Benford's Law, A Growth Industry

Often data in the real world have the property that the first digit 1 appears about 30% of the time, the first digit 2 appears about 17% of the time, and so on with the first digit 9 appearing about 5% of the time. This phenomenon is known as Benford’s law. This paper provides a simple explanation, suitable for nonmathematicians, of why Benford’s law holds for data that has been growing (or shrinking) exponentially over time. Two theorems verify that Benford’s law holds if the initial values and rates of growth of the data appear at random. INTRODUCTION. To get started, consider the sequence 2, 4, 8, 16, 32, 64, 128, . . . , 21000. Each of these numbers has a first digit, starting out with 2, 4, 8, 1, 3, 6, 1. About how many of these numbers do you think have first digit 1? What about the other eight possible first digits? Consider the same questions for 9, 81, 729,. . . , 91000, where the first few first digits are 9, 8, 7, 6, 5, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2. We will return to these questions after Table 3. Way back in 1881, Simon Newcomb observed that certain sets of data from the real world have the property that the first digits of the numbers do not appear uniformly often. In fact, the first digit 1 appears about 30% of the time, while the first digit 9 appears about 5% of the time. As often happens in science, this observation was forgotten and rediscovered later. Frank Benford observed this phenomenon in 1938 [2], and it has become known as Benford’s law or the “first-digit law.” He gave numerous examples of data from many sources, including newspaper items, areas of rivers, street addresses, cost data, and populations. We give another example in Table 1, namely data based on the populations of 117 cities in Indiana, from Wikipedia based on the 2000 census.1 For a good history up to 1975 with many references, see Raimi [17]. More references will be given at the end of the paper. This is the first and only state that I examined for this purpose.