BENFORD’S LAW

The results were far from what they expected. The first digits of all the numbers in every set of numbers they looked at always followed the same distribution:

 

 

In other words, the number one appeared nearly 3 times more than was expected and the number 9 appeared more than 2 times less, with the other digits in a descending distribution from 2 to 8. The researchers tried to find out what this meant. What they discovered was that natural number datasets always had this distribution, without fail (obviously depending on the size of the dataset).

In fact, they realised that you could use this distribution, from that point known as Benford’s Law, to audit datasets where you may suspect that fraud was taking place in. The best case for this was the auditing of financial records. Government income and revenue offices could analyse the distribution of first number digits of accounting records to see if they followed Benford’s Law or whether another hand had been used in creating the numbers seen.

 

 

It appears that humans are very bad at selecting numbers at random. A randomly guessed number will favour higher digits skewing the distribution back towards a 11.1% probability. Humans don’t seem to select 1 as a commonly picked number. A few quizzes of work colleagues when asked to pick a random number tend to favour higher numbers. One is seldom selected, two probably even less. When deliberately cheating a numerical system, numbers like 5, 7, and 9 would be “guessed” randomly and entered onto the expenses claim. Larger numbers may also be favoured due to the desire to get a higher remuneration

Tax and revenue auditors now constantly scan returns for anomalies that are related to non-Benford conforming mathematics. Deeper audits of the returns that don’t follow Benford’s Law, with unbelievable accuracy, yield errors that have been deliberately (or accidentally) made. 

It is so accurate, that as a forensic tool, it has been used across multiple areas where auditing of suspicious datasets is required. The results of the US 2020 Presidential Election were audited using the Benford’s Law distribution to try to pick up fraud. Nothing out of the ordinary was discovered in multiple studies, suggesting that the counts were not fabricated by humans, with high precision. Both sides claimed they could prove using Benford’s Laws that the results were/weren’t fraudulent, but detailed mathematical analysis has found those claiming fraud were misusing the data – the Election, by Benford’s Distribution, was conformant.

 

So how can this be applied to people working in the tanneries and factories of the leather industry? At a minimum they can be used to check the results of financial data sets where numbers are required to be entered (particularly claiming of expenses). In the tannery, datasets must be chosen quite carefully as some of the numbers that appear would never follow Benford’s Law, e.g., final pH values like 3.8. These kinds of numbers have what is called an attractor (they must try to meet the last digit of 8) – so would naturally disfavour a Benford’s distribution, but the distribution of numbers linked to values that are not targeted, will shift to a Benford’s distribution. 

It could be that a first digit number is always fixed (or is consistent) like the area of leather so assessment of the last digit (less common in Benford auditing), but possible could diagnose whether the last digit is guessed or read. The natural distribution strangely should favour a higher percentage of lower digit numbers and is uncannily accurate in the measurement of natural systems. Intuition on the number of chicks, or offspring, a breeding pair of birds would have would naturally select for 1, or 2, and the frequency of nine chicks would follow the Benford’s distribution.

The number of first grades unfortunately do not follow this distribution (which would be nice) so there is a limit to this tool, so choose the datasets that this can be applied to carefully, but it would be interesting to see the full range of what kinds of datasets (the tanneries and shoe factories) would follow the distribution. 

 

Analysis in the factory that Benford’s Law is known to work for are as follows:

When the average of the data is greater than the median (the midpoint of a number distribution)

Data that relate to sales or paying out

Data that results from quantity and price calculations

Where numbers are assigned sequentially, numbers linked to human marketing or strategising (e.g., R10.99), those that target focussed numbers (minimums and maximums or key performance indicators), or that do span the full magnitude of digits 1 to 9, will not follow the distribution.

Mathematicians are finding all sorts of weird instances where a Benford’s Law distribution is discovered. The simplest maths problem in the world (that cannot be solved) – the Collatz’s Conjecture - has an unbelievable link to the Benford’s Law. In the problem the final numbers coming out follow Benford’s Law with exceeding precision suggesting that the Law could be a natural inherent part of the way that humans use mathematics.

Good luck in working out (using the Law)  if any of the change is missing after someone used your money to buy the round of drinks.