There are plenty of constants in our lives. Death is always one. Taxes are another. But we can also rely on gravity, cell reproduction, and math. Math is a very reliable constant. 1+1 will always equal 2. Our trust in math allows us to build skyscrapers, create new technologies, and even have some fun. Case in point: Benford’s Law.

Benford’s Law is sometimes called the first digit law. It applies data sets in explaining the frequency of how often an individual unit in the data set starts with a 1, 2, 3, 4, etc. If numbers were truly random there would be about an 11% chance of any number starting with any one of the 1-9 digits. But the number 1 is much more frequent than that, occurring 30% of the time. And then it drops all the way to 9 which occurs much less than 11% of the time.

I was skeptical that this was the case so wanted to check it for myself. I went to my trusty friend the FRED site and downloaded the monthly numerical change in employment from 1939 until now.  Then I plotted the frequency of the first digit in the each month and found this:

Benfords law employment changes

That’s pretty much in line with Benford’s law.  Look how much more common 1 and 2 is than the other numbers.  But I wasn’t sold yet.  Next I went to Wikipedia and looked at the populations of the top 250 most populous cities in the country.
Benfords law city populations

This is really starting to get scary now.  How can this be?  The secret is in how we look at data and count.  For something to grow from 1 to 2 it must double.  To go from 2 to 3 it only needs to grow 50%.  3 to 4 is just 33%.  As the numbers climb the growth rate shrinks.  Here’s an easier way to think about it.  It’s a lot harder to grow your salary from $100,000 to $200,000 than it is to grow from $800,000 to $900,000.  If you think back to high school math, this is partially what logarithmic math does.

But is this anything more than a party trick, an interesting anecdote?  Yes.  Because this distribution of numbers is naturally occurring in our numerical system, it can be used to pick out data that isn’t natural.  Take the financial reporting of public companies.  Analysts and investors poor over the financial reports looking for some kind of an edge.  Regulators and investors alike are also looking for signs of fraud.  Executives may fake their reports to hide bad losses or pump up the stock price.  And some researchers have discovered Benford’s Law works here as well.

The researchers found that the financial statements of companies later found to be lying in their reports had fallen out of the distribution under Benford’s law.  Because made up numbers can start with anything, they’re a lot less likely to start with 1 or 2 than under Benford’s Law.  Once again skeptical, I went to the annual report of Yahoo! and looked for the frequency of $1, and $2, and so on.

Benfords law financial reports

Benford’s Law won again.  So I looked at Goldman Sachs.  Also in compliance with this mathematical law.  Finally I went searching for a company that may be out of compliance with financial reporting: Herbalife.  Herbalife has had some accusations thrown at it that they’re cooking the books and running a scheme.  A more complex analysis would be warranted, but using the same screen as above, even Herbalife falls in line with this law.

Read: The Simple Mathematical Law That Financial Fraudsters Can’t Beat (Forbes)

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categories: economics, investing, personal finance