Pretty much the title.
I understand this is a law but can there be a reason why this happens?
For background: for almost all real life data like population, GDP and other real word stats, probability of it having 1 is like almost 30% which keeps decreasing with 9 to be least probable.
But why this happens, is this just a fascinating pattern in randomness?
In: 130
For many things, units are arbitrary. What I mean by that is that, for example, there is no particular reason why we measure lengths in metres. We could use *any* length as a base for units. If there is a “normal” way first digits are distributed, then that distribution shouldn’t change if change all our measurements to he based on another unit.
Eg let’s say we are measuring the lengths of something which can vary by many orders of magnitudes – eg the lengths of rivers. If we measure them and express their lengths in metres, we can look at the distribution of leading digits. But we can then convert them to any other unit of measurement, by multiplying all the lengths by some factor. So if we double all the measurements, then we are expressing the lengths in units of “half-a-metre”.
Now, let’s see which numbers in the first set of measurements map to numbers beginning with 1 in the second set of measurements. Every number beginning with 5,6,7,8 or 9 will map to a number beginning with 1.
So if there is a standard distribution of first digits, 1 must appear as a first digit equally as often as 5,6,7,8 and 9 combined!
By considering other multiples, we can find the distribution which is impervious to multiplication by a factor, and that is the one given by Benford’s Law. To be precise, 1 appears in a proportion of log(2/1), 2 with a proportion of log(3/2), 3 with a proportion log(4/3) etc – where all the logs are done base 10.
We can, in fact extend Benford’s law to other bases, by using the same formulas but changing the base of the logarithms.
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