Benford’s law, also called the Newcomb–Benford law, the law of anomalous numbers, or 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 digit is likely to be small. In sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. If the digits were distributed uniformly, they would each occur about 11.1% of the time.
For example, consider all baseball pitchers who have won at least 100 games in their major league careers:
There is one whose number of career wins begins with a 5: 1: Cy Young (511)
With a 4: 1: Walter Johnson (417)
With a 3: 22
With a 2: 94
With a 1: 503
What’s going on is that winning a major league baseball game is difficult, so the number of men drops off the higher you count. Many social statistics look like this, following Benford’s Law.
On the other hand, if you look at career winning percentages among major league pitchers with at least 100 career decisions, only two begin with a 7: Albert Spalding, the 1870s pitcher who retired at 27 to go into the sporting goods business (a rather successful man) and Spud Chandler, the Joe DiMaggio Era Yankees pitcher. But then are 125 pitchers in the .600s, with Clayton Kershaw of the current Dodgers the highest at just below the .700.
I suspect there are more pitchers in the .300s than in the .200s and more in the .200 than in the .100s. That’s because they don’t let you stick around in the majors if you are really bad. Among pitchers with 100 career decisions, only two are in the .200s, while many more are in the .300s.
Many other social statistics looks like this, not following Benford’s Law.
On the third hand, if you want to know the maximum number of players on the roster of a major league team through August, it’s almost always going to be a number beginning with the digit 2. That’s because the rules allowed 25 players on the roster until 2020, when it was increased to 26.
And many social statistics look like this.
You see a lot of splitting or lumping due to administrative decisions.
For example, if a voting precinct grows so large that the physical facility gets overcrowded with voters and voting slows down, the precinct is likely to eventually be split into two. On the other hand, if the number of voters in a precinct turns out to be too small to justify the fixed costs of voting facility, the precinct is likely to be lumped in with another precinct.
Hence, there’s little reason to expect the number of votes recorded in precincts to follow Benford’s Law.