Following are results from three hypothetical corporate tax returns. Each purports to be a list of expenditures, in dollars, that the corporation is claiming as deductions. Two of the three are genuine, and one is a fraud. Which one is the fraud? Write at least 6 complete sentences explaining why you think your chosen return is a fraud. 17 ORE 80.535 3.037 132.056 59.727 38.154 137.648 79.386 203.374 11.967 100239 46.428 7012 957359 SS1 284 97.439 780.216 22443 1.023 738,527 634,814 850.840 1.279 91.404 323.547 194.288 24,146 695.236 160.546 1.393 47,689 75.854 5.195 53,079 7.791 93,401 129,906 568,123 4,693 21.902 337,122 162.182 7,942 31,121 165.648 601,981 262.971 65,407 6.892 743.151 45,054 83.821 228.976 913.337 252378 2 581 538 342 99.613 78.175 64.888 1.643 812,618 126,811 13,545 22332 29,288 81,074 401.437 3,040 244,676 49.273 112111 56.776 262359 374.242 12138 14.204 31.484 1.818 104.625 34178 3684 11.665 15 376 541.894 65.92 250.601 650316 90.852

Here is the probability distribution of digits as predicted by Benford’s law: Digit Frequency 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046 The surprising nature of Benford’s law makes it a useful tool to detect fraud. When people make up numbers, they tend to make the first digits approximately uniformly distributed; in other words, they have approximately equal numbers of ls, 2s, and so on. Many tax agencies, including the Internal Revenue Service, use software to detect deviations from Benford’s law in tax returns.

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2633.66 3168.83 3301.11 3754.09 3834.44 5117.12 6448.27 7908.25 9181.43 11497.12 10786.85 10021.50 8341.63 10453.92 10783.01 10717.50 12463.15 13264.82 8776.39 Here is a frequency distribution of the first digits of the stock market averages: Digit Frequency For the stock market averages, the most frequent first digit by far is 1. The stock market averages give a partial justification for Benford’s law. Assume the stock market starts at 1000 and goes up 10% each year. It will take 8 years for the average to exceed 2000. Thus, the first eight averages will begin with the digit 1. Now imagine that the average starts at 5000. If it goes up 10% each year, it would take only 2 years to exceed 6000, so there would be only 2 years starting with the digit 5. In general, Benford’s law applies well to data where increments occur as a result of multiplication rather than addition, and where there is a wide range of values. It does not apply to data sets where the range of values is small

Virginia 7,567,465 6,287,759 Washington West Virginia 1,816,856 Wisconsin 5,536,201 509,294 Wyoming Here is a frequency distribution of the first digits of the state populations: Digit Frequency For the state populations, the most frequent first digit is 1, with 7, 8, and 9 being the least frequent. Now here is a table of the closing value of the Dow Jones Industrial Average for each of the years 1974–2008 First Digit Year 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 Average 616.24 852.41 1004.65 831.17 805.01 838.74 963.98 875.00 1046.55 1258.64 1211.57 1546.67 1895.95 1938.83 2168.57 2753.20

Indiana 6,271,973 lowa 2,966,334 Kansas 2,744,687 Kentucky 4,173,405 Louisiana 4.523,628 Maine 1,321,505 Maryland 5,600,388 Massachusetts 6,398,743 Michigan 10,120,860 Minnesota 5,132,799 Mississippi 2,921,088 Missouri 5,800,310 Montana 935,670 Nebraska 1,758,787 Nevada 2,414,807 New Hampshire 1,309,940 New Jersey 8,717,925 New Mexico 1,928,384 New York 19,254,630 North Carolina 8,683,242 North Dakota 636,677 Ohio 11,464,042 Oklahoma 3,547,884 Oregon 3,641,056 Pennsylvania 12,429,616 Rhode Island 1,076,189 South Carolina 4,255,083 South Dakota 775,933 Tennessee 5,962,959 Texas 22,859,968 Utah 2,469,585 Vermont 623,050

One of the most surprising probability distributions found in practice is given by a rule known as Benford’s law. This probability distribution concerns the first digits of numbers. The first digit of a number may be any of the digits 1, 2, 3, 4, 5, 6, 7, 8, or 9. It is reasonable to believe that, for most sets of numbers encountered in practice, these digits would occur equally often. In fact, it has been observed that for many naturally occurring data sets, smaller numbers occur more frequently as the first digit than larger numbers do. Benford’s law is named for Frank Benford, an engineer at General Electric, who stated it in 1938. Following are the populations of the 50 states in a recent census. The first digit of each population number is listed separately. State Population First Digit Alabama 4,557,808 Alaska 663,661 Arizona 5,939,292 Arkansas 2,779,154 California 36,132,147 Colorado 4,665,177 Connecticut 3,510,297 Delaware 843,524 Florida 17,789,864 Georgia 9,072,576 Hawaii 1.275,194 Idaho 1.429,096 Illinois 12,763,371