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## The Math

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

Dr. Frank Benford was a physicist with the General Electric Company who noticed an unusual numeric pattern in lists of numbers.  Legend has it that Dr. Benford noticed the logarithm pages starting with the number one were much dirtier, implying more use, than the pages whose logarithms started with later numbers.  Whether this anecdote is true or not has no bearing on the result of Dr. Benford’s finding, which is that, in certain situations, numbers are much more likely to start with lower digits than higher ones.

In lists of data such as a random sampling of today’s stock quotes, points scored per player, per year in the NBA, street addresses chosen from the yellow pages or the lengths of rivers, Benford’s law holds true.  Moreover, the law is independent of units.  For instance, it holds true for lengths of rivers measured in miles, feet, or kilometers.  So, what’s the law?

In probability, absolute certainty is defined as 1 and impossibility is defined as 0.  For Benford’s Law, the probability that an element in a data set will start with the digit n is given by

When does it work? Benford’s law only holds true for sets of data with specific attributes. One is that the data must have dimensions (ie. units). The second is the data must have sufficient magnitude, or room for the numbers to grow from small to large.  Following is a list of sets that we could collect and test for Benford’s law. Your task is to determine if Benford’s law will hold or not and explain why. Here we go…

1. The height of all 5th grade students in the U.S. measured in inches.
2. The zip codes of all American CEOs.
3. The number of "hits" a random 6-digit number has in a Google search.
4. Random numbers guessed by 100 incoming college freshmen.
5. The population, at 1 hour intervals, in a bacterial culture.

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