Benford\’s law, also known as the law of first digit, is practically used in various applications in order to identify the reliability of the numerical data. Therefore, this paper will review one of the decisive laws in mathematics that is still present and used in our daily life by many countries and authorized places such as governments.
This paper highlights the quantitative data obtained from the social media platform Instagram into an analytical graph by the use of the software application, spreadsheet.
As a result, this study included the observation of 116 accounts. Hence, the leading digit 1 was found to be the highest as predicted and following numbers were found decreasing in an irregular sequence. The dramatic decrease between the leading digit 1 (36.20%) and 2 (21.55). As well as that, the leading digit 8 is identified to be greater than both the leading digits 6, 7 and equal to the value of the leading digit 5.
Overall,the graph obtained obeys Benford\’s law, however it is essential to examine the data and assure the accuracy of the naturally occurring numbers.
Key words: Benford’s law, leading digit, data, mathematics
In 1881, Simon Newcomb was a Canadian-American astronomer. As well as that, a professor of mathematics in the United States Navy and at Johns Hopkins University. He discovered the phenomenon of one of the well-known laws that are considered essential in many applications, such as but not limited to preparing taxes, decisions, population, and even criminal justice. Moreover, it all started when the astronomer Simon acknowledged the earlier pages of the logarithm table were significantly more worn. Later then, in 1938, the Physicist Frank Benford noticed the same observation, then tested it on a variety of different data. Besides, Benford’s law could also be called Newcomb–Benford law, the law of anomalous numbers, or the first-digit law. Therefore, this mathematical law is used by a significant number of countries to identify the accuracy of valid numeric data by collecting a suitable amount of naturally occurring numbers. Then distributing them in groups matching the first digit, then calculating the data in the sub-groups of each integer. As a result, leading digit number 1 will be recognized as the greatest among others, creating the other numbers to be lower as in a pattern. That will actively illustrate that number one will be the leading digit is an absolute data set of numbers thirty percent point one of the time; the numeral two will be the leading digit seventeen percent point six of the time. Furthermore, this method is like a lie detector that identifies whether the given data is biased or not. I believe this study is objective and crucially important, and the reason for this occurrence is that it reflects the relationship of how our life is naturally related to Benford\’s law.
As shown in table 1, the data was assembled from the social media platform called Instagram. Performing the analysis involved observing random account followers which were in total of 116. After that, record the number of followers from every account that follows the examined account. Following that, the numeric data were inputted in spreadsheet software to be manipulated and organized. Then, with the use of the function COUNTIF, the total number of followers within the leading digit was calculated. Also, the actual percentage was calculated by dividing the account over the total and multiplying it by a hundred. As a result, in figure 2 a line graph is represented which illustrates the differentiation between actual percentage and Benford\’s law predicted percentage.
Overall, the leading digit 1 is considered the digit with the highest number of counts amongst other numbers and has the highest percentage which is relatively higher than the prediction by 6.1%. Moreover, the leading digit 2 recorded the second-highest percentage with a difference of 3.95%, and then the trend decreased downwards smoothly until in the leading digit 8, where it is acknowledged greater than both the leading digits 6,7 and equal to the value of the leading digit 5. However, leading digit 8 is the only digit which is the lowest value of change (actual percentage – Benford\’s predicted %) which means it\’s relatively closer to Benford\’s law predicted % from any other value in table 1. Finally, the figure 2-line graph obeys Benford\’s law.
Although Benford\’s law can be used in a various number of applications, some constraints need to be taken into consideration in order to be within the law of anomalous numbers predicted percentages:
- Numeric data
- Randomly generated numbers:
– Not restricted by maximums or minimums
– Not assigned numbers
- Large sets of data
- The magnitude of orders 
As we know, this theory has always been applied to digital analysis. Moreover, it is important to acknowledge that the large data set should be randomly generated. For instance, the example in table 1 reflects a large set of numbers ranging from different values with the presence of maximum and minimum value. Therefore, the IT team should ensure that the numbers collected are randomly generated without any real or artificial restrictions.
The IT auditor should be careful in extracting a sample and then using Benford’s Law on the sample. That is especially true for directed samples in which the amount is part of the factor allowing a transaction to be chosen. This is because the sample is not truly random. For small entities, using a data set for the whole month, or a random day of each month is a better sample for Benford\’s Law purposes.
It is essential to know that when using Benford\’s law, the results must not be considered definitive in any case; because the process of applying this mathematical law will never decidedly prove if fraud is occurring. However, the results obtained are an analytical tool that may give the observer a wake-up call to work on the additional investigation.
Several steps need to be taken into consideration to obtain reliable results:
- Reconsider the data stability: before announcing the occurrence of manipulation in the numeric data, reexamine the results, and check the possibilities of bias records.
- Consider the source: was the data obtained from reliable sources? A source with an official license? Ask these questions and take action by changing the source if necessary.
- Rethink internal controls. Consider whether reliable controls are in place to detect or prevent improprieties.
- Analytical reviews: ask some expertise and professional analytical individuals to review the data in detail. Their job also includes comparing the new set of data with an old one, consider the possibility of an unusual event occurred that resulted in the manipulation of data, and using the ratios to analyze the data.
To conclude, Benford\’s law is an essential mathematical theory that illustrates a third of the time you choose a number the leading digit will most likely be one and not nine. The law has been conducted to work with a respect to the distribution chosen from a span of loads of orders of magnitude. Alternatively stated, the data need to be collected professionally to assure the accuracy to have a reliable diagram that obeys Bedford\’s law predicted digits. The line graph will tend to follow the Benford\’s law trend only if the data is a naturally occurring number not artificial, and the analysis who is responsible for analyzing the data should be aware of the presence of bias in any case and consider them so then the appropriate actions will take place.
- AbeBooks. 2021. “A Treatise on Money: Volume I: The Pure Theory of Money by Keynes, John Maynard: Good Hardcover (1930) | Tiber Books.” Abebooks.com. 2021. https://www.abebooks.com/servlet/BookDetailsPL?bi=30750640007&searchurl=an%3Dkeynes%26fe%3Don%26sortby%3D17%26tn%3Da%2Btreatise%2Bon%2Bmoney&cm_sp=snippet-_-srp1-_-title1.
- “The First Digit Phenomenon.” n.d. Accessed January 24, 2021. https://hill.math.gatech.edu/publications/PAPER%20PDFS/TheFirstDigitPhenomenonAmericanScientist1996.pdf.
- Berger, Arno, and Theodore P. Hill. 2011. “Benford’s Law Strikes Back: No Simple Explanation in Sight for Mathematical Gem.” The Mathematical Intelligencer 33 (1): 85–91. https://doi.org/10.1007/s00283-010-9182-3.
- Kruger, Paul, and Sarma Yadavalli. 2017. “The Power of One: Benford’s Law.” ResearchGate. unknown. September 2017. https://www.researchgate.net/publication/319949720_The_power_of_one_Benford’s_Law.
- Newcomb, Simon. 1881. “Note on the Frequency of Use of the Different Digits in Natural Numbers.” American Journal of Mathematics 4 (1/4): 39. https://doi.org/10.2307/2369148.
- “Understanding and Applying Benfords Law.” 2011. ISACA. 2011. https://www.isaca.org/resources/isaca-journal/past-issues/2011/understanding-and-applying-benfords-law.
- Collins, Carlton. 2017. “Using Excel and Benford’s Law to Detect Fraud.” Journal of Accountancy. Journal of Accountancy. April 2017. https://www.journalofaccountancy.com/issues/2017/apr/excel-and-benfords-law-to-detect-fraud.html#:~:text=Briefly%20explained%2C%20Benford’s%20Law%20maintains,leading%20digit%20with%20decreasing%20frequency..
- Numberphile. 2013. “Number 1 and Benford’s Law – Numberphile.” YouTube Video. YouTube. https://www.youtube.com/watch?v=XXjlR2OK1kM&t=460s.