Logic>
Patrick Corn, Hua Zhi Vee, Christopher Williams, and 2 others contributed
Benford's law is an observation about the leading digits of the numbers found in real-world data sets. Intuitively, one might expect that the leading digits of these numbers would be uniformly distributed so that each of the digits from 1 to 9 is equally likely to appear. In fact, it is often the case that 1 occurs more frequently than 2, 2 more frequently than 3, and so on. This observation is a simplified version of Benford's law. More precisely, the law gives a prediction of the frequency of leading digits using base-10 logarithms that predicts specific frequencies which decrease as the digits increase from 1 to 9.
This phenomenon occurs generally in many different instances of real-world data. It becomes more pronounced and more likely when more data is combined together from different sources. Not every data set satisfies Benford's law, and it is surprisingly difficult to explain the law's occurrence in the data sets it does describe, but nevertheless, it does occur consistently in well-understood circumstances. Scientists have even begun to use versions of the law to detect potential fraud in published data (tax returns, election results) that are expected to satisfy the law.
Here is a histogram of the areas of 196196 countries (data taken from Wikipedia). The units are \text{km}^2km2.Here is a table with percentages. The "BL prediction" column is the percentage that Benford's law predicts for each digit. (These numbers will be explained in the full statement of the law in the next section.)First digitNumber of countriesPercentageBL prediction15629%30%23719%18%32312%12%42211%10%5116%8%6168%7%7126%6%884%5%9116%4%
Here is a histogram of the population of each of the 3,142 counties or county equivalents in the United States (data taken from Wikipedia).Here is a table with percentages.First digitNumber of countiesPercentageBL prediction195630%30%259319%18%338012%12%430110%10%52257%8%62036%7%71776%6%81595%5%91485%4%
So Benford's law appears to predict the data in both examples quite well.
A set of 10001000 numbers between 11 and 10^6-1106−1 generated by a random number generator will produce a distribution that does not follow Benford's law. In fact, the first digits of these numbers should be evenly distributed between all nine possibilities. (Computer science exercise: write a program to generate such a set, and verify that the first digits of the results are roughly equally distributed.)