Use Of Benford's Law In Fraud Investigation - Essay Example

USE OF BENFORDS LAW IN FRAUD INVESTIGATION Table of Contents 3 Introduction 4 Benford’s Law – Percepts and Background 5 Applications of Benford’s Law 10 Significance and Use of Benford’s Law in Detecting Accounting Fraud 11 Benford’s Law in Fraud Detection – Cases 15 Implications of Using Benford’s Law 17 Conclusion 21 References 23 Abstract The application and potential use of Benford’s law in detecting fraud, particularly in accounting and audit situations, is gaining popularity in the recent years. The law is based on a simple observation that lowers digits appear more frequently than others in certain data sets. The technological developments in the past years and the validation of the law by eminent mathematicians, coupled with the increasing instances of white-collar crimes have resulted in considerable research in the area; a review of the research including real life applications, suggests that Benford’s law is an effective tool in investigating fraud. The study presents the background and percepts of Benford’s law and its limitations and application of the law. The central purpose of the research is to understand the potential use of Benford’s law in fraud investigation and detection, with special emphasis on detecting fraud in accounting data. Real life cases demonstrating the use of Benford’s law in detecting fraud is also presented. The study further explores the implications, including possible errors and limitations, while applying the law in accounting and audit situations. In short, the paper presents the usefulness of Benford’s law in detecting fraud, and cautions auditors of the need for caution and prudence in using Benford’s law in fraud investigation. Introduction Science and technology has impacted almost every aspect of human life and business, any further discussion on the topic would seem trite. Yet, when old scientific principles present innovative and potential applications in important business situations, the academic community cannot underestimate the need to understand them and their potentialities, not to mention the cost and consequences of applying them. One such case is the potential application of the Benford’s Law in fraud investigation and detection in corporate accounting systems. A simple statistical law devised in the 1920s and 1930s and first observed by mathematicians such as in 1881, Benford’s law, helps to scientifically predict the occurrence of a digit in a data set. The probability distribution given by the law suggests that certain digits will show up more often than others in certain sets of data – digit “1” should appear as the first digit about 30-31 percent of the time, “2” as the leading digit about 17-18 percent of the time and so on, the percentage gradually reducing to less than 5 percent for “9” as a leading digit. The law works almost as a gimmick for certain random data sets, negating the native logic that given the basic numbers 0 to 9, the probability of the occurrence of each number in data sets should be equal. While the law was mostly applied by statisticians, physicists, engineers and mathematicians in analyzing data sets in the past, the potentialities of the law and the advanced applications of technology in businesses, coupled with the increasing incidences of accounting and auditing frauds as exposed by the Enron debacle and many others in the recent years, are facilitating accountants and auditors in applying the law to investigate and detect fraud in accounting and auditing situations. Durtschi, Hillison and Pacini (2004) assert that, if used properly, Benford’s law, which facilitates digital analysis on transaction level data rather than aggregated data, can assist auditors by identifying specific accounts in which fraud might reside so that they can then analyze the data in more depth. Others like Dr. Theodore Hill and Dr. Mark Nigrini are also understood to be stanch proponents of the application of Benford’s law in fraud detection. It is understood that, today income tax departments of several nations and several states, including California, as well as a multitude of large companies and accounting are using fraud detection software based on Benford’s Law. As a prelude to understanding the significance of the law in fraud investigation, it would be imperative to understanding the law and its percepts, including the limitations. Benford’s Law – Percepts and Background Before presenting the Benford’s law in its mathematical form, an understanding of the background of the law and its development would be worthwhile and add to the perceptive application of the law in significant situations. The statistical representation of Benford’s Law, as expressed later in the section, was the result of simple, yet discerning observation by Dr. Frank Benford, during his career as a physicist at the General Electric Research Laboratories in Schenectady, New York Company, during the 1920s and 1930s. During these years, physicist and mathematicians were largely dependent on logarithm tables – tables expressing number as exponents of some base, such as 10 and vice versa -- for faster computations. [Becker, 2000] By 1938, Benford noticed that the first few pages of his logarithm book corresponding to numbers starting with the numeral 1 and other low digits were more worn and dirtier as compared to the last pages which gave the logarithm of higher numerals. The observation obviously led to the conclusion that logs of numbers starting with lower digits were frequently encountered in a data set, nonetheless, it proved unscientific to conclude that scientists and engineers had a special preference for logarithms starting with number 1. The realization instigated the scientist in Benford to explore further and to formulate the mathematical phenomenon underlying the observation. [Benford, 1938] In attempting to do so, Dr. Benford performed a mathematical analysis of the first digits of 20,229 sets of numbers, gathered from 20 lists of data gathered from diverse and discrete sources including scientific, geographical and demographic data including the areas of rivers, baseball statistics, numbers in magazine articles such as Reader’s Digest and the street addresses of the first 342 people listed in the book “American Men of Science.” [Benford, 1938] Benford found that the different set of numbers, despite their incongruence in number and nature, presented similar first-digit probability pattern as that suggested by his logarithm tables. His analysis revealed that about 30.1 % of the numbers had 1 as the first digit, 17.6 % had 2, and only 4.6 % had 9 as a first digit. [Benford, 1938] Based on scientific assumptions about the distribution of naturally occurring data and, applying principles of integral calculus, Benford computed the expected frequencies of the digits and digit sequences. He further derived a mathematical/ statistical formula for predicting the frequency of the numbers in numerical and statistical data sets. Mark J. Nigrini provides tabulation [Figure. 1] of the expected frequencies of the digits in the first four positions. The chart shows a greater chance of occurrence of low digits in the first position, in any number; the probability that the first digit is 1, 2 or 3 is 60.2%. Benford’s Law As suggested, based on the frequency analysis, Dr. Benford devised a mathematical formal to determine the probability of the occurrence of a number in the data set. [Benford, 1938] He explains that if absolute certainty is defined as 1 and absolute impossibility as 0, then the probability of any number “d” from 1 through 9, being the first digit, in many real world applications, follow the distribution: P (d) = log10 (1 + 1/d) for d = 1, 2, 3, …,9. The probability distribution gives P (d = 1) = log 2 = .301, P (d = 2) = log(3/2) = .176, on up to P( d = 9 ) = log(10/9) = .046, and explains the high expected frequency of lower digits and low expected frequency of higher digits. Though the law became known by Dr. Benford’s name, it is interesting to note that astronomer and mathematician Simon Newcomb made the same observation and deduced the probability pattern in 1881, however received little attention due to the apparent unscientific nature of the observation. [Durtschi, Hillison and Pacini, 2004] Apparently Dr. Frank Benford’s observation was independent of Newcomb’s observation and he went further to establish the probability through empirical analysis. However, despite the expediency of the law and its verification, the scientific community relegated Benford’s law also for a long time, as solid probability proof remained elusive. [Becker, 2000] Theodore P. Hill, Professor of Mathematics at Georgia Institute of Technology, analyzing the probability distribution of Benford’s law in the early 1990s has validated the rule through rigorous mathematical proofs, granting Benford’s law the due credibility. Hill demonstrated the applicability of the Law to stock market data, census and certain accounting data [Hill 1995]. Hill observed that Benfords distribution, like any normal distribution, is an empirically observable phenomenon, and confirmed that the data sets that conform to Benford’s distribution are second-generation distributions –those derived from combinations of other distributions. This implies that while the individual distributions may not strictly follow the law, the digit frequencies of the combined samplings -- random samples taken from randomly selected distributions – would match Benford’s probability pattern. [Hill 1998] As Hesman observes, combining unrelated numbers gives a distribution of distributions, a law of true randomness that is universal [Hesman 1999, Cited Durtschi, Hillison and Pacini, 2004]. Prof. Hill observed that Benford’s law was base invariant and Rutgers University Professor Roger Pinkham had, in 1961, already found that the law was scale invariant. The invariance in scale and base implies that the rule applies to any sample, irrespective of whether the numbers are based on the dollar prices of stocks or their prices in yen or marks or whether the numbers are in terms of stocks per dollar. The assortment of data was crucial; analysis confirmed that random samples from arbitrarily selected diverse distributions would always conform to Benfords Law. [Becker, 2000] Though computers and calculators have replaced logarithm tables and logarithmic rules in routine calculations, logarithms, the basic premise of Benford’s law, continue to be significantly applicable in technical and scientific applications, computerized and otherwise, as the law helps scientists and data analysts in predicting the frequencies of numbers and/or in analyzing discrepancies and errors in data sets. Benford’s Law presents itself as a significant tool in analyzing numerical data in a wide range of applications, however, as in the case of any natural rule, the Law has limitations. An understanding of the limitations would be imperative to examining the significance and use of law in practical life applications. Limitations of Benford’s Law Benford’s law is not applicable universally and is invalid in certain situations; for example, the law does not apply: To uniform distributions - when the numbers are uniformly distributed in a set, as in a lottery, every number has an equal probability. [Browne, 1998] When data are all about the same magnitude, as in the case of height measurements of adults [Author Unknown, 2000 ] When there is a set maximum or minimum to the data set If the numbers are assigned such as credit card numbers, policy numbers etc. If the numbers are multiples of a unit rate, such as price break point in service sales data. [Coderre, 1999; Durtschi, Hillison and Pacini, 2004]. The situations suggest the dependence of the law on the randomness of the selection, as explicated by Dr. Hill. Any human intervention, which influences predictability adversely impact the applicability of Benford’s Law. David Coderre presents guidelines for determining whether the data will comply to Benford’s Law – based on the last three items as above – and suggests that data sets that meets the three criteria but does follow the expected frequency pattern suggested by Benford may include fraudulent items; [Coderre, 1999] the proposition suggests the potential application of Benford’s law in fraud detection. However, Nigrini suggests a drawback of the law, when applied to fraud detection cases, that some of the tests may present too many false positives. Several anomalies, which may not necessarily be fraudulent representations, can also surface during the investigation for innocent reasons [Browne, 1998] Applications of Benford’s Law Despite the limitations of the law, Benford’s probability pattern presents many real-life applications. While the law is a potential statistical tool presents extensive application in mathematical and statistical analysis, the research shall be essentially limited to studying the application of Benford’s Law in investigating and detecting fraud. Dr. Theodore Hill, an avid researcher of Benford’s law is convinced that the “astonishing” mathematical theorem “is a powerful and relatively simple tool for pointing suspicion at frauds, embezzlers, tax evaders, sloppy accountants and even computer bugs.” [Browne, 1998] Dr. Mark J. Nigrini, an accounting consultant affiliated with the University of Kansas and a faculty of Southern Methodist University in Dallas, has applied the law in many real world situations, and is understood to be among the first to apply Benford’s probability pattern to investigate and detect fraud. He had done his Ph.D. dissertation on the potential use of Benfords Law to detect tax evasion, and developed successful computer programs based on the rule to detect cases of fraud. The theory underlying his program was that if the numbers in a set of data, such as a tax return, did not match the frequencies and ratios predicted by Benford’s Law, the data were probably faked. The effectiveness of the program was put to test, in detecting cases of admitted fraud in tax audits at Brooklyn; according to Robert Burton, the chief financial investigator for the Brooklyn District Attorney, Nigrini’s program could “correctly spot all seven cases (of admitted fraud) as involving probable fraud.” [Browne, 1998] Dr. Nigrini’s academic studies and speeches, since his validation of Benford’s law, have spawned considerable interest among accountants and auditors, as the potentialities of the law in accounting and audit situations are being assessed, the law being beneficially applied, especially with reference to fraud detection. The following sections shall address the rising significance of fraud detection in audits and accounts, and the potential application of Benford’s Law in investigating and detecting fraud. A few case studies, real life situations, where the law has been put to use effectively in detecting fraud, are also included, as proof of the use of law in fraud investigation. Significance and Use of Benford’s Law in Detecting Accounting Fraud Fraud may be committed in a variety of situations-- not only in accounts and audits, but also in many other real life situations involving the use of numbers, such as faking clinical test results, manipulating the results of examinations and other competitive tests, where alteration of data presents potential benefit to the impostor. The frequency pattern suggested by Benford’s law may be applied effectively to detect frauds in any of these situations if the data set satisfies the criteria discussed earlier. Yet, the most significant and potential application of the Law in detecting fraud pertains to accounting and auditing situations, perhaps due to the increasing incidences of occupational frauds and fraudulent accounting by corporations. The basic premise of applying the law in detecting accounting and fraud is the idea that if a data set, which satisfies the criteria of randomness, does not conform to Benford’s distribution presents a probable fraud. The increasing instances of fraudulent actions by corporations such as Enron, WorldCom, Tyco etc and the failure of the auditing community to detect and check the frauds has led to the issuance of Statement on Auditing Standard No. 99, Consideration of Fraud in a Financial Statement Audit requiring the auditors to use appropriate analytical tools and audit methods during the planning phase of the audit to identify the occurrence of unusual transactions, events and trends, with the objective of detecting fraud. [Author Unknown, 2004] The precautionary suggestion in the Statement that the use of traditional analytical procedures, usually performed on highly aggregated data, can provide only broad indications of fraud, and the contemporary validation and promotion of Benford’s law, by eminent professionals as Nigrini and Hill, as an effective analytical tool in detecting fraud explains the rising significance of the law in specifically detecting frauds, particularly in accounting and auditing situations. Over the past 10 years, a considerable number of articles have been published, which promote the use of the Law as a simple, effective digital analysis tool/method for auditors to discover fraud in accounting numbers as well as towards identifying operational errors and discrepancies. [Durtschi, Hillison and Pacini, 2004] Boyle has demonstrated that data sets follow Benford’s law when the elements in the set are random variables taken from divergent sources that have been multiplied, divided, or raised to integer powers – distribution of distributions. [Cited in Durtschi, Hillison and Pacini, 2004] This feature explain the reason why certain sets of accounting data are often found to closely follow Benford distribution and certain sets do not. As Durtschi, Hillison and Pacini explains, accounting data usually result from mathematical process, as in the case of accounts receivable data -- the data set represents the number of items sold (as per one distribution) multiplied by the price per item (as per another distribution), and hence found to follow Benford’s law. [Durtschi, Hillison and Pacini, 2004 ] However in certain cases, such as the case of a courier company which prices its services on the basis of single price break point (say $10 for items less than 1 kg), the law may not apply, since the distribution is not random. [Coderre, 1999] There are other instances as well, where Benford’s law may not be applicable, the same are discussed in later sections of the paper. Benford’s distribution presents potential fraud detection capabilities in many auditing situations, including external and internal audits in business, as well as in governmental audits Research suggests that a person making fraudulent accounting operations is likely to enter the same transaction many times, and auditors can effectively identify probable fraudulent transactions from the resulting deviation of first and second digits from the Benford distribution [Cleary and Thibodeau, 2005] Though faking numbers seem simple and easy, professional statisticians and mathematicians are convinced that the faked distribution does not comply with the natural frequency patterns as expounded in Benford’s and other statistical distributions. Dubinsky explains the reasons why human psychology works against realistically faking numerical data in random distribution: Human logic appeals to even distribution of digits and tend to do so subconsciously Tendency to select specific numeric sequences consistently A programmed avoidance of repetitions of numbers A natural preference of some digits to others [Dubinsky, 2001] Dubinsky, in line with the suggestions of Cleary and Thibodeau as discussed earlier, claims that even if someone committing a financial fraud may properly distribute the first digit’s frequency, the chances that the faked numbers come up with appropriate two-, three-, and four-digit sequences would be nearly impossible. He explains that subsequent digits follow similar, but essentially different, rules and it is beyond the capability of the human mind to adhere to these rules. Whether computer programs may be devised to fake numbers realistically, so as to follow Benford’s law, is a probability that needs to be considered by accountants, who seriously consider fraud detection using the law. Nonetheless, today, the Law helps not only in effectively detecting fraud but also in identifying data discrepancies and errors; and is basically a more complex form of digital analysis, that auditors normally use for analyzing data. The complexity is due to the obvious reason that the law considers each data, perhaps digits, in the entire account to determine if the numbers conforms to the expected distribution. However, the advanced capabilities of computing technologies have made it easier for auditors; Nigrini and Mittermaier explains how auditors can use software programs based on Benford’s Law in data analysis to effectively identify unexpected figures, which may be the result of fraud, in transactions. [Nigrini and Mittermaier, 1997] Though the potential use of Benford’s law as a test of the honesty or validity of random scientific data in a business and social science context have been proposed by researchers since the 1970s and 1980s, and the law have been applied in rare occasion to detect earnings manipulation, perhaps the complexity and the perceived irrationality of the law have limited accountants and auditors in applying the law effectively accounting and audit situations. [Durtschi, Hillison and Pacini, 2004] The mathematical validation of Benford’s Law by Prof. Hill and the development of software programs based on the Law have effectively resolved the issues, as auditors and accountants are increasingly adopting Benford’s Law in detecting frauds and discrepancies in data sets and financial statements. Technological developments in the recent past have facilitated the development of many user friendly software programs such as ACL (Audit Command Language) and CaseWare 2002, and also enhanced the import large volumes of financial data from corporate accounting systems, enabling forensic accountants and auditors to put Benford’s law to effective use in fraud investigation. Yet, it is significant to note that the applicability of the law is limited to certain categories of financial data and presents errors – false positives and negatives – in data analysis. It does not also offer to detect all kinds of frauds related to accounting information [Durtschi, Hillison and Pacini, 2004]; these implications are discussed later in the paper. Also, while the tool can effectively isolate unexpected results within the context of financial data in a company, the results of such analysis alone cannot prove fraud in the legal context; to detect and establish fraud in legal terms, the forensic accountant would need to further investigate so as to obtain the necessary evidence to establish that fraud occurred and also identify the perpetrator. [Dubinsky, 2001] While a discussion of the mathematical validation of the law and the intricacies of the software program may be beyond the scope of the paper, an exposure to real life cases where the Law has been put to beneficial use may reveal the potentiality of the law in detecting fraud. Benford’s Law in Fraud Detection – Cases Subsequent to Mark Nigrini’s Ph.D. dissertation on the use of Benford’s Law to detect tax evasion, the law has been used effectively in a number of real life situations including legal cases for detecting financial and other frauds. Research suggests that today many fraud examiners, internal auditors and forensic accountants, as well as income tax departments of many states including California [Becker, 2000]are employing new investigative methods that rely on Benford’s Law to detect acts such as tax evasion, bogus vendor payments, expense account fraud etc. A significant number of the Fortune 500 U.S. companies are also understood to be using fraud detection methods based on Benford’s Law of late. [Dubinsky, 2001] Bruce G. Dubinsky, a forensic accountant who specializes in the detection of fraud, presents real life situations, as detailed below, when Benford’s law has effectively helped forensic accountants in detecting fraud. In the first case, digital frequency analysis of the health insurance claims in a travel company using the Law revealed that the claims made by a supervisor showed a deviation in the frequency pattern, the numbers beginning with the digits 65 was recurring, the pattern deviating from Benford distribution. The follow-up audit by the forensic accountants showed 13 fraudulent checks for between $6,500 and $6,599 relating to fraudulent heart surgery claims. The analysis is reported to have also exposed other fraudulent claims worth around $1 million. [Dubinsky, 2001] The second instance that Dubinsky demonstrates is a 1993 legal case -- State of Arizona v. Nelson (CV92-18841). The case involved the issuance of 23 checks to a bogus vendor by the accused, Mr. Nelson, a manager in the office of the Arizona state treasurer. He claimed that he had simply diverted funds to a bogus vendor due to the lack of necessary safeguards in a new computer system used by his office. However, a digital frequency analysis of the 23 checks, , showed that the amounts he had selected at random did not conform to Benford’s Law; the digit distribution were found to be almost opposite to those of Benford’s Law. The forensic accountant, who investigated the case could have easily detected that these numbers, invented to seem random, fell outside of the expected patterns and thus warranted further examination. The investigation revealed that the accused was guilty of defrauding the state of nearly $2 million. [Dubinsky, 2001] In yet another case presented by David Coderre, the auditors for a company were investigating possible fraud in the contracting section, where a considerably large number of contracts were raised every month. A system based on Benford’s Law was used to analyze the frequency patterns of the first two digits of the contract amount. The digital analysis revealed that the digits 49 were occurring in the data more frequently than expected, as suggested by Benford distribution. A subsequent analysis revealed that the contracting manager was raising contracts for $49,000­ to avoid contracting regulations. Contracts under $50,000 could be sole-sourced; contracts greater than $50,000 had to be submitted to the bidding process. While the first analysis revealed that the contracting manager was raising contracts just under the financial limit to evade a protracting contracting regulation, further investigation revealed his fraud – he was directing the contracts below $50,000 to a company owned by his wife. [Coderre, 1999] The cases discussed above and many other prospective applications put forth by researchers in fraud detection suggest that Benford’s law offers itself as a beneficial and effective tool in fraud investigation and detection. However, researchers examining the application and use of the law in detecting fraud in financial data suggests chances of potential errors – classified as Type I and Type I, as detailed in the subsequent section, that present significant consequences, which need to be considered by accountants before embarking on fraud detection using systems based on the law. Also, significant is the need to understand the situations when the law may not applicable due to the innate nature of the data, and also the type of frauds that cannot be detected by the law. The following sections present a discussion on these crucial implications. Implications of Using Benford’s Law As suggested earlier, the digital frequency analysis are prone to errors, such as Type I and Type II as well as others and the auditors should be subjecting the data to theoretical testing during the preliminary analysis stage of applying Benford’s Law. ‘Type I errors’ – false positives -- leads to the conclusion there was evidence of fraudulent activity, when in reality the entries were legitimate. ‘The Type II error’ – false negative -- refers to the situation when a real fraudulent activity goes undetected. [Author Unknown, 2004; Durtschi, Hillison and Pacini, 2004] The relative ease and generality of using standard audit software packages software such as ACL – perceivably an auditor choosing to use the Benford command, when operating the system, just to need to identify the appropriate data field within the appropriate data file (e.g., invoice amount within client accounts receivable file) for an automated analysis-- calls for consideration as to whether additional audit testing should be considered for data that did not conform to the Benford’s Law probability distribution. [Author Unknown, 2004] A statistical null hypothesis testing is considered to help the auditor in effectively analyzing data using Benford’s law. The basic assumption in “null research hypothesis,” H0: First digits in a data set are distributed according to Benford’s law, is that there is no fraudulent activity. Digital analysis of data presents four possible explanations that the digits are not behaving as one might expect; the possible explanations being: Type I errors – The data set generally follows Benford’s law, but due to random chance a particular set of observations does not. The criteria and the assumption of adequate size of the data set is not satisfied The recurrence of some particular first digit may be reasonable explained – for example in a sales report, if a frequently shipped item has a value of $75, the 7 as the first digit in the data will deviate the Benford distribution. The probability that the entries in the list are fraudulent. [Author Unknown, 2004] Thus, if the auditor rejects the statistical null, H0 the chance of fraudulent entries is one of the possible explanations. The situation implies additional cost consequences, in terms of money, time and effort, to the auditor, so as to determine which of the four explanations, explains the reality of the audit situation appropriately, given the specific client situation. The different categories of Type I error, presents potential cost consequences, and a detailed analyses of the errors only occasionally reveal that an actual fraud has been committed. The implications of the chances of the occurrence of Type II errors are also significant. In statistical analysis for fraud detection, a type II error occurs when the auditor fails to reject a null hypothesis that is false, which implies that he failed to identify a find a fraudulent scheme when one actually occurred. [Author Unknown, 2004] This occurs when the fraudulent the transactions, such as bribes, kickbacks or asset thefts, are not recorded. The law also fails to effectively detect duplicate entries and may shell fraudulent companies and employees. Benford analysis also fails to detect such frauds as contract rigging and defective deliveries. [Durtschi, Hillison and Pacini, 2004] It is understood the Benford command in standard software packages as ACL automatically produces an analysis completed on a “digit by digit” basis rather than on a “case by case.” Research suggests that though “digit by digit” analysis helps in identifying the specific fraudulent data, the chances of Type I errors are much more, as compared to the “case by case” analysis, traditionally employed by statisticians, which provides only broad indications of fraud. [Cleary and Thibodeau, 2005] This means that while a “case by case” test controls the probability of Type I error, the results may not be as informative in the case where fraud actually has occurred. Though a “digit by digit” approach increases the chances of a Type I error, it enhances the auditor’s ability to detect actual fraudulent entries. Considering the significant cost implications in “digit by digit” analysis, auditors using Benford’s Law, with a view to detecting fraud, may choose a prudent approach – initiating the audit with an overall analysis using a “case by case,” approach such as traditional chi-squared test, and proceed to use Benford’s law for a “digit by digit” analysis, if there is an indication of possible fraud in this overall analysis. [Author Unknown, 2004] Despite the costs, the potentiality of Benford’s law in detecting specific fraud, in the event of a probable fraud in overall analysis, suggest the usefulness of the law in fraud investigation and detection. Apart from the Type I and Type II errors as detailed above, forensic accountants and auditors using Benford’s law, for fraud detection and investigation, need to aware of the specific accounting-related data, do not conform to Benford’s distribution. Studies confirm that most accounting-related data can be expected to conform to a Benford distribution, and thus will be appropriate candidates for digital analysis [Hill 1995]. However, certain data sets, though apparently related to accounting, do not follow Benford’s Law, and it is important forensic accountants and professional detecting fraud should not be mislead by the apparent relation and profundity of the law towards fraud detection. These are covered in the general limitations of the rule, however, are reiterated for clarity in application, with respect to detecting accounting frauds. Certain specific accounting related data sets are found not conforming to Benford distribution. Instances that do not follow Benford’s law include the list of assigned numbers, such as check numbers, purchase order numbers, or other numbers that are influenced and generated as a result of human involvement and thought, such as product or service prices, and more importantly ATM withdrawals [Nigrini and Mittermaier 1997]. ATM withdrawals present a significant case of fraud, however, the data set do not conform to Benford’s law as they are often in pre-assigned, even amounts. As suggested by Durtschi, Hillison and Pacini pre-assigned numbers should follow a uniform distribution rather than a Benford distribution. Their study presents a good idea of instances and reasons why these data sets and others do not conform to Benford’s Law. Any human intervention in data setting presents psychological barriers, which affects the distributions. Certain assigned numbers such as price are often set to fall below psychological barriers, companies often set price just below a whole number -- $1.99, $ 2.99 etc rather than $2.0 and $3.0 respectively as the former set are perceived as much lower than the latter—the result is that price data and sales data tend to show clusters of these digits psychological barriers. Certain firm-specific data and data sets with built-in maximum or minimum value also do not conform to Benford’s law and calls for prudent intervention and judgment by auditors before embarking on a detailed digital frequency analysis. [Durtschi, Hillison and Pacini, 2004] Conclusion In conclusion, it is surmised that when used appropriately and prudently, Benford’s Law presents itself as useful tool in investigating fraud, particularly in relation to accounting and auditing situations. The mathematical validation of the law and the technological advances in the recent past, which facilitate faster and easier programs for digital analysis, have enhanced the usefulness of the law in detecting fraud, as the law is increasing used by forensic accountants and auditors. The cases discussed in the course of the paper demonstrate the usefulness of the law in detecting fraud in real life situations. However, it is important to note that the detection and establishment of fraud in the legal sense of the term calls for further analysis and Benford’s law is only a facilitator in detecting fraud, albeit an effective facilitator. Given the caution in SAS No. 99 that traditional statistical methods only provide broad indications of fraud, and the increasing incidences of white-collar crimes, the usefulness and applicability of Benford’s law in fraud investigation assumes greater significance. Benford’s law, is particularly useful as it conducts “digit by digit” analysis and helps in identifying the fraud exactly, despite the huge scale and size of data. However, since the costs involved, particularly due to the increased probability of Type I errors, are significant, a prudent approach should be chosen in applying the law. Fraud detectors using the law should not apply the law to data sets that do not conform to Benford distribution; should be careful in interpreting the statistical results of the test; and should aware of the fact that not all types of fraud in accounting data are detectable using Benford’s law. [Durtschi, Hillison and Pacini, 2004] Yet as Dubinsky remarks, “in the never-ending crusade against white collar crime and corporate fraud, Benford’s law is a very valuable tool in the forensic accountant’s arsenal.” [Dubinsky, 2001] References 1. Author Unknown (2004) “Applying Digital Analysis Using Benford’s Law To Detect Fraud: A Practice Note About The Dangers Of Type I And Type II Errors” American Accounting Association Available at: Http://Aaahq.Org/Audit/Midyear/04midyear/Papers/Dangers%20of%20Applying%20Benfords%20Law%20-%20Paper%20-%20December%202003.Doc Accessed 10/22/05 2. Becker T.J. (2000) “Sorry, Wrong Number,” Research Horizons Available at: Http://Gtresearchnews.Gatech.Edu/Reshor/Rh-F00/Math.Html Accessed 10/22/05 3. Benford, F. (1938) The Law Of Anomalous Numbers. In Proceedings Of The American Philosophical Society 78:551-572. 4. Browne, M.W. (1998) “Following Benfords Law, Or Looking Out For No. 1” (August 4) Available at: http://www.math.yorku.ca/Who/Faculty/Brettler/bc_98/benford.html Accessed 10/22/05 5. Cleary R., Thibodeau J. C., 2005, Applying Digital Analysis Using Benford’s Law to Detect Fraud: The Dangers of Type I Error, Auditing Vol 24(1): 77-81 6. Coderre , David 1999 Auditing : Computer-Assisted Techniques For Fraud Detection The CPA Journal (August) Available at: http://www.nysscpa.org/cpajournal/1999/0899/departments/D57899.HTM Accessed 10/22/05 7. Dubinsky, B. G. (2001) Math Formula Fights Fraud Legal Times (February 26) Vol.XXIV(9) Available at: http://www.kd-cpa.com/kdanews/math-formula-fights-fraud.pdf Accessed 10/22/05 8. Durtschi, C., Hillison W. & Pacini. C. (2004) The Effective Use of Benford’s Law to Assist in Detecting Fraud in Accounting Data, Journal Of Forensic Accounting Vol.V:17-34 9. Hill, T. P. (1998) The first digital phenomenon. American Scientist. 86(4): 358-363. 10. Hill, T. P. 1995. A statistical derivation of the significant digit law. Statistical Science 10(4): 354-363. 11. Nigrini, M. J. (1999) “Fraud Detection: I’ve Got Your Number” Journal Of Accountancy (May)Vol. 187(5) Available at: http://www.aicpa.org/pubs/jofa/may1999/nigrini.htm Accessed 10/22/05 12. Nigrini, M. J. and L. J. Mittermaier. (1997) The use of Benfords law as an aid in analytical procedures. Auditing: A Journal of Practice & Theory. 16(2):52-67. Read More