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Alicia Iannucci’s Precalculus Class Applies Benford’s Law

Photo of Seattle Academy Upper School Math 2022 Class discussing Benford's Law

Alicia Iannucci’s precalculus class went from basic exponential and logarithmic functions to cracking the math law that governs if an individual is cheating on their taxes, fudging attendance records at Seattle Academy, faking accounts on social media, and generally proves that the world is not as random as it seems. In just one little project on Benford’s Law, students compiled their own data sets (essentially 50-100 numbers of any kind, spanning several orders of magnitude) to see if the distribution of numbers as outlined in Benford’s Law applies to sports, grades, food consumption, weather patterns — and even the number of Teslas in Seattle (as junior Peter Ratzlaff humorously points out). 

Their discoveries are insane. What appeared to be random last week, this week appears to be Benford’s Law. If you know Benford’s Law, then you understand how truly revolutionary this math lesson is. If you don’t, hang on, because this is going to be a big moment for math aficionados. 

To be clear, Benford’s Law was not discovered by Seattle Academy students. It was discovered by Simon Newcomb in 1881, and rediscovered by Frank Benford in 1938 when it began to grow in awareness. Let’s just say SAAS students are bringing that discovery to the forefront of discussion today. 

What is Benford’s Law? Benford's Law, according to Cypress McDonald '23, is “the idea that if you take almost any group of numbers and single out the first digit in the series, 30% of the time it will be 1, followed by 2,3,4, etc. creating an exponential regression curve.” 

Or, in other words, “Benford's Law is this law saying that if you look at a group of numbers, the number 1 will pop up more than the number 2, and the number 2 will pop up more than the number 3 and so forth,” Jamie Williams '23 puts it simply. 

This is significant because almost every single data set — spanning all industries, all natural disasters, all histories  — follows Benford’s Law. Believe it or not. 

So, using the mathematical equation of p(d) = log10(1+1d), we can determine if data sets have been altered, manipulated, or faked, simply by identifying those data sets that do not follow Benford’s exponential curve. Which looks like this:

Photo of Seattle Academy Upper School Math 2022 Class discussing Benford's Law

Ask Aiden Carroll '23. He looked at the number of trees per person in each country. He gathered data from 81 countries and studied the leading digit to see if Benford’s Law fits. 

Madeline Bunnell '22 applied Benford’s Law to the average hours of sleep middle and high schoolers get.

This math law can be, and is being, applied to election monitoring, news reporting, tax fraud, climate change, and really, so much more. To understand this is to understand “chance” and redefine our understanding of random. 

Interested in knowing more about Benford’s Law? You can check out this short Numberphile” video, or this longer “Digits” episode on a Netflix series called “Connected.”


Fun Fact: Benford’s Law was initially discovered because of greasy fingertips on an old book of logarithm tables. One day, Astronomer Simon Newcomb observed that the pages in the front of the book were noticeably more handled than the pages in the back of the book. This sparked his inquiry into why leading digits of 1s, 2s, 3s, and so forth, appear significantly more than 8s, and 9s. In fact, 30% of numbers start with 1s, and 17% start with 2s, as so forth as shown in the exponential curve above.

 

 

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