Cecily Zamora
Benford's Law, also known as the Newcomb–Benford law, the law of anomalous numbers, and the first-digit law, is a statistical tool used to detect fraud. It is based on the probability of the leading digit of a number being n, which can be calculated as log10(1+1/n). Benford's Law can be used to identify anomalies and discrepancies in a transaction dataset, which can indicate data manipulation or intervention. It is a simple method that does not require complex algorithms or coding. By applying Benford's Law analysis, auditors can identify potential fraud and verify the integrity of the data. However, it is important to note that the results of Benford's Law analysis are not definitive, and further investigative work may be necessary.
| Characteristics | Values |
|---|---|
| Basis | Based on base-10 logarithms |
| Leading digit probability | The probability that the leading digit of a number will be n can be calculated as log10(1+1/n) |
| Leading digit distribution | The leading digit is likely to be small, with the number 1 appearing as the leading digit about 30% of the time, and 9 less than 5% of the time |
| Applicability | Can be applied to a wide variety of data sets, including electricity bills, street addresses, stock prices, house prices, population numbers, death rates, lengths of rivers, and physical and mathematical constants |
| Accuracy | Tends to be most accurate when values are distributed across multiple orders of magnitude, especially if the process generating the numbers is described by a power law |
| Data set size | Works better with larger data sets, with some experts recommending a minimum of 500 data points |
| Data set characteristics | The data set must contain data in which each number 1 through 9 has an equal chance of being the leading digit, and the numbers should be randomly generated without any real or artificial restrictions |
| Results interpretation | The results should not be considered definitive, but rather an analytical tool to gauge whether further investigation is warranted |
| Tools | Can be applied using Microsoft Excel, with functions such as LEFT and COUNTIF, along with the Column Chart tool |
| Use cases | Has been used to detect accounting and expenses fraud, voter fraud, and loan fraud |
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What You'll Learn
Benford's Law is based on base-10 logarithms
Benford's Law, also known as the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is based on base-10 logarithms. It is an observation that in many real-life sets of numerical data, the leading digit is likely to be small. In other words, the probability that the leading digit of a number will be 'n' can be calculated as log10(1+1/n).
The law was discovered in 1881 by Canadian-American astronomer Simon Newcomb, who noticed that in a book of logarithm tables, the pages were worn and smeared more towards the front of the book and less so towards the end. This observation led to the conclusion that the first few pages were accessed more frequently than the others.
Fifty-seven years later, physicist Frank Benford made the same observation. He tested 20 different sets of data, including the populations of over 3000 US cities and over 100 physical constants, and found that all of them followed Benford's Law.
According to Benford's Law, the odds of a digit appearing as the first digit are not random. For example, there is a 12.5% chance that the first digit will be a 3 and a 30.1% chance it will be a 1. This means that in a set of three-digit payment amounts, the first digit is more likely to be 1 than 9. Benford's Law can be used to detect fraudulent activity, as it is difficult to fabricate a set of falsified data that conforms to the law.
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It can be used to detect manual intervention in automated transaction activity
Benford's Law is a powerful tool for detecting manual intervention in automated transaction activity, which can indicate potential fraud. This law is based on base-10 logarithms, which predict the likelihood of each digit, from 1 to 9, appearing as the leading digit in a number. According to Benford's Law, in a naturally occurring set of numbers, the leading digit is more likely to be smaller rather than larger.
For example, in a data set of three-digit payment amounts, one might assume that each digit, 1 to 9, has an equal chance of being the first digit. However, Benford's Law states that this is not the case. In reality, the first digit is far more likely to be a 1, occurring around 30% of the time, while 9 is the leading digit less than 5% of the time. This law is useful for fraud detection because it is difficult for fraudsters to create a large set of fabricated data that follows this law.
When applied to transaction data, deviations from Benford's Law can indicate manual intervention in otherwise automated transaction activity. For instance, in tax fraud, manipulated data for tax evasion purposes will likely deviate from Benford's Law. Tax authorities can use Benford's Law analysis to determine whether data on tax returns have been tampered with. This can also assist with resource allocation, as deviations in specific data sets could warrant further scrutiny of particular industries or transaction types.
Benford's Law can be a valuable addition to auditors' and fraud examiners' toolkits, especially with the increasing complexity of organisations and the vast number of transactions recorded daily. By exploiting audit trails in ERP systems, Benford's Law analysis can be used to continuously monitor transactions and identify suspicious activity, errors, or irregularities. While Benford's Law analysis does not provide definitive proof of fraud, it is a useful indicator for further investigation.
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It can be used to identify general ledger irregularities
Benford's Law can be used to identify irregularities in general ledgers, which may indicate fraud. This is because Benford's Law gives the expected proportions of the first, first-two, first-three, and so on, digits in tabulated data. Accounting data, including journal entry amounts, have been found to conform reasonably closely to these expected proportions.
Deviations from Benford's Law could indicate that the general ledger includes large counts of fictitious journal entries that are below the auditor's testing threshold. For example, in the HealthSouth fraud, accounting personnel abused the testing threshold of $5,000 by creating thousands of fraudulent journal entries that were just below that threshold. Benford's Law-based testing could have detected this fraud scheme.
Benford's Law can also indicate that the general ledger includes unusually high duplications of same-dollar transactions. These irregularities are not necessarily fraudulent but might be due to factors such as processing inefficiencies or high duplications of amounts that are per diem travel allowances.
Benford's Law can be used to detect manual intervention in otherwise automated transaction activity. This is because offenders rarely consider Benford's Law when creating false transaction documents. For example, if a company's financial statements show an unusually high proportion of numbers starting with the digit 9, it could be a red flag for fraudulent activity.
It is important to note that the results obtained using Benford's Law analysis should not be considered definitive, and the process of counting leading digits will never decidedly prove the absence or presence of fraud. When Benford's curve fails to materialize, CPAs should step up their efforts to verify the data.
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It can be used to detect tax fraud
Benford's Law can be used to detect tax fraud. It is a statistical tool that can flag anomalies and discrepancies in a transaction dataset to detect interventions or compromises that undermine a dataset's integrity. Tax authorities can apply a Benford's Law analysis to determine whether data on tax returns have been manipulated.
Benford's Law is based on base-10 logarithms that show the probability that the leading digit of a number will be n can be calculated as log10(1+1/n). By substituting the numbers 1 through 9 for n, you can calculate that each subsequent number 1 through 9 has a diminishing probability that it will be the leading digit. According to Benford's Law, the number 1 is the first digit about 30% of the time, while 9 is the leading digit less than 5% of the time. If a dataset is significantly different from the expected distribution of the first digits, this may be an indication that it has been altered or fabricated. For example, if a company's financial statements show an unusually high proportion of numbers starting with the digit 9, it could be a red flag for fraudulent activity.
Benford's Law can be used to uncover fictitious numbers in random datasets because it detects manual intervention in otherwise automated transaction activity. Data manipulated for tax evasion purposes will likely deviate from Benford's Law. A Benford's Law analysis presents a null hypothesis and an alternative hypothesis. The null hypothesis prevails when there is no statistically significant difference between the observed and expected frequencies of the first digit, suggesting that the data are not compromised. The alternative hypothesis prevails when there is a statistically significant difference between expected and observed frequencies, indicating potential fraud.
Benford's Law-based testing could direct auditors to large counts of journal entries that are below their testing threshold, which could be an indicator of a certain type of fraud mechanism. In the HealthSouth fraud, for instance, the accounting personnel abused the testing threshold of $5,000 by creating thousands of fraudulent journal entries that were just below that threshold. Deviations from Benford's Law could indicate that the general ledger includes large counts of fictitious journal entries that are below the auditor's testing threshold. They could also indicate that the general ledger includes irregularities in the form of unusually high duplications of same-dollar transactions.
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It can be used to detect fraud in any given dataset
Benford's Law is a statistical tool that can be used to detect fraud in any given dataset. It is based on the probability of the occurrence of leading digits in a series of numbers. The law states that in naturally occurring datasets, the first digit of a number is usually small rather than large. For example, the number 1 is the first digit about 30% of the time, while 9 is the leading digit less than 5% of the time. This is contrary to the assumption that each number from 1 to 9 has an equal chance (11%) of being the first digit.
Benford's Law can be used to detect anomalies and discrepancies in a dataset, indicating potential fraud. For instance, if a company's financial statements show an unusually high proportion of numbers starting with the digit 9, it could be a red flag for fraudulent activity. This is because fraudsters tend to submit fake invoices for larger amounts, which upsets the natural order of the way numbers should occur.
The law can be applied to various types of data, including accounting data, journal entry amounts, tax returns, and financial statements. It is a useful tool for auditors and fraud examiners to identify potential fraud or irregularities. However, it is important to note that Benford's Law analysis should not be considered definitive proof of fraud, but rather an analytical tool to gauge whether further investigation is warranted.
Additionally, Benford's Law-based testing can help uncover certain types of fraud mechanisms, such as the creation of multiple entries below a specific testing threshold. This was evident in the HealthSouth fraud case, where accounting personnel manipulated thousands of journal entries just below the $5,000 threshold. By applying Benford's Law, auditors can identify such deviations and anomalies, even in large datasets, without the need for complex algorithms or coding.
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