Chasing Benford’s Law down an election rabbit hole

Benford’s Law is a fun statistical phenomenon that this blog has explored a couple of times, most notably here. Benford has gained a sudden new popularity among 2020 election conspiracy sites, alleging huge vote rigging, but only in states where Donald Trump lost. However, this technique is invariably misused and misunderstood in these applications, and so, this is my attempt at some clarity.

We will start out with a good application of Benford’s Law, but if you want to skip right to the bad election vote-counting use, just drop down to the second header.

A good application of Benford’s Law before the bad one

Benford’s law says that, in certain random populations (that “certain” is key here), the first digits of the observed numbers take on a seemingly non-random characteristic. “First digit” means that whether the value is in the range of 10 to 19, 100 to 199, or 1,000,000 to 1,999,999, you will count that observation as a “1”. The same rule applies to the other eight digits as well (90-99, 9000-9999, etc., all count as “9”).

When you do that extraction of the first digit, you will likely (and probably unexpectedly) find more ones than twos, more twos than threes, and so on, down to having more eights than nines. In fact, the percentage of occurrences of each digit will be nearly constant as well. That seems counter-intuitive to most people. We expect “randomness” to be less predictable and more uniform.

I recently came across a database of populations of towns and cities in my birth state of Michigan. There are 1773 communities in that list, ranging from Pointe Aux Barques, at the tip of Michigan’s famous “thumb” with a population of 10, to Detroit, which is listed in this data set as having a population of 713,777. If I apply that “first digit rule” to this data set and then graph my results against the Benford’s Law predicted count of each first digit, I get this graph, which is a pretty good Benford prediction:

Benford's Law - Michigan Towns

Benford’s Law predicts that 30.1% of the populations will start with the digit 1, and we overshoot a tad here with 32.5%. Benford predicts 4.6% of the towns will start with 9, and Michigan comes up with 4.3%, pretty darn close. 6 seems low, but you can compute a “margin of error” for Benford and it probably still fits.

The quick answer as to why Benford’s Law works here is that town populations, just like populations of coronavirus outbreaks, each grow (or sometimes shrink) at roughly an exponential rate, say, 5% per year, rather than a linear rate of, say, 1000 people per year. If you think about it (which may not be advisable) a certain percentage of people in the town are “reproducing” at any one time, just like the coronavirus is.

Benford’s Law also works with a lot of financial reporting data for a similar reason. Money, in large (or small) aggregations like your IRA balance, also grows or shrinks exponentially. The math here is commonly called the time value of money, and so “present and future value of money” computations are key to every college finance course. Benford’s Law also makes for a good auditing tool. If you try to “cook the books” using random numbers, and do not take Benford probabilities into account, your fake numbers will stand out like a sore thumb, as they did for Bernie Madoff.

The appropriate model for measuring of an exponentially-growing population is not a linear ruler, but rather the sorely missed slide rule, which is based on a logarithmic scale where the physical distance between the smaller numbers is greater than the distance between the larger ones.

Slide Rule

If a town’s population is growing at 10% per year, it will take over seven years to double from, say, 1000 to 2000 people (1100, the first year, 1210 the second, 1331 the third, etc.). But if the town has only 800 people, it will reach 900 in just a little over one year (880 the first year, then 968 the second year). Humans invented a linear counting scale based on the number of our fingers and toes, but nature more often counts exponentially. It is the human counting scale that is “re-using” digits at a non-constant “Benford” rate, mostly ones and twos; nature does not re-use digits.

Chambered nautilus

Source: Wikipedia

Politics makes for bad Benford’s bedfellows

It would seem that vote totals in states or precincts might also show some Benford characteristics, and when conspiracy theorists found that their graphs did not match the blue line in the Michigan example above, they started screaming “vote rigging.” I first became aware of this because of increased traffic to a Benford post on my site on election night 2020 and the days following. One sports fantasy league discussion site seemed to especially be pushing the theory of vote rigging in favor of Joe Biden, “proven” by Benford’s Law.

This particular site was particularly concerned with what appeared to be “bumps” in the Benford graph, where certain digits occurred “too many” times in selected vote counts. It must be vote fraud! So, let’s look at some past voting data to see how an uncontested election looks in a Benford analysis.

First of all, you need a lot of data to see (or not see) a Benford trend. Fifty state vote counts, or even 100 counties, won’t cut it in order to demonstrate a Benford trend or problem. Well then, how about precinct level data? Let’s try that. The state of Iowa has a nice data set of state-wide votes by precinct for the 2016 Clinton-Trump presidential election. I think that was a relatively clean race in Iowa. This set shows the vote totals in each of 1680 voting precincts in the state. They show votes for Hillary Clinton ranging from 6 in one precinct up to 1644 votes in her best precinct. Donald Trump had a range of precinct votes from 27 to 1894.

Here is Hillary Clinton’s Benford graph, and you can see noticeable “bumps” at “5” and “8”:

Benford's Law - Clinton 2016

What is going on here? My take is that precinct sizes are not great Benford data sets. When precincts get too big, cities break them up into multiple new precincts per community. Thus, precincts do not have as nice a clean random size as the entire town populations would. We see some correlation, but not close enough to base any conclusions upon, especially voting conspiracies. Certain digits will inevitably seem over- or under-represented.

Indeed, the Trump graph in the same election is worse:

Benford Trump Iowa results

In fact, it is just not correlated to the Benford probabilities much at all. My take of the data here is that, while Hillary Clinton may have captured votes in the 100-199 range in the many small rural precincts in Iowa, Donald Trump was more likely to get 300 or 400 votes in the same precincts, and thus those numbers are over-represented. Again, precinct-level votes are not good Benford data. When we get better 2020 statistics, we will likely find similar non-conspiracy data anomalies.

Unless, there was some huge voting conspiracy in Iowa back in 2016 that brought Donald Trump into office and we all missed it! Hmmm…..

The reality is that if all states had counted votes the way Florida did, with early and mail votes pre-processed for faster vote counting, the election would have been called as a Biden blow-out by late on Tuesday night, with a five-million-plus popular vote edge and a 50+ point electoral college difference. We would have all slept better.


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