Thread by @dollarsanddata

\1 Tweetstorm on How to Detect Financial Fraud Using Simple Math (aka Benford’s Law or the Leading Digit law)


If you haven’t heard of this, it will BLOW YOUR MIND.

First, let’s consider something mundane: The revenue of every company in the S&P 500 in 2017.

\2 For example, Walmart had 2017 revenue of $485 billion, Exxon Mobile had $237 billion, and Amazon had $177 billion. Now, let’s take the LEADING DIGITS of these numbers. So, the leading digit for Walmart’s revenue is 4, for Exxon is 2, and for Amazon is 1.

\3 Question: If we took the leading digits for revenue for ALL the companies in the S&P 500 would you expect the distribution of leading digits to be equal? For example, is a leading digit of 8 more likely than 5? Is 2 more likely than 3?

\4 This may sound silly to you because they should be equally likely, right?

NOPE.

In fact, over 30% of the revenue numbers will have a leading digit of 1, while 5% will have a leading digit of 9. More of the numbers will start with 1 than 2, 2 than 3, and so on to 9.

\5 Wait…how is this possible? Simple: let’s do some easy math to see why.

Start by counting from 1 to 9. By the time you hit the 9, the distribution of leading digits across all numbers counted is equal. There is one 1, one 2, etc. However, once you hit 10 things change...

\6 As you count 10, 11... all of the numbers start with a 1. This means the distribution of leading digits from 1 to 19 has far more 1s than 2s, 3s, etc. You only reach an equal number of leading digits when you count to 99. But, then you hit 100 and the pattern breaks again.

\7 What this shows is that ANY NUMERIC SERIES THAT GOES ACROSS MULTIPLE ORDERS OF MAGNITUDE will exhibit a skewed distribution of leading digits. This is Benford’s Law and it has been demonstrated many times throughout the world.

\8 For example, the height of 60 tallest buildings in the world have far more leading digits of 1 for their height, regardless of the unit of measure (i.e. feet or meters).

See here: en.wikipedia.org/wiki/List_of_t…

\9 Also, the number of Twitter followers across a sample of accounts also follows this law.

Data from here: testingbenfordslaw.com/twitter-users-…

\10 If we look at the distribution of leading digits of “Cash and Cash Equivalents” of all the S&P 500 companies, it perfectly fits Benford’s Law. These data come from 2015:

\11 This works because cash at companies goes across multiple orders of magnitude and it is not something that can be easily manipulated by humans. You either have cash at your bank or you don’t.

\12 However, something that humans can tamper with is the income tax they pay. As a result, if we look at the distribution of leading digits on income tax paid by all companies in the S&P 500, it does NOT conform to Benford’s law as closely:

\13 Additionally, if we look at the leading digits on common iPhone passcodes, we see that these do not fit Benford’s law as well either (data also from here: testingbenfordslaw.com/most-common-ip…):

\14 Why do these deviate from Benford’s law? Because PEOPLE can directly affect them. There are biases in our society that make people choose certain numbers for certain things. This is more obvious for iPhone passcodes than income tax, but my point stands.

\15 Why is this interesting?

Because WHEN PEOPLE MANIPULATE DATA, IT IS POSSIBLE TO TELL. Benford’s law is one way to start digging.

BEWARE THOUGH: Benford’s law is not the be-all and end-all for fraud detection, but it can be helpful, read this: rss.onlinelibrary.wiley.com/doi/full/10.11…

\16 With that being said, happy data mining from Of Dollars and Data and thank you for reading!

If you are interested in learning more on this topic, check out my blog post here: ofdollarsanddata.com/fantastic-lies…