I've Got Your Number

How a mathematical phenomenon can help CPAs uncover fraud and other irregularities. EXECUTIVE SUMMARY * BENFORD'S LAW PROVIDES A DATA analysis method that can help alert CPAs to possible errors, potential fraud, manipulative biases, costly processing inefficiencies or other irregularities. * A PHYSICIST AT GE RESEARCH LABORATORIES in the 1920s, Frank Benford found that numbers with low first digits occurred more frequently in the world and calculated the expected frequencies of the digits in tabulated data. * CPAs CAN USE BENFORD'S DISCOVERY in business applications ranging from accounts payable to Y2K problems. In addition, subset tests identify small lists of serious anomalies in large data sets, making an analysis more manageable. * DIGITAL ANALYSIS IS WELL SUITED to finding errors and irregularities in large data sets when auditors need computer assisted technologies to direct their attention to anomalies. Is it possible to tell that a number is wrong just by looking at it? In some cases, you bet. Using Benford's law--a mathematical phenomenon that provides a unique method of data analysis--CPAs can spot :irregularities indicating possible error, fraud, manipulative bias or processing inefficiency. Benford's law is used to determine the normal level of number duplication in data sets, which in turn makes it possible to identify abnormal digit and number occurrence. Accountants and auditors have begun to apply Benford's law to corporate data to discover number-pattern anomalies. For large data sets, CPAs use highly focused tests that concentrate on finding deviations in subsets. EUREKA! Frank Benford made a simple observation while working as a physicist at the GE Research Laboratories in Schenectady, New York, in the 1920s. He noticed that the first few pages of his logarithm tables books were more worn than the last few and from this he surmised that he was consulting the first pages--which gave the logs of numbers with low digits--more often. The first digit of a number is leftmost--for example, the first digit of 45,002 is 4. (Zero cannot be a first digit.) Benford extrapolated that he was looking up the logs of numbers with low first digits more frequently because there were more numbers with low first digits in the world. Benford then tested this idea by looking at the first digits of 20 lists of numbers with a total of 20,229 observations. His lists came from varied sources, such as geographic, scientific and demographic data. One list contained all the numbers in an issue of Reader's Digest. He found that about 31% of the numbers had 1 as the first digit, 19% had 2, and only 5% had 9 as a first digit. Benford then made some physics-related assumptions about the distribution of naturally occurring data and, using integral calculus, he computed the expected frequencies of the digits and digit combinations. The expected frequencies of the digits in the first four positions can be seen in exhibit 1, page 80, which shows a large bias in favor of low digits in the first position. The probability that the first digit is either a 1, 2 or 3 is 60.2%. Exhibit 1: Benford Law-- Expected Digital Frequencies Position of digit in number Digit First Second Third Fourth 0 .11968 .10178 .10018 1 .30103 .11389 .10138 .10014 2 .17609 .10882 .10097 .10010 3 .12494 .10433 .10057 .10006 4 .09691 .10031 .10018 .10002 5 .07918 .09668 .09979 .09998 6 .06695 .09337 .09940 .09994 7 .05799 .09035 .09902 .09990 8 .05115 .08757 .09864 .09986 9 .04576 .08500 .09827 .09982 Example: The number 147 has three digits, with 1 as the first digit, 4 as the second digit and 7 as the third digit. The table shows that under Benford's law the expected proportion of numbers with a first digit 1 is 3. …