EXAMPLE 21 The Mean of the Probability Model for benford’s Law
In Self Check 8, you were asked to create the first digits for 20 fictitious invoice numbers. To ensure these numbers were random, you used a random digits table. That way each integer, 1 through 9, was equally likely to be chosen as a first digit. The table below shows the probability model governing your selection of first digits.
| First digit | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Probability |
The mean of this model is
377
If, on the other hand, legitimate records obey Benford’s law, the distribution of the first digit is
| First digit | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Probability | 0.301 | 0.176 | 0.125 | 0.097 | 0.079 | 0.067 | 0.058 | 0.051 | 0.046 |
The mean of Benford’s model is
The comparison of means between Benford’s law and random digits, , reflects the greater probability of smaller first digits under Benford’s law. Probability histograms for these two models appear in Figure 8.21. Because the histogram for random digits (Figure 8.21a) is symmetric, the mean lies at the center of symmetry. We can’t determine the mean of the right-skewed Benford’s law model precisely by simply looking at Figure 8.21b; calculation is needed.
