# First Digits Rule!

(Benford's Law)

Numbers can show when you are cheating!

So says *Benford's Law.*

## First Digits

How often do you expect a **"1"** to be the first digit in a set of numbers?

### Example: you are looking at a list of expenses, with numbers like:

- $65.20 (first digit is
**6**) - $35.00 (first digit is
**3**) - $7.50 (first digit is
**7**) - $12.50 (first digit is
**1**)

Would there be as many **1**'s as **2**'s for the first digit?

Well **1** is just a digit, like **2** to **9**, right?

So maybe we could expect it as a first digit about **1 out of 9** times (about 11%):

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

11% | 11% | 11% | 11% | 11% | 11% | 11% | 11% | 11% |

**But no!**

A man called Dr. Frank Benford discovered that in many cases, the number **1** is the first digit **about 30% of the time**.

And poor old number **9** is the first digit **only 5%** of the time.

The story is that a man called Simon Newcomb noticed a book of logarithms was **very worn at the start** but not at the end.

*"Why are people more interested in 1's and 2's than 8's and 9's?"*

He decided to investigate! (Would you investigate something odd?)

Dr. Benford found this amazing thing also happened with baseball statistics, areas of rivers, population sizes, street addresses and many more cases.

## Why is this?

Well, let's think about street addresses:

### What are the first digits of house numbers?

- some streets are short: 1,2,3,4,5,6
- some streets are longer: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 (notice how many have 1 as the first digit?).
- other streets are a bit longer, with numbers from 1 to 30 (many "1"s and "2"s)
- And when the streets are very long we have lots of them starting at 100.

The result is that numbers starting with 1 are more common, 2 is also fairly common and 9 least of all.

### Example: Stock Prices

Let us say a price starts at 1.00, and goes up 10% each time:

Price | First Digit |
---|---|

1.00 | 1 |

1.10 | 1 |

1.21 | 1 |

1.33 | 1 |

1.46 | 1 |

1.61 | 1 |

1.77 | 1 |

1.95 | 1 |

2.14 | 2 |

2.36 | 2 |

2.59 | 2 |

2.85 | 2 |

3.14 | 3 |

3.45 | 3 |

3.80 | 3 |

4.18 | 4 |

4.59 | 4 |

5.05 | 5 |

5.56 | 5 |

6.12 | 6 |

6.73 | 6 |

7.40 | 7 |

8.14 | 8 |

8.95 | 8 |

9.85 | 9 |

Lots of **1**'s, quite a few **2**'s, less **3**'s, and only one **9**

## The Result

Benford figured that the probability of a first digit being **d** is:

P(d) = log_{10}(1 + 1/d)

### Example: the probability of a first digit of 2:

_{10}(1 + 1/2)

_{10}(1.5)

And these are the probabilities:

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

30.1% | 17.6% | 12.5% | 9.7% | 7.9% | 6.7% | 5.8% | 5.1% | 4.6% |

### Example: Sam went through a list of 100 work expenses for the year.

There was $1.95 for a pen, $4.95 for a marker, etc. Here are the counts of the **first digits**:

First Digit: | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Count: | 26 | 19 | 10 | 11 | 9 | 15 | 2 | 5 | 4 |

It follows Benford's Law pretty well.

Except there are a lot of **6**'s, because printer paper costs $6 and they buy a lot of it.

## Lotteries

Lottery numbers **don't** follow this rule, because they aren't the size or amount of anything, they are really just symbols (and a lottery would work as well using letters or pictures).

## Finding Cheaters

When people try to fake numbers they often choose the first digit randomly and end up with as many **1**'s as **9**'s.

But a computer program can go through all the numbers and count first digits to see how often a "1" appears compared to a "5" or "9".

If it looks suspicious ... watch out!

This can help uncover tax cheats, election rigging and more.

### Your Turn

Gather a list of 100 numbers from a category of your choosing. Make sure the numbers count or measure something (and aren't just symbols).

Here are some suggestions:

- House Numbers
- City populations
- Supermarket prices
- Used car prices

Find their first digits and complete this table:

First Digit: |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |

Count: |

What did you find?

### Bonus Activity

Get some friends to make up pretend shopping lists with how much each item costs. Find the first digits and put them in a table:

First Digit: |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |

Count: |

What did you find?