The Benford's Law feature in IDEA can provide a valuable reasonableness test for large data sets.

- IDEA only tests positive numbers 10 and over in the data file.

- For negative numbers, values greater than minus 10 are excluded (exclude -9, -8, . . . -1).

- These steps eliminate immaterial items from the analysis.

- Positive and negative numbers are analyzed separately.

The positive and negative numbers are evaluated on their own due to the fact that positive numbers behave very differently from negative numbers. For example, where positive earnings are manipulated for management bonuses, there is motivation to increase the earnings, moving away from zero toward larger numbers. Where there are losses and management wishes to improve stock prices, there is incentive to move the larger negative number to a smaller one toward zero.

IDEA can apply most of the Benford's Law tests and can also display suspicious results in graphical format. Tests provided in IDEA are the first digit, first two digit, first three digits, second digit, last two digits, second order, and summation tests as shown in Figure 5.6.

Results that show a poor fit with Benford's Law should be examined, as they are an indicator of excessive duplications and anomalies.

FIGURE 5.6 Applying the Benford's Law Feature in IDEA

This first two digits primary test output from IDEA indicates that it marginally conforms in Figure 5.7. The graph highlights the three most highly suspicious numbers and the three most suspicious items. By placing the cursor over any bar, such as the

FIGURE 5.7 First Two Digits Test Benford's Law Output with Suspicious Numbers Highlighted

highly suspicious 62 bar, options for extracting or displaying the records are offered. Field statistics may also be displayed.

FIGURE 5.8 Benford's Law Summation Test Example

The summation test (Figure 5.8) analyzes the first two digits in the data by grouping the records of the first two digits together and then computing the sum of each group. Amounts with the same first two digits, such as 1200, 125, 12, 1234, and so on, are added together. Using the computed and summed values, the process determines whether a uniform distribution is followed. The summation test identifies excessively large numbers as compared to the rest of the data. The test is based on sums rather than on counts, as in the other Benford's Law tests. In theory, the sums of numbers with the same first two digits should be equal in distribution. However, in normal data sets, there are regular abnormal duplications of large numbers that may be caused by a few very large numbers or a high volume of moderately large numbers. Additional analysis will be needed.

The second order test is also based on the first two digits in the data. The data is sorted from the smallest to the largest and the differences between each pair of consecutive records are checked to determine whether they conform to the expectations of the first two digit distribution. Numbers of even tens—10, 20, 30, 40, 50, 60, 70, 80, and 90—on their own are expected to conform to the Benford's Law distribution and the rest of the numbers on their own are also expected to conform, as displayed in Figure 5.9.

The last two digits test groups the last two digits and computes the frequency. The grouped frequency is matched against the expected uniform distribution. In the accounts payable file shown in Figure 5.10, numbers ending in .00 significantly exceed the expected distribution but were found to be normal based on their actual payments.

FIGURE 5.9 Benford's Law Second Order Test Example

FIGURE 5.10 Benford's Law Last Two Digits Test Example

Found a mistake? Please highlight the word and press Shift + Enter