The Second-Order and Summation Tests
Chapters 5 and 6 dealt with forensic analytic situations where we expected our data to follow Benford's Law. Those chapters included guidelines for assessing which data sets should follow Benford's Law and a review of methods for measuring conformity to Benford's Law. In contrast, this chapter deals with two Benford's Law tests that do not rely on the data conforming to Benford's Law. One of these tests is called the second-order test. This relatively new test can be applied to (almost) any set of data. The second-order test looks at relationships and patterns in data and is based on the digits of the differences between amounts that have been sorted from smallest to largest (ordered). The digit patterns of the differences are expected to closely approximate the digit frequencies of Benford's Law. The second-order test gives few, if any, false positives in that if the results are not as expected (close to Benford), then the data do indeed have some characteristic that is rare and unusual, abnormal, or irregular.
The second of these new tests is called the summation test. The summation test looks for excessively large numbers in the data. The summation test is an easy extension to the usual first-two digits test and it can be run in either Access or Excel. This chapter also introduces Minitab as a possible tool for forensic analytics. The second-order test uses some of the mathematics from the prior chapter. The summation ...