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The First Digit Law

Pick up an almanac, the Guinness Book of World Records or any newspaper and write down all the numbers that you find. Most people would expect that the first digits of these numbers would be uniformly distributed from one to nine. However, amazingly, you are likely to find that 30% of these numbers begin with the digit 1!

This unusual phenomenon is known as Benford’s Law, which is not a rigid mathematical law, but more of an observation. Rather than an equal 11% chance of each digit appearing as the first digit in a set of real-world data, the distribution is matches more with a logarithmic scale, with a 30.1% chance that the digit one appears as the first digit of a data point and only a 4.6% chance that the digit nine appears as the first digit of a data point.

There are reasons for Benford’s Law, but here are some logical reasons. If we take a random number and list the numbers less than it, the digit 1 will always appear the most. For example, 40% of the numbers less than 3,000,000 start with the digit 1 and while this percentage gets smaller as the number you choose gets larger, it will at the least equal that of the numbers. Another reason is that the percentage change between numbers gets smaller as you use bigger and bigger numbers. If a stock price was at 100, it would take a 100% increase to get to 200. However, if the stock was at 800, it would only take a 12.5% increase to get to 900. Thus numbers in data tend to stay within the range of numbers that start with one the most often.

Benford’s Law also has some great practical applications. It can be used to show whether a set of real-world data is authentic or not. In fact, it was used to detect fraud in the 2009 Iranian elections, where the data did not quite match up with the distribution given by Benford’s Law. So the next time someone shoves a bunch of statistics at you, don’t just gobble it up. It is simple enough to count the numbers and see whether the data could possibly be authentic.

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