Benford's Law - Limitations

Limitations

Benford's law can only be applied to data that are distributed across multiple orders of magnitude. For instance, one might expect that Benford's law would apply to a list of numbers representing the populations of UK villages beginning with 'A', or representing the values of small insurance claims. But if a "village" is defined as a settlement with population between 300 and 999, or a "small insurance claim" is defined as a claim between $50 and $100, then Benford's law will not apply. More generally, if there is any cut-off which excludes a portion of the underlying data above a maximum value or below a minimum value, then the law will not apply.

Consider the probability distributions shown below, plotted on a log scale. In each case, the total area in red is the relative probability that the first digit is 1, and the total area in blue is the relative probability that the first digit is 8.

For the left distribution, the size of the areas of red and blue are approximately proportional to the widths of each red and blue bar. Therefore the numbers drawn from this distribution will approximately follow Benford's law. On the other hand, for the right distribution, the ratio of the areas of red and blue is very different from the ratio of the widths of each red and blue bar. Rather, the relative areas of red and blue are determined more by the height of the bars than the widths. The heights, unlike the widths, do not satisfy the universal relationship of Benford's law; instead, they are determined entirely by the shape of the distribution in question. Accordingly, the first digits in this distribution do not satisfy Benford's law at all.

Thus, real-world distributions that span several orders of magnitude rather smoothly (e.g. populations of settlements, provided that there is no lower limit) are likely to satisfy Benford's law to a very good approximation. On the other hand, a distribution that covers only one or two orders of magnitude (e.g. heights of human adults, or IQ scores) is unlikely to satisfy Benford's law well.