Benford’s distribution
Benford’s distribution specifies that the probability of the first significant digit x1 of a numeric value x will be d (d = 1, 2, …, 9)
P(x1 = d) = log10(1 + 1/d)
Here the first significant digit of 4509 and 0.048 is 4. For numeric values that follow this distribution, the probability that the first significant digit will be 4 is log10(1 + 1/4) = .09601.
The probabilities for each digit are shown in Figure 1. Note that the probabilities in this table add up to 1 as expected.
Figure 1 – Benford probabilities
The distribution can be applied to more than the first significant digit. E.g. the probability that the first two significant digits are 43 (such as 4357 and 0.0004301) is given by the same formula. Thus, the probability that the first two significant digits are 43 is log10(1 + 1/43) = .009984.
Benford’s law
Numerical values that follow Benford’s distribution are said to obey Benford’s law.
It turns out that numeric values in many real-world situations obey this law. E.g street addresses and values in a tax return. It is often applied to fraud detection. E.g. if too many of the key numbers in a balance sheet or tax return start with higher digit numbers, then this may be an indication that the numbers are fraudulent (since according to Figure 1, the first digit of such numbers should favor 1, 2, and 3 over 7, 8, and 9).
Notably, random numbers between 0 and 1 don’t follow Benford’s law.
Goodness-of-Fit
There are a number of goodness-of-fit techniques for determining whether a set of numbers follows Benford’s law. We will consider three such techniques: the chi-square test, the Kolmogorov-Smirnov test, and the Anderson-Darling test. Click here for an explanation of these goodness-of-fit tests.
References
Wikipedia (2022) Benford’s law
https://en.wikipedia.org/wiki/Benford%27s_law
Morrow, J. (2010) Benford’s law, families of distributions and a test basis
http://www.johnmorrow.info/projects/benford/benfordMain.pdf
Lesperance, M., Reed, W. J., Stephens, M. A., Tsao, C., Wiltons, B. (2016) Assessing conformance with Benford’s Law: Goodness-of-fit tests and simultaneous confidence intervals. PLoS ONE
https://doi.org/10.1371%2Fjournal.pone.0151235
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4809611/