Benford's Law Quiz | Math

Quiz Answer Key and Fun Facts

1. Apart from the Newcomb-Benford law and the law of anomalous numbers, what other alternative name is Benford's law known by?

Answer: The first-digit law

Benford's law provides the frequency distribution of the first digits of all the numbers that appear in big data sets. The alternative name of Benford's law, namely the first-digit law, implies that the first digits of all the numbers are taken into consideration in producing the frequency distribution that can be explained by the law.

2. Benford's law is named after a celebrated American physicist, who is best remembered for his remarkable works in optical measurement. This scientist shares the same first name with the 32nd president of the United States and an American singer who is known for its signature song "My Way". Who is the physicist that is being described?

Answer: Frank Benford

Frank Albert Benford Jr. was born in Johnstown, Pennsylvania, in 1883. He received his tertiary education from the University of Michigan. He worked for General Electric for almost four decades.

Franklin Delano Roosevelt (1882-1945) was the 32nd president of the United States. Frank Sinatra (1915-1998) was a famous American singer and producer.

Mark Benford is the name of the main character from the TV series "FlashForward" (2009-2010). The character is played by Joseph Fiennes.

Tommy Benford (1905-1994) was an American jazz artist, known for his drumming skills.

Gregory Benford is an American astrophysicist. He also penned the science fiction novel "In the Ocean of Night" (1977).

3. Intuitively speaking, if the leading digits of all numbers in a data set are distributed uniformly, what is the expected frequency of each of the nine numbers (1-9)?

Answer: 11.1%

In general, a numerical figure can start with 1, 2, ..., 9, giving a total of nine possible combinations. If each of these nine numbers is equally likely to appear as the leading digit of a number, then the probability of each of the nine digits (1, 2, ..., 9) appearing as the leading digit in a number is given by 1/9 = 11.1%.

4. Benford's law, albeit counter-intuitive, states that this digit is the most likely to appear as the leading digit in any numbers contained in a big data set. Which lonesome number is being described?

Answer: 1

The probability of the number 1 to appear as the leading digit in a big data is 0.301. The theoretical probability values for the remaining eight digits (2, 3, ..., 9) are 0.176, 0.125, 0.097, 0.079, 0.067, 0.058, 0.051, and 0.046, respectively. It can be observed that a greater digit would result in a lower probability value.

Let d denote the leading digit of a number. A mathematical function that captures this relationship is given by P(d) = log (base 10) (1+1/d). It can also be proven that the summation of all the possible values of d, namely 1, 2, ..., 9, over the probability mass function, is 1.

5. In 1881, Simon Newcomb discovered an earlier version of Benford's law. He observed that the first few pages of certain mathematical tables were more worn compared to the later pages. This implied that the leading digits of all the numbers in big data sets are not evenly distributed across all the possible values. Which mathematical tables, whose concept was introduced by John Napier, were being described?

Answer: Logarithm

Simon Newcomb (1835-1909) was a Canadian mathematician and astronomer. He noticed that the first few pages of tables of logarithms were more worn than the remaining pages. This led him to postulate that in a big data set, more numbers would begin with the digit 1.

John Napier (1550-1617) was best known for his work in logarithm - a mathematical function that captures the inverse of exponentiation.

6. Besides the decimal numeral system (base 10), Benford's law is also applicable to numbers expressed in other bases.

Answer: True

For numbers that are expressed in decimal (base 10), the formula P(d) = log (base 10) (1+1/d) is used, where d is the value of a leading digit, and P is the probability of observing the digit d to as the leading digit of a number.

A more generalized function is given by P(d) = log (base b) (1+1/d), where d and P are defined similarly, and b is the value of the base. For example, b can be 2 (binary number system) or 16 (hexadecimal number system).

7. Which of the following data sets is the least likely to follow Benford's law?

Answer: Numbers generated from pseudo random number generators

As the name implies, pseudo random number generators (PRNGs) would generate numbers that are sufficiently random. Therefore, the generated numbers would not follow Benford's law because each of the nine digits (1, 2, ..., 9) would be equally likely to appear as the leading digit in the numbers generated.

All the other three options (population of countries, Fibonacci numbers, and atomic weight of elements) would follow Benford's law closely.

8. Benford's law has found widespread applications in different fields. Specifically, the works of Mark Nigrini, a scholar from South Africa, rely heavily on this law. Which field of application, similar to the one depicted in a 2016 movie starring Ben Affleck, does Nagrini apply Benford's law in?

Answer: Accounting fraud detection

Mark Nigrini has also authored a book entitled "Forensic Analytics: Methods and Techniques for Forensic Accounting Investigations".

In "The Accountant" (2016), the character of Christian Wolff, played by Ben Affleck, managed to identify fraudulent transactions using Benford's law.

9. Benford's law has also been used to analyze election data. A famous case study involves a presidential election held in 2009 in an Asian nation, where the number 7 was found to the main leading digits in a portion of the numbers collected during the election, contrary to what Benford's law would predict. Which Middle Eastern country, also known as Persia, is being described?

Answer: Iran

During the 2009 Iranian presidential election, Mahmoud Ahmadinejad won the election, beating his opponents Mir-Hossein Mousavi, Mohsen Rezaee, and Mehdi Karroubi.

Boudewijn Roukema from the Nicolas Copernicus University in Poland has pointed out that the figures collected for the candidate Mehdi Karroubi during the election do not conform to Benford's law's prediction.

The use of Benford's law in analyzing election data has received mixed reactions from statisticians - some applaud the innovative method, but some found it not scientifically sound.

10. Benford's law is also featured in American crime drama TV series. In the episode entitled "The Running Man", the main character, Charlie Eppes, played by David Krumholtz, used the law as an analogy to help solve a series of burglary cases. What is the name of the TV show that is being described?

Answer: Numbers

The TV series "Numbers" (2005-2010) featured the use of mathematics as a tool in solving crimes.

"The Running Man" is the title of the 15th episode of the second season of the show. The episode featured a bio-terror weapon that was stolen from a laboratory.

Source: Author Matthew_07

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Benford's Law Trivia Quiz

Benford's law gives a neat mathematical description about the frequency distribution of leading digits of numerical figures that appear in large data sets. The law, albeit incomprehensible at first, has found many useful applications in real-world data.

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