. As you read through this imaginary paper, imagine isolating every number that comes up whether it be a percentage, population, or description. If we take all these numbers and isolate the first digits of each: a pattern will become evident.
According to Benford’s law, in many untampered data sets, 30% of the numbers will be 1s, 18% of the numbers will be 2s, 13% of the numbers will be 3s and so forth. There is an exact pattern to any untampered data set. Now what do I mean by untampered? Any data set that is random and not manipulated by humans follows Benford’s law. So, back to the newspaper example; Out of all the first digits of all the numbers in a newspaper, 30% of these numbers will be a 1, 18% of these numbers will be a 2, and 13% of these numbers will be a 3. Weird, right? But what makes it even more mind blowing is that Benford’s law is everywhere. Take COVID-19. If you look at the first digit of the number of COVID-19 cases in the United States, China, and Italy, they all follow Benford’s Law, illustrated clearly here. 30% of the first digits of the number of coronavirus cases are 1s, 18% are 2s, and 13% are 3s.
I’m sure you’re wondering why you haven’t heard about Benford’s Law before. If it’s as astonishing as I make it out to be, why don’t more people know about it? In the case of Benford’s law, a majority of the general population remains ignorant. This might be because Benford’s law is the political community’s best tool for solving the problem of tax fraud, or it might just be that math is hard and Benford’s Law has been ignored by common core math curricula. Although speculations have been brushed off by the IRS, the European Union has openly admitted to using Benford’s law as a way to detect large scale tax evasion or fraud.
By taking a population and taking the first digit of the total amount of taxes paid by every person, the European Union along with other countries can effectively detect tax fraud. Since Benford’s law applies to all natural data sets, when a person’s taxes don’t follow Benford’s law, the government suspects it must be manipulated and begins investigating. Because Benford’s law applies to other data sets as well, governments also use it for one more important event: elections.
When speculations arise about fraud in the elections, Benford’s Law can be used to validate these accusations. If the frequency of the first digit of the number of ballots in each precinct followed Benford’s Law, the election could be called clear. However, if the number of ballots don’t follow Benford’s Law, then further investigation is done.
The history of Benford’s Law does not seem to fit with its uses. In 1881, Canadian-American astronomer Simon Newcomb observed that the first pages of the logarithmic book were more used than the middle and final pages in. The first pages contained the logarithmic combinations of the numbers 1, 2, and 3. As he went through the pages, the pages became less worn out – signifying that there was a pattern to what numbers were most used. Although this was just a small discovery, his theory soon evolved into the statistical pattern of Benford’s Law.
It might be tempting to think that Benford’s Law can be used for anything. However, you should be careful when using Benford’s Law to try to gain an unfair advantage. You cannot use this same technique with lottery tickets, because the lottery ticket numbers are not random. The ticket numbers increase by 1, and since there is already a defined pattern to these numbers, Benford’s Law can’t be used to predict the winning ticket. On the contrary, you could use Benford’s Law to better understand the size of the lottery pool.
I know you’re thinking, this is impossible. What else does it work for? Well, just searching up “Benford’s Law in _____ (fill in any data you like, from NBA player rankings to average SAT Score)”, will reveal the Benford’s Law patterns behind almost every data set. The population of cities around the world follows this pattern. The price of stocks follows Benford’s Law. Benford’s Law even predicts natural hazards. This pattern is key to better understanding the economy, human geography, and even nature. Benford’s Law reveals mathematical order in the world.
Lee, Kang-Bok, et al. “COVID-19, Flattening the Curve, and Benford’s Law.” Physica A: Statistical Mechanics and Its Applications, vol. 559, Dec. 2020, p. 125090, 10.1016/j.physa.2020.125090. Accessed 11 Oct. 2020.
Sarkar, Tirthajyoti. “What Is Benford’s Law and Why Is It Important for Data Science?” Towards Data Science, Towards Data Science, 25 Oct. 2018, towardsdatascience.com/what-is-benfords-law-and-why-is-it-important-for-data-science-312cb8b61048.
“Using Excel and Benford’s Law to Detect Fraud.” Journal of Accountancy, Apr. 2017, http://www.journalofaccountancy.com/issues/2017/apr/excel-and-benfords-law-to-detect-fraud.html.
Weisstein, Eric W. “Benford’s Law.” Mathworld.Wolfram.Com, mathworld.wolfram.com/BenfordsLaw.html.