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## Introduction (cont.)

•  Introduction  •  Introduction (cont.)
•  Benford's Law and the IRS  •  The Math  •  The Answers

In the following graph, we have what your bar graph should look like plotted alongside what Benford’s law predicts for the data set.

The following table presents the results of Benford’s Law.  The “First Digit” refers to the leading digit in any number. The “Chance of Occurrence” is how often one can expect a number in a set of particular data to start with the indicated digit.

 First Digit 1 2 3 4 5 6 7 8 9 Chance of Occurrence 30.1% 17.6% 12.5% 9.7% 7.9% 6.7% 5.8% 5.1% 4.6%

Does this surprise you?  Shouldn’t the digits one through nine be distributed evenly, with no digit having any more chance than another of showing up first? Actually, many naturally occurring sets (including tax returns) diverge from this and follow Benford’s Law.

Why is this so?  Benford’s law can be thought about in terms of percent change. For instance, for \$100 to increase to \$200, it takes 100% growth, but from \$200 to \$300 it is 50% growth, and \$300 to \$400 is 33% growth.  As we get to \$900 to \$1,000, it only takes 11% growth.  When we start back at \$1000, it is again 100% growth until we get to numbers that start with 2. (Remember, we’re only concerned with the first digit.) This pattern can be repeated over and over again and will always have the leading digits dominated by 1, then 2, then 3, and so on.

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