An educational video written for Rocketboom about Benford's Law.
Benford’s Law Report
[Keghan to camera] Mathematicians have known a secret fact that they’re not telling you. But I’m about to let you in an exclusive piece of information that may change your life. The truth is: not all numbers are created equal. You could almost say that nature loves some numbers more than others. Nature’s favorite number? [still of the number “1” on screen] One. [back to Keghan] The simple fact is that one shows up a lot. Sorry, Aimee Mann fans. Turns out that, when you’re looking at large collections of numbers, [v/o a table of data, for example http://nationalatlas.gov/articles/people/IMAGES/gender-tbl3.gif] you can look at the first digit and see a lot of ones. [Ones that occupy the first digit of a number on the table are circled in blue (there should be eleven)] Ones occur the most frequently. The least frequent are, as you can guess, nines. [nines that occupy the first digit are circled in red (there should be four)] See? [back to Keghan] Ones are cooler than nines. [Lower third on Keghan: “1 > 9”] This interesting fact was first discovered by Simon Newcomb in 1881. However, his publicist wasn’t great, so it was rediscovered by Frank Benford in 1938, which is why it’s called Benford’s Law, which is in and of itself an example of Stigler’s law, which is a whole ‘nother story. Benford’s law works on the world’s tallest buildings [ v/o table from http://en.wikipedia.org/wiki/List_of_tallest_buildings_and_structures_in_the_world#Tallest_structure_by_category)]
in both metric and standard, [v/o a monthly bank summary] financial figures, [back to Khegan] lengths of rivers, birth rates, death rates, street addresses, molecular weights, and almost every collection of data compiled, ever! [Lower third: “*Notable exceptions include the 1974 telephone book for Vancouver, Canada”] So why does this happen? [v/o a number line from 0 to 9] When a number grows arithmetically from zero to nine, the probability that the first digit is a one is, well, one in nine…once you hit nine. [v/o of a number line from 0 to 19] Once you get past the number nine, the first digit is a one again. So from one to nineteen, [the ones on the number line are circled in red] eleven of the numbers begin with a one! In order for the first digit to flip over to a two, the number has to be twice as large as it was when we first moved into the two-digit numbers! [back to Keghan] In other words, you spend a whole lotta time with a one in your first digit before you can get to a two. [v/o of a number (not a number line) that starts at 90 and counts to 91, 92, etc...] And if a number does get all the way into the nineties, it can only hang around there for ten units. Add one more and it flips to a hundred... [v/o the number hits 100] ...and now it has to hang around for another hundred units until the first digit becomes a two! And that’s just arithmetic growth. Data that relies on geometric and exponential growth - also follows the law! [v/o a courtroom scene] Benford’s law is so well established that it’s even been used to determine whether financial figures have been falsified in court cases, divorce proceedings, [v/o a picture of Greece on a map] and even the macroeconomic data submitted by Greece before joining the Euro Zone. [back to Khegan] Spoiler alert: it wasn’t real! So the next time you’re going to try and defraud the IRS or join an economic trade block, keep that finger on the 1 key.