Area |
Objective |
Comments |
Income Tax |
Compliance |
NY AG identified suspicious corporate returns |
Automobile Insurance Claims |
Potentially fraudulent claims |
Unusual patterns were identified |
Accounts Payable |
Identify Questionable or Unusual Disbursements |
Confirmed no unusual or questionable payments |
Images |
Classify images as created (man made) vs. photograph |
Photographs of real images will have pixel values distributed according to Benford’s Law |
Medical Claims |
Potentially fraudulent claims |
Unusual patterns can be identified |
We took several sources of data in order to test their compliance with Benford's Law. The populations, and the variables tested came from several sources. The first population tested, was census data available at the North Carolina Child Advocacy center. We expected that this data would conform with Benford's Law. We also looked at various medical insurance claims. We were unsure to what extent this data would conform with Benford's Law.
| Description | Test Code |
|---|---|
| Test of the first (leading) digit | F1 |
| Test of the first two (leading digits) | F2 |
| Test of the first three (leading digits) | F3 |
| Test of the last two (trailing digits) | L2 |
| Test of the last digit | L1 |
| Test of the second digit | D2 |
For each test, we had the software compare the expected results (according to Benford's Law) with the actual results, and quantify the difference using Chi Squared. This resulted in a “probability” score (inverse chi square) that quantified the probability that any difference was due to chance alone. The software used was EZ-R Stats for Windows. We used the command language version of the product, although there is also a graphical version called Test Compliance With Benford's Law available without charge.
The initial testing was done for the comparison of census data between the year 1990 and the year 2000. The field V0371990/ V0372000 which counts the number of persons under the age of 18 who are Hispanic, was tested with the following results:
| Test | Year | Raw Chi Square | Probability that difference is due to chance |
|---|---|---|---|
F1 |
1990 |
1.67 |
98.9% |
F1 |
2000 |
1.917 |
98.0% |
F2 |
1990 |
1.665 |
98.9% |
F3 |
1990 |
(insufficient size population) |
(N/A) |
F2 |
2000 |
1.25 |
99.6% |
L2 |
2000 |
1.91582 |
99.99% |
L1 |
2000 |
1.96194 |
99.178% |
As expected, the results indicate that the census data conforms with what would be expected using Benford's Law. Note that the three digit test had insufficient population size to enable a Chi Square test (which requires that each cell should have an expected count of at least 5). (Note: the scripts used, and the detail reports are available at www.ezrstats.com/zip/Benford.zip ).
We then took a look at some medical insurance claims data, primarily pharmacy claims. The population consisted of approximately 1.6 million pharmacy claims paid for service during one month. The purpose of the tests was to determine the feasibility of using Benford's Law for fraud detection. We performed all of the tests of conformance with Benford's law and obtained the following results:
| Distribution of Paid Amounts |  |
| First Digit |  |
| First Two Digits |  |
| Three Digits |  |
| Last Digit |  |
| Last Two Digits |  |
| Second Digit |  |
These results indicate that the Internal Auditor/Researcher can identify potential areas for further review. Although these procedures may not be able to detect potential fraud in medical claims, they can highlight anomalies or areas which merit a more in-depth analysis.