to obey Benford's law When the mean is greater than the median and the skew is positive Numbers that result from mathematical combination of numbers: e.g. quantity × price Transaction level data: e.g. disbursements, sales Distributions that would not be expected to obey Benford's law Where numbers are assigned sequentially: e.g. check numbers, invoice numbers Where numbers are influenced by human thought: e.g. prices set by psychological thresholds ($9.99) Accounts with a large number of firm-specific numbers: e.g. accounts set up to record $100 refunds Accounts with a built-in minimum or maximum Distributions that do not span an order of magnitude of numbers. Benford’s law compliance theorem Mathematically, Benford’s law applies if the distribution being tested fits the "Benford’s law compliance theorem". The derivation says that Benford's law is followed if the Fourier transform of the logarithm of the probability density function is zero for all integer values. Most notably, this is satisfied if the Fourier transform is zero (or negligible) for n ≥ 1. This is satisfied if the distribution is wide (since wide distribution implies a narrow Fourier transform). Smith summarizes thus (p. 716): Benford's law is followed by distributions that are wide compared with unit distance along the logarithmic scale. Likewise, the law is not followed by distributions that are narrow compared with unit distance … If the distribution is wide compared with unit distance on the log axis, it means that the spread in the set of numbers being examined is much greater than ten. In short, Benford’s law requires that the numbers in the distribution being measured have a spread across at least an order of magnitude. Tests with common distributions Benford's law was empirically tested against the numbers (up to the 10th digit) generated by a number of important distributions, including the uniform distribution, the exponential distribution, the normal distribution, and others. The uniform distribution, as might be expected, does not obey Benford's law. In contrast, the ratio distribution of two uniform distributions is well-described by Benford's law. Neither the normal distribution nor the ratio distribution of two normal distributions (the Cauchy distribution) obey Benford's law. Although the half-normal distribution does not obey Benford's law, the ratio distribution of two half-normal distributions does. Neither the right-truncated normal distribution nor the ratio distribution of two right-truncated normal distributions are well described by Benford's law. This is not surprising as this distribution is weighted towards larger