Radiometric dating graph
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.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.But since this operation is not allowed to change the probability of leading digits, that means that the probability of having a leading digit of 1 must be the same as the probability of having a leading digit of any of 5, 6, 7, 8 or 9.This property is satisfied by the formula given above, since Of course, there is nothing special about multiplying the numbers from our random source by 2, so a similar property must hold regardless of what we multiply our numbers by.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.
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.
the left most non-zero digits) of one of these numbers will be is approximately equal to .