### Fooling the IRS - Benford’s Law

Sorry for once again posting about a math related issue. I’ll try some biology or physics next time…

Let’s say you think the IRS shouldn’t take so much of your hard earned money. So, you sit around and just fake the numbers. Do you think you can get away with it?

Benford's Law, also called the first-digit law, states that in lists of numbers from many real-life sources of data, the leading digit is 1 almost one third of the time, and larger numbers occur as the leading digit with less and less frequency as they grow in magnitude, to the point that 9 is the first digit less than one time in twenty. This is the distribution of the numbers according to their first digit:

1 - 30.1%

2 - 17.6%

3 - 12.5%

4 - 9.7%

5 - 7.9%

6 - 6.7%

7 - 5.8%

8 - 5.1%

9 - 4.6%

This is based on the observation that real-world measurements are generally distributed logarithmically, thus the logarithm of a set of real-world measurements is generally distributed uniformly. This counter-intuitive result applies to a wide variety of figures, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature). The result holds regardless of the base in which the numbers are expressed, although the exact proportions of course change.

Real numbers on tax reports follow this pattern, but since this law is very counter-intuitive, people that fake the numbers almost always create different patterns. Many people will create an almost even distribution of first digits, or make up more numbers that start with the digits 4, 5, or 6. These patterns can be easily tested, and the fake report will immediately pop up, suggesting tax fraud. So what do you do when you wanna fake it? You Benford it!

~~~

p.s. They think they proved the Four Color Theorem; I say let them try it on the Middle East.

Benford's Law, also called the first-digit law, states that in lists of numbers from many real-life sources of data, the leading digit is 1 almost one third of the time, and larger numbers occur as the leading digit with less and less frequency as they grow in magnitude, to the point that 9 is the first digit less than one time in twenty. This is the distribution of the numbers according to their first digit:

1 - 30.1%

2 - 17.6%

3 - 12.5%

4 - 9.7%

5 - 7.9%

6 - 6.7%

7 - 5.8%

8 - 5.1%

9 - 4.6%

This is based on the observation that real-world measurements are generally distributed logarithmically, thus the logarithm of a set of real-world measurements is generally distributed uniformly. This counter-intuitive result applies to a wide variety of figures, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature). The result holds regardless of the base in which the numbers are expressed, although the exact proportions of course change.

Real numbers on tax reports follow this pattern, but since this law is very counter-intuitive, people that fake the numbers almost always create different patterns. Many people will create an almost even distribution of first digits, or make up more numbers that start with the digits 4, 5, or 6. These patterns can be easily tested, and the fake report will immediately pop up, suggesting tax fraud. So what do you do when you wanna fake it? You Benford it!

~~~

p.s. They think they proved the Four Color Theorem; I say let them try it on the Middle East.

## 2 comments:

Fascinating!

I didnt know that . After reading your blog , i went ahead and checked my telkephone bill , electricity bill . Thye both started from 2

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