what to do with benfords law z score
| EXECUTIVE SUMMARY | |
 Benford's law holds that the leftmost digit in many types of numerical data is a 1 nearly one-third of the time, with probability inversely proportional to the value of each increasing digit. Marking Lehman, CPA, Ph.D., and Marcia Weidenmier Watson, CPA, Ph.D., are associate and assistant professors, respectively, of accountancy at Mississippi State Academy, Starkville, Miss. Their email addresses are mark.lehman@msstate.edu and mweidenmier@cobilan.msstate.edu , respectively. Tim Jones, MPA, is a graduate student at Mississippi State University. |
Worldwide fraud is on the ascent. The magnitude of the trouble prompted the AICPA and the Association of Certified Fraud Examiners to create the Establish for Fraud Prevention. To assist in detecting fraud, auditors need to employ innovative techniques like Benford'south law, which predicts the frequency of digits i through 9 in the showtime four places of any number.
American physicist Frank Benford in the 1930s observed that lower digits, showtime with 1, appear more frequently than higher ones, starting from the leftmost position of many types of collections of numbers. Provided with a large information set, auditors tin can use Benford's constabulary to help notice fraud by analyzing all account transactions to come across if they fall into the expected blueprint (see " I've Got Your Number," JofA , May 99, folio 79). Some other JofA article ("Turn Excel Into a Fiscal Sleuth," Aug. 03, folio 58) presented a Benford analysis using a "Fraud Buster" Excel template on aggregate data.
To pump upwards auditors' detective powers, we present an Excel application that simultaneously applies the technique to each employee, and can be easily adapted for other groups including customers and vendors. A Big Four business firm is evaluating this approach, which has been successfully used by an internal auditing department of a large international company. Our application has received positive feedback because information technology is easy to use and tin can assistance identify fraudsters, particularly when used in combination with other methods. An internal auditor from an international retail concatenation said the procedure allowed him to "piece of work smarter instead of harder—the central to success when you are dealing with vast amounts of data."
The method could help auditors in such activities as SAS no. 99'south requirement that they discuss the potential for a fabric misstatement in financial statements due to fraud (encounter " Auditors' Responsibility for Fraud Detection," JofA , Jan. 03, page 28).
| Astronomer's Keen Eye Paves Way for Law Who would've thought that an astronomer and a physicist could develop a tool for use in accounting? What came to be called Benford's law was discovered in 1881 by the American astronomer Simon Newcomb, who observed that the pages of printed logarithmic tables starting with the number 1 were much more than worn than later pages. Newcomb and then analyzed how numbers were distributed in naturally occurring data and derived the frequencies of what is now chosen Benford's law. Unfortunately, Newcomb's discovery went unnoticed until 1938, when Frank Benford, a physicist, rediscovered the same worn design of the logarithmic tables. Benford analyzed 20,229 sets of numbers, including baseball statistics, areas of rivers and numbers in magazines. Surprisingly, these number sets all follow the same starting time-digit pattern. |
PUMPING UP WITH Z-SCORES
Benford'south law is used by several countries and states (for case, California and New York) to identify tax defrauders. Several software companies—for example, Noon Analytix and Price Recovery Solutions—use Benford's law to place suspicious vendors. Using such software, internal auditors at a big international manufacturing house run a Benford's constabulary test for each vendor based on amounts that exceed user-defined exception levels. I of these auditors indicated that the company was experimenting to observe the optimum level of deviations from Benford's constabulary.
Exactly how practice we increase the detective ability of Benford's police? For each employee, customer, vendor or other party to a transaction we calculate a z-score for each leading digit (1 to 9). The z-score is a statistical mensurate of how many standard deviations a number is from the mean and allows the auditor to empirically determine—not guess—whether deviations from the pattern are statistically significant. The larger the z-score, the less likely it is that unexpected frequencies are the result of gamble. The auditor selects the maximum allowable z-score corresponding to the level of fault that he or she is willing to accept. For example, if the auditor is willing to accept a v% chance of drawing the wrong conclusion, the accountant would set the maximum allowable z-score to 1.96. Any z-score that exceeds the auditor's maximum allowable z-score may indicate fraud and must be investigated further. Thus our approach eliminates the demand to experiment with the appropriate exception level—we simply leave it upwardly to statistics!
HOMING IN ON False SALES RETURNS
While our workbook can be used in many situations, we selected the setting used past the internal auditor quoted above, who investigated sales returns at a retail store. Selected employees are authorized to process sales returns, with management approval required for returns of $500 or more than. Our dataset includes 56,000 hypothetical sales returns over a six-calendar month period and contains the following fields: transaction number, engagement, employee number, sales return amount, and, if required, manager number.
Exhibit one shows the event of a Benford assay with the ordinarily used ACL commercial information analysis software, performed on the entire population of sales returns. The z-score (Zstat ratio) amounts are relatively modest, less than ACL'due south 1.96 default z-score. Therefore, an accountant would not doubtable any fraud, because the sales return counts fall inside the expected pattern and the count of each leading digit falls within its acceptable range. However, the dataset contains a fraud perpetrated by Amy who wrote threescore fake returns (1.1% of her 5,320 returns) for amounts just under the $500 managerial level. To increase the chance of detecting fraud, we advise an automatic method that applies Benford's law to each employee past analyzing the frequency of the starting time digit of every transaction amount.
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EXCEL TO THE RESCUE (AGAIN)
The steps include (one) extracting the get-go digit using 2 text functions ( Exhibit 2), (ii) creating a PivotTable to summate the actual frequencies for each employee using the COUNT function (Exhibit iii), and (3) calculating a z-score using the actual counts and expected frequencies for each combination of digits for each employee (Exhibit four). The spreadsheet displays "yes" for any z-score that exceeds the maximum allowable z-score in cell K2, currently set at 1.28. Based on this z-score value, we can conclude that there is less than a 20% chance that frequencies identified in column L with a "yes" are the result of chance.
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Desire to increment the power of your examination? Just change the z-score. If y'all are willing to have only a 10% probability that the unexpected frequencies are the result of adventure, then set the z-score to ane.65. For a v% or i% chance, fix the z-score to 1.96 or 2.58, respectively. Amy's fraud (issuing fake returns most $500) has a maximum z-score of two.59 for the 4 digit, equally shown in cell E17. Given that Amy'due south score exceeds 2.58, at that place is less than a 1% probability that her bodily return frequencies are due to run a risk—indicating that fraud is very likely.
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WORKING WITH YOUR Information
Got a large dataset? Although Excel easily handled our 56,000-line dataset, yours may overpower Excel. No problem! Programs such as Microsoft Access and ACL tin can identify the leading digit and create a cross-tabulation that can be copied into our workbook'south Actual worksheet. Access users can create a crosstab query to replace Excel's PivotTable. ACL users can substitute that plan's LEADING function for the formula in stride 1 to excerpt the first digit, then create a cross-tabulation table.
A MORE DYNAMIC Analysis
Internal auditors, external auditors and managers are under increasing pressure to identify fraud. The combination of the three text functions and Excel's PivotTable allows a more dynamic information analysis when using Benford'due south law. However, similar nearly audit tests, Benford'south constabulary cannot be relied upon to catch all frauds, which oft requires a combination of approaches. For example, Benford'due south police did non identify the fraud committed by Tom, who scattered his 100 imitation returns (2.1% of his iv,833 returns) randomly between $100 and $500 (see employee 1981 in the workbook). Certainly, Tom's maximum z-score of 0.88 (for the four digit) is larger than other employees' but not big enough to exceed the acceptable z-score. Benford's police does not work with small datasets, information with assigned numbers or artificial minimums and maximums, and numbers that are truly random. Despite these limitations, Benford's law can withal exist a powerful financial sleuth. Both auditors and managers must employ tools such as Benford's law in combination with their professional person judgment and investigative skills to uncover fraud. 
| AICPA Resource JofA manufactures Publications CPE For more data or to identify an lodge, get to www.cpa2biz.com , or call the Institute at 888-777-7077. OTHER RESOURCES Web sites |
" Worldwide: Looters Take a Foothold," January. 07, page 32. Source: https://www.journalofaccountancy.com/issues/2007/jun/flexingyoursuperfinancialsleuthpower.html
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