Benford’s Law: Real Data Behaves as a Descending Logarithmic Curve

Benford's law, also called the first-digit law, is an observation about the frequency distribution of leading digits in many real-life sets of numerical data. The law states that in many naturally occurring collections of numbers, the leading significant digit is likely to be small. For example, in sets which obey the law, the number 1 appears as the most significant digit about 30% of the time, while 9 appears as the most significant digit less than 5% of the time. By contrast, if the digits were distributed uniformly, they would each occur about 1 in 9 or 11.1% of the time.

Examples of real-life sets of numbers and how the number 1 is the leading digit in an average of 30% of the numbers, occured in the following data sets:

• area of rivers
• populations
• constants
• newspapers
• heat
• pressure
• molecule weight
• drainage
• atomic weight
• square roots
• design
• Digest
• cost data
• x-ray volts
• American League baseball
• addresses
• square numbers

We used the 2017 World Almanac and looked at real data with electoral college and popular votes for Trump and Clinton and many other pages of data, including the results of the 2012 election, football statistics, 2016 olympics, population of US cities, mathematical formulas, and most famous rollercoasters. The Almanac has 1008 pages of data. The children were able to prove Benford’s Law worked with any of these pages.

The tally sheets allowed them to see that the distribution of numbers were not equal as with random numbers but logarithmic. Numbers with a leading digit of 1 occurred one in three times or about 30%, 2: one in five times or about 18%, 3: one in seven times or about 13%, 4: one in eleven times or about 9% and continued to descend until numbers with a leading digit of 9 occurred only one in twenty times or about 5%. 

This phenomenon is still used 80 years later by the Federal Bureau of Investigation to identify fraud. Since numbers that are fabricated (a lie) tend to be created randomly, they behave like the numbers we looked at last week, about one in nine for each leading digit of 11%. So if the data behaves randomly and not like Benford’s law, the FBI concludes that the data is fraudulent.

The children should pick up a newspaper and look at the sports page, the weather page, the stock market page or any other page of numbers and test for Bedford’s law. This phenomenon is so consistent.