your Favorite social media platform Pay attention to how many friends and followers you have. Specifically, pay attention to the first digit of this number. For example, if she has 400 friends, the first digit is 4, and if she has 79, it is 7. Let’s say you asked a lot of people to do this. Common intuition suggests that the number of friends should be somewhat random, so the numbers 1 through 9 should be treated equally, and the leading digits should also be random. Oddly enough, this is not what we find. Instead, we see a sharp imbalance. half Only 10% of people start with 1 or 2 friends, while others start with 8 or 9.
This strange over-representation of 1 and 2 extends beyond friends and followers to likes and retweets, and far beyond social media, to the population of countries, the length of rivers, the height of mountains, and deaths. It spans countless corners of the world of numbers, including rates, stock prices, and even diverse collections.numbers found in a typical issue of Scientific AmericanNot only are lower leading numbers more common, but they follow an exact and consistent pattern.
As might be expected, if all digits are represented equally, they each appear 1 in 9 (approximately 11.1%). But in a staggering number of real-world datasets, a staggering 30.1% of entries start with 1 and 17.6% start with 2. This phenomenon is known as Benford’s law. The law persists even if you change the units of the data. Measure rivers in feet or furlongs, stock prices in dollars or dinars, anyway These exact ratios of leading digits persist as you measure. Mathematicians have proposed some clever reasons for why the pattern appears, but its simple ubiquity evades simple explanations.
It may seem like a mild observation, but Benford’s law has been used to great effect to put people in jail and detect large-scale fraudulent activity.
Before calculators, people outsourced their hairy calculations to reference books called tables of logarithms. In 1881, astronomer Simon Newcomb noticed that the first page of the table of logarithms (corresponding to numbers starting with 1) was dirtier and worn than the later pages in the early days. He speculated that lower leading numbers must be more common in natural data sets, and published the correct percentages. Physicist Frank Benford made the same observation in his 1938, and to demonstrate its universality he compiled over 20,000 data points to popularize the law. Aside: Benford’s eponymous work is an example of Stigler’s Law, which claims that no scientific discovery is ever named after its original discoverer. It was claimed by Robert K. Merton long before Stephen Stigler got the name.
Benford’s Law is more than just a statistical oddity. FINANCIAL, his adviser Wesley Rose, threw investors out when prosecutors argued in court that his documents did not match the expected distribution of leading numbers and were therefore likely fabricated. Convicted of fraud. This principle later helped computer scientist Jennifer Golbeck discover his network of Russian bots on his Twitter. She found that for most users, follower counts followed Benford’s law, but artificial accounts deviated significantly from the pattern. She used similar methods to catch people buying fake retweets. There are many examples of the Benford method being applied to fraud detection. For example, Greece manipulated macroeconomic data in its application to join the Eurozone, and voter fraud in the 2009 Iranian presidential election. The message is clear: organic processes generate numbers that favor small numbers, but naive methods of falsifying data do not.
Why does nature produce a shortage of nines and an excess of ones? First, it is important to state that many data sets do not conform to Benford’s law. Adult height, measured in feet, most often starts at 4s, 5s, 6s. The roulette wheel has the same chance of reaching numbers starting with 1 and numbers starting with 2. This law is more likely to arise from data sets spanning several orders of magnitude that evolve from some kind of random process.
Exponential growth is a particularly intuitive example. At first he imagined an island inhabited by 100 animals. That number doubles every year. 200 after 1 year, 400 after 2 years. I’ve already noticed something odd about the leading digits. For the entire first he-year period, the first digit of the island’s population size was 1. Meanwhile, year two population numbers spanned the 200s and his 300s over the same period, with less time for each major population. The reigning digit. This continues, with 400-800 in year three, with the higher digits retiring even earlier. Going from 1,000 to 2,000 requires a doubling, while going from 8,000 to 9,000 is only a 12.5% increase, and the trend resets with every new digit increase. There is nothing special about the parameters chosen for the island example. For example, you can start with a population of 43 animals and grow it by 1.3 times a year, with the same exact pattern of leading digits. Almost all of this kind of exponential growth tends towards Benford.
The law’s stubborn indifference to units of measurement gives another hint as to why this pattern is so common in nature. River lengths follow Benford’s law whether recorded in meters or miles, but converting non-Benford-compliant data, such as adult height, to meters results in the leading digit of The distribution changes radically. (Surprisingly, Benford that’s all Distribution of leading digits unaffected by such unit changes. ) You can think of changing the units as multiplying all the values in the data set by a certain number. For example, multiply a set of lengths by 1,609.34 to convert from miles to meters. Benford’s law is actually resilient to much more general transformations.Take and multiply Benford-compliant data each entry by different Using a number that is independent of the data (rather than a fixed number like 1,609.34) keeps the distribution of the leading digits intact. This means that when natural phenomena arise from the product of several independent sources, one The overall result is that Benford’s Law is cannibalistic. This is roughly the same as multiplying many numbers, only one of which must be zero for the result to be zero overall.
These explanations explain the many occurrences of the pattern, but do not explain the reason for the diverse collection of numbers drawn from a single problem. Scientific American Illustrate Benford’s law. These numbers don’t grow exponentially and you’re not multiplying them. Mathematician Ted Hill discovered what many consider to be the definitive proof of the law of leading numbers. His argument is unfortunately very technical, but the simplification is that if you choose a large number of random numbers from a large number of random data sets (probability distributions in mathematics terms), they tend to approach Benford’s law. There is. In other words, we’ve seen countless datasets exhibit Benford’s pattern, but the most reliable way to achieve it is to pull numbers from a variety of sources, such as those you see in newspapers. to pull it out.
I’ve spent a lot of time thinking about Benford’s Law.Despite the tapestry of explanations, it still amazes me how often it occurs. You may start noticing it when you pay for .
This is an opinion and analysis article and the views expressed by the author or authors are not necessarily Scientific American.
