Benford’s Law in Analyzing Accounting Data
Benford law named after physicist Frank Benford, also called first-digit law, is an observation about the frequency distribution of leading digits in natural sets of numerical data.
Benford‘s Law states that natural sets of numbers follow a predictable pattern, no matter what their origin or subject. The digit 1 will be the leading digit approximately 30% of the time. A leading digit is simply the left-most digit in a number (0 cannot be a leading digit). In fact, digits 1, 2, or 3 lead approximately 60% of the time. 9 as a leading digit appears only 4.5% of the time.
Mark Nigrini developed five tests to use Benford law in analyzing accounting data.
1. First digit test
2. Second digit test
3. First two digit test
4. First three digit test
5. Last two digit test
The first-order tests include the first digit test, the second digit test, and the first two digits test.
The first-order tests are usually run on either the positive numbers or the negative numbers, but not both in the same analysis. This is because the incentive to manipulate is opposite for positive and negative numbers.
The first and second digit tests are high-level tests of reasonableness and are merely used to determine whether the data set appears reasonable. If the first and second digit tests indicate that the data set is significantly different from Benford’s Law, the first two digits and first three digits tests will be performed to select audit targets. The last two digits test is used to detect rounding.
First Digit Test
The first digit test compares the actual first digit frequency distribution of a data set with that developed by Benford. It is an extremely high-level test and will only identify obvious anomalies. It should not be used to select targets for sampling, as the sample size will be too large.
Second Digit Test
The second digit test is also a high-level test designed to test conformity or reasonableness. Expected second digit proportions are less skewed than expected first digit proportions. Because this test results in a large sample selection, it should not be used to select audit samples.
First Two Digit Test
The first two digits test combines the previous two tests and identifies deviations that warrant further review. To that end, it can be used to select efficient audit samples for testing.
First Three Digit Test
The first three digits test is a highly focused test that is also used to select audit samples. While the first two digits test tends to indicate broad categories of abnormality, such as payments made just below an authorized limit, the first three digits test tends to identify unusual amounts that have been duplicated.
Both the first two and first three digits tests tend to identify overused digit patterns indicative of fraud, representing erroneous inputs, or the duplicate processing of the same invoice on multiple occasions.
Last Two Digit Test
The last two digits test is used to identify fabricated and rounded numbers. This test is especially handy because it might be all the examiner needs to select audit targets in populations smaller than 10,000. Since the expected proportion of all possible last two digit combinations is .01, it is very easy to identify abnormalities via a graph. This test is especially useful if financial statement figures have been rounded, thereby suggesting that the figures are estimates rather than actual amounts.
Since this test results in small and efficient sample sizes, it can be used to identify patterns that might not be evident when using the previous four tests.
Benford law analysis can only be done on natural data sets like vendor invoices, sales data, journal entries, customer refunds etc. It cannot be applied to numbers which follow a predefined sequence like cheque numbers, data restricted by maximum or minimum number etc.
You can download the template to do Benford analysis in Excel Benford analysis in Excel here. The template is not configured to do the analysis for last two digit test.
If you come across any error in the template, please let me know in the comment section.