Radiometric dating graph
By dating rocks of known ages which give highly inflated ages, geologists have shown this method can’t give reliable absolute ages.Many geologists claim that radiometric “clocks” show rocks to be millions of years old.Hence, what we are doing is sampling from multiple distributions each of which is the same except for a scaling factor, and then pooling those samples together, at which point we calculate the probability of the various leading digits (how often is 1 the first non-zero digit, how often is two the first non-zero digit, etc).The result in every case shown is that this leads to a distribution of leading digits that fits Benford’s Law quite well.Hence, we should already expect the law to approximately hold in some real world scenarios.More importantly though, as was demonstrated by the probabilist Theodore Hill, if our process for sampling points actually involves sampling from multiple sources (which cover a variety of different scales, without favoring any scale in particular), and then group together all the points that we get from all of the sources, the distribution of leading digits will tend towards satisfying Benford’s Law.In fact, it seems that the law was first discovered due to a realization (by astronomer Simon Newcomb in 1881) that the pages of logarithm tables at the back of textbooks are not equally well worn.What was noticed was that the earlier tables (with numbers starting with the digit 1) tended to look rather dirtier than the later ones.
the front pages of newspapers, tables of physical constants at the back of science textbooks, the heights of randomly selected animals picked from many different species, etc.), the probability that the leading digits (i.e.the left most non-zero digits) of one of these numbers will be is approximately equal to .That means the probability that a randomly selected number will have a leading digit of 1 is which means it will happen about 30.1% of the time, whereas the probability that the first two leading digits will be 21 is given by which means it will occur about 2.0% of the time.Besides just being generally bizarre and interesting, Benford’s Law has lately found some real world applications.
For certain types of financial data where Benford’s Law applies, fraud has actually been detected by noting that results made up out of thin air will generally be non-random and will not satisfy the proper distribution of leading digits.Benford’s Law indicates that in base 10, the most likely leading digit for us to see is 1, the second most likely 2, the third most 3, the fourth most likely 4, and so on.But why should this be true, and to what sorts of sources of random numbers will it apply?On the other hand though, a great deal of data has been collected (e.g.