Benford's law also known as the first-digit law deals with the distribution of natural numbers in which it tends to major into their importance regarding significance. The law states that given any arrangement of natural numbers, the smallest number always seems to be the most significant while the largest number in the collection is always the least significant. Therefore, number 1 appears to be the most significant in any given combination of natural numbers 30% of the times. While on the other hand, number 9 appears only 5% of the times as the most significant number. The law states that the probability of the leading number been of certain value is given as indicated below.
Consequently, the probability of the second digit from logarithmic population is given by the equation shown here.
Subsequently, the equation will undergo several changes up to the nth value of which the equation will be as below.
The law is very interesting since it claims that once the numbers are in the uniform arrangement, they will all have equal chances of significance at 11.1%. Additionally, Benford's law predicts the distribution of second digits, third digits and many others including digit combinations (Miller 6). This law has some applications in the real world whereby its uses are possible to experience in various fields globally. The law is applicable in the street addresses, determination of electricity bills, stock prices, and lengths of rivers, determining population growth, and many other instances where natural numbers are involved. From Benford’s law the probability of the first digit been a certain value is as shown in the table below.
History of the Benford's Law
Furthermore, the law goes back to the 1880s in which the American Astronomer Simon Newcomb discovered the law in a rather funny and interesting manner (Kossovsky & Alex 21). The scientist derived inspiration for discovering the law from common logarithmic tables. He noticed that the tables' pages worn out in a peculiar order in which the first pages were the most drastically affected by the condition. The first pages were older in the real sense than the last pages of the tables.
Noteworthy is that the basic observation prompted the scientist to conduct an investigation into the manner in which objects involving numbers rearrange themselves on any given scale (da Cunha et al. 2). He then published the very first law concerning the probability of a number becoming the most significant at any given instance. Therefore, Newcomb first law of any number N being the first number is expressed as log (N + 1) - log N
This phenomenon was again identified in the period around 1938 the much-known physicist Frank Benford. I particular, the scientist received diversified credits with the discovery of the first-digit law owing to his testing of the digits by working on 20 different domains. The data set Benford used in his quest for the correct definition of the law to include though not limited to surface areas of 335 rivers, 1800 molecular weights, 5000 entries obtained from some random mathematical handbooks and 418 death rates.
Frank Benford extended his discovery of the law from 1 digit to 2nd, 3rd and even higher digits using a simple formula. The formula is as shown below in which d is any digit from 0 to 9 and k on is the position of the digit.
Consequently, the digital frequencies that will probably arise from the above equation are as in the table below.
Mathematical Statement
Expression of Benford's law exists in the form of theoretical explanation as well as the mathematical form. Mathematically, it is expressible in a very simple but rather inclusive manner hence expresses volumes of meaning within the mathematical context (Fox, & Theodore 3). Any set of digits is taken to be satisfying Benford's law if the leading digit d [d is approximately equal to (1,,,, 9)]
P(d)= log10 (d+1)- log10(d) =log10(d + 1/d) = log10 (1 + 1/d)
This mathematical expression of the law helps any individual in understanding the concept behind the law whenever any valid explanation is required (Miller 8). For instance, the numbers are normally given to a distribution involving the categories between 1 and 9 hence a thorough analysis of the numbers will be very easy to undertake (Barjaktarović et al. 701). This is because the law majorly aimed at dealing with such numbers hence cases like those for example population densities and death rates are easily manageable.
This mathematical expression of the Benford law is very crucial because it enables users to express and utilize the law in its simplest form. The law when expressed mathematically is useful in a variety of cases to estimate the accurate results and output required to make valuable decisions in the day-to-day life. For example, the law proves to be of significant importance when used to examine the height of the top 60 tallest building in the world. The example below proves that indeed the number 1 is the most significant number of all.
Multiplicative Fluctuations of the Benford's Law
The law states that a variety of numbers usually increases by a certain amount, which is possible to derive by multiplying the original number by some random value, say 0.99 or 1.01. This feature of the natural numbers is usually applicable in the field of the stock exchange in which the closing figures differ from the opening value by some very minute differences (Miller 12). In that case, therefore, the Benford's law will be of very great importance because it will assist in the prediction of the possible probability of the next value.
The multiplicative phenomena of certain natural numbers are unique on its own since it leads to the extraction of the Benford's law unlike the additive feature of numbers that later leads to the probability distribution. The uniqueness of the multiplicative feature of natural numbers lies in the logarithmic property of the numbers. The logarithm of such numbers usually undergoes a random walk hence over some period the probability of its distribution will become rapidly smooth and broad.
Applications of the Benford's Law
Accounting Fraud Detection. In real life, human beings tend to arrange numbers in certain orders and distribution likewise in accounting professionals arrange the order of the accounting digits according to specific arrangements (Kossovsky & Alex 81). The arrangement can take the form of natural numbers and hence, in the end, there must be some form of order in the manner in which data collected and presented is brought forward. The Benford's law will then be of help in determining whether there is any form of anomalous results indicated in the presentation.
Frauds are always committed in some form of suspicious arrangements in which the people associated with the crime tend to hide the fraud in some form of mathematics. New figures will finally have to be part of the original and perhaps final data in the books of accounts. On the other hand, Benford's law states that natural numbers must always indicate some form of arrangement and distribution as per the indication of the law. Any disagreement between the Benford's law and the financial data provided will then lead to an investigation into the reliability of the source of the financial data.
Benford's law has application in auditing and forensic accounting as a tool used to detect fraud expenses and accounting mistakes (Kossovsky & Alex 81). However, in many of these accounting applications, the concept involves utilizing more than just the first digit principle.
Election Data. Benford's law is also applicable in detecting election frauds and rig as it can compute the distribution of the data presented as the final tally of election results. The law was for instance, practically used in the Iranian election fraud in which massive rigging was witnessed and helps a lot detecting flaws within the election management criteria. The votes counted are placed in the specific arrangement and a clear method used to categorize the votes into clear and elaborate blocks.
Benford's Law in Price Digit Analysis. The law is usually present in the calculation and approximation of the prices of goods and services. The basic principle applicable to the determination of the final prices of goods is simple since it incorporates the logic first digit, second and third digit nominal market prices. The Benford's law utilization in the marketing field was rather a huge step in curbing unnecessary expenses incurred to investigate market changes (Barjaktarović et al. 696).consequently; the Benford law has been of very particular importance in aligning the price of commodities with the changing value of currency. For example in the countries that incorporate the use of dollar or rather a euro, it will be necessary for the market experts to equip themselves with the basic concepts of the Benford's law.
Benford's Law in Auditing. Benford's law has made possible for auditors to shift their expertise from the previous mainly manually operated audit exercise to modern day digital auditing. The digital auditing utilizes technology in contacting even the most basic activities in the industry (da Cunha et al. 1170). The technology applied in this auditing is the application of Benford's law in detecting fraud and gross mismanagement of shareholder's money in both large and small companies. For example in the case below, Benford’s law has applied in analyzing a particular fraudulent action done by an individual with intentions of defrauding the state of millions of moneys:
Benford's law claims that the most significant number in the naturally occurring distribution of numbers is number 1. Therefore, considering, for instance, some form of investment in stock exchange it will be possible to derive this fact right into the understanding of all the intended people. For example, if a person has $100 to invest at let say 5% annual return then it will take him or her about 15 years for the value to stand at $200. That is to say, the first digit changes to 2 while it will be approximately eight years for the first digit to be 3. Subsequently, an additional six years to change the first number to 4. The formula that is defines the pattern followed by the above phenomenon is as shown below.
nd = log {(d+1)/ d} / log 1.04 where nd is the number of years taken to move from (d) dollars to (d+1) dollars. Likewise, the percentage of time spent on the first digit is expressed as in the equation below considering n as number of time it takes to move from $1 to $10, then 1. (1.04)n= 10
{log (d+1)/ d /log 1.04} / {log 10 /log 1.04} = [log d+1 /d] /[log 10] = log10 {d + 1 /d}.
The above interesting pattern leads to some form of conclusion in regards to Benford's law concerning the behavior of natural numbers regardless of the situation at hand (Barjaktarović et al. 550). That is to mean, even when the first investment amount was $1000 it will take the same period to change the first value to just exactly the one above it. It is this unconstrained pattern of numbers that makes it possible for the Benford's law to be utilized in auditing to detect frauds in books of accounts.
Psychological marketing and analysis of natural numbers claims at some points that some individuals usually tends to pick the mid numbers whenever they are obliged to do so. The mid numbers between 1 and 9 are (5, 6, and 7). A thorough analysis of these three numbers is carried out to determine the possibility of any form of fraud occurring. Since they are the most likely numbers to picked, fraudsters will prefer picking the numbers in cases where cooking of figures is essential.
Constraints of the Benford's Law
Generation of Numbers should be Random. Benford's law states that the numbers should be random which is to say that there should be no any form of constraining or external interference to the source of the numbers. In so doing the law will not be applicable in cases where the data to be analyzed has some form of minimum and maximum limits. Due to this basic constraint of the Benford's law, all data to be analyzed using Benford's law should always be from a trustable source that does not have any form of constraining whatsoever.
Data such as hourly wages demand that there must always be some minimum limit to the amount a person is payable on a daily basis. It is also mandatory to have the upper limit been the maximum wage a person can earn per day per the specific conditions of the job. In such instances, the Benford's law will not be of any importance due to the inability of the data generation to be random.
Another very common example of the data involving constrained numbers is the United States telephone numbers that usually consist of three accurately designed numbers and four last randomly selected numbers. This phenomenon will not be possible to apply the Benford's law on it because of the human interference with the first numbers. It is, therefore, necessary for the user of the Benford's law to first assess the numbers to ensure that its generation is random before applying the law to them.
The Data Set Should Be Large. The Benford's law applies to a large amount of data such as large quantities of invoices to customers and some disbursements. The law will not be applicable in cases where the data set at hands is small hence will be limited to only large companies(Park et al. 286). The recommended size of data set is 1000 records or more for the law to function efficiently.
Applicable to Non-Uniform Numbers Only. Non-uniform distributions obey the Benford's law from their random selection from a large group of existing numbers. On the other hand, uniform distributions do not obey Benford's law because such numbers have some limitations regarding its order and boundaries. It is, however, worth noting that the ratio distribution of two uniform distribution data perfectly obeys the Benford's law (Fox & Theodore 5).
Conclusion
In conclusion, the Benford's law has proven beyond reasonable doubt that it is essential in the day-to –day activities across various sectors of an economy. The law has undergone some improvements right from its inception in the 1880s to its modern day utilization in the various sectors outlined above. Benford's law is crucial in applications involving naturally occurring numbers usually between 1 and 9 especially in numbers selected randomly. It is this feature of the Benford's law, which has made it possible for the drastic reduction in the fraud committed by the various departments within the accounting field.
The mathematical representation of the Benford's law makes it very simple to use in the computing of distribution frequencies as well as in determining the required calculation involving numbers. The law applies to various sectors in the auditing, election, and marketing owing to the fact all these areas involve the use of random numbers. The law, however, has some limitations in its applicability in the various sectors. The limitations or rather constraints are not so detrimental to the application of the law since they still have some allowances for the applicability of the Benford's law.
Works Cited
Barjaktarović, Lidija, Marko Milojević, and IvicaTerzić."Results of applience of Benford's law on serbian companies." OF MACEDONIA 568 (2014): 696.
da Cunha, Flavia C. Rodrigues, and Mauricio S. Bugarin. "Benford's Law for audit of public works: an analysis of overpricing in Maracanã soccer arena's renovation." Economics Bulletin 35.2 (2015): 1168-1176.
Fox, Ronald F., and Theodore P. Hill. "Hubble's Law Implies Benford's Law for Distances to Stars." arXiv preprint arXiv:1412.1536 (2014).
Kossovsky, Alex Ely. Benford's Law: Theory, the General Law of Relative Quantities, and Forensic Fraud Detection Applications.Vol. 3.World Scientific, 2014.
Miller, S. "A quick introduction to Benford's Law." (2015): 3-18.
Park, Junghyun A., Minki Kim, and Seokjoon Yoon."Evaluation of Large-scale Data to Detect Irregularity in Payment for Medical Services." Methods of information in medicine 55.3 (2016): 284-291.
