have a look at a table of physical constants, which are numbers spanning
a wide range, and count the occurrence of the digit "1", of the digit
"2" and so on. you will find, that the digit "1" occurs more often than
"2", with decreasing frequency down to "9". is that odd?

This is because of scale invariance and Benford's Law.

ReplyDeleteIf every digit would have the same probability to occur, rescaling by a factor of e.g. 2 would lead to a nonuniform distribution of first digits. In that case the former first digits 5-9 would become 1, 4 would become 8 or 9,... Rescaling by another factor would lead to another distribution.

If the compared numbers cover a wide range and have different scale factors (or units) naturally the first digit distribution will become nonuniform. Instead the first digits will form a logarithmic distribution.

I completely agree with the answer: this fact is known as Benford's law and refers to the logarithmic uniform distribution of the numbers, because there's no scale involved (and there can't be any scale involved because the choice of units is free, as pointed out by the previous post!). Benford discovered the law, weirdly enough, empirically by noticing that the corners of the logarithmic tables for the numbers starting with "one" were more strongly worn out than the other numbers. Benford's law has even an application in certified accountancy, as the distribution of sums in money transfers in large company has the same logarithmic uniform distribution, and a deviation from that would indicate fraudulent transfers with forged cheques.

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