How Numbers Work In The Real World

I really like numbers. I understand that not everyone likes numbers as much as I do and that’s OK. I still like you.

Radiolab did an awesome podcast on Numbers a few years ago. There’s a lot of great stuff in there and it’s accessible even to people that are afraid of math. One of the most interesting bits is how our natural sense of numbers is different from formal math. Come to think of it, that may be why some people have a problem with math.

The core idea is something they refer to as natural counting. We’ve all been conditioned by math classes to think of counting as 1,2,3, etc. However, in the natural state, people act differently. On the show, they told of an experiment with an Amazon tribe where people do not count and don’t have numbers beyond five. They showed the tribesmen a line. On one side they placed 1 object and on the other side, they placed 9 objects. The experimenter asked, “What number is exactly between 1 and 9?” The response was “3.” The reasoning works like this:

  1. First person has 1 stick
  2. Middle person has 3 sticks (3 times as many as the first person)
  3. Last person has 9 sticks (3 times as many as the second person

So the second — the one in the middle has 3 sticks.

Let’s look at another example. Imagine you’re giving out bonuses at work and you have a pool of $500,000 and you want to fairly distribute the pool. You could give everyone the same amount of money. But what if someone made $1 million and 4 people made $100,000. You could give everyone $100,000 as a bonus. But that doesn’t feel fair. To be fair you’d give everyone about the same percentage of their salary as a bonus — which in this case would be about a 36% bonus for everyone.

Another non-intuitive concept on Radiolab is Benford’s law.  Benford’s law says that in the real world, you’ll see the number 1 appear many times more as the first digit than the number 9. This happens for naturally occurring phenomena like money in your bank account, size of countries or views on YouTube.

Radiolab has a great story on how this is used in the real world. Say someone was trying to commit a fraud on tax returns. They would be trying to create “random” numbers by having the first digit of each number evenly distributed (equal numbers of 1,2,3, etc.) But Benford’s law says that you should have more 1’s as the first digit than 9’s. So if numbers end up looking more random than they would be in the real world, that’s a sure sign of fraud.

So why does this happen? For the same reason that we saw above, things like to grow by percentages rather than units. Think about the bonus example. Say things are growing by 10%. If you start with the number 1 the next numbers are 1.1, 1.21, 1.33. 1.46, 1.61, 1.77, 1.94. That’s 8 numbers that start with a 1. For 9, you have 9 and 9.9 so that’s only 2 numbers that start with a 9 (and then you’re back to another number that starts with a 1).

For a more detailed explanation of Benford’s law including a pretty technical mathematical explanation, check out Singing Banana.

So why is this important? Even though we think about the world in 1, 2, 3’s, it’s actually more about the changes from where we are now. It’s 10%, another 10%, another 10%, etc. By understanding the world in these terms it’s more intuitive and more useful.