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Various probability distributions are shown (the first five that I happened to think of) together with the frequency of leading digits (from 1 to 9) that I got when sampling 100,000 points from that distribution (where the frequencies are depicted by the blue bars).For each, the pink line shows what we would expect to get if Benson’s Law held perfectly.Some insight into Benford’s Law can be gleaned from the following mathematical fact: If there exists some universal distribution that describes the probability that numbers sampled from be the formula given above.The reason for this is because if such a formula works for all sources of data, then when we multiply all numbers produced by our source by any constant, the distribution of the likelihood of leading digits must not change. Now notice that if we have a number whose leading digit is 5, 6, 7, 8, or 9, and we multiply that number by 2, the new leading digit will always be 1.For the technical details and restrictions, check out Hill’s original 1996 paper.Perhaps the best way to quickly convince yourself that Hill’s result is true is to look at the graphs found below.

decimal) we would simply replace the in the formulas above with .Radiometric dating measures the decay of radioactive atoms to determine the age of a rock sample.It is founded on unprovable assumptions such as 1) there has been no contamination and 2) the decay rate has remained constant.As you can see, for some distributions we get a good fit (e.g.the exponential and log normal distributions) whereas for others the fit is poorer (e.g. What the third graph in each table shows is the distribution of leading digits that we get when, instead of sampling just from one copy of each distribution, we sample from 9 different copies (with equal probability), each of which has a different variance (in most cases chosen to be proportional to the values 1 up to 9).

Besides just being generally bizarre and interesting, Benford’s Law has lately found some real world applications. 