Here's one way to present the idea:
Suppose you send 100 peasants wandering through 1000 acres of woodland in Southern France for 10 days randomly collecting sticks, and you have someone carefully measure the lengths of these sticks in millimeters, Angstroms, cubits, and nautical miles. You then use these measurements to generate a histogram by counting the total number of measurements whose most significant digit is 1 and letting that (very large integer) be the height of the first "bar" in the histogram. Then take the number of measurements whose most significant digit is 2, and let that be the height of the second "bar" in the histogram. Do the same for digits 3, 4, 5, 6, 7, 8, and 9, and you've got a histogram with 9 very tall bars.
Would you expect these bars to have the same height? In other words, would you expect to have the same number of measurements with most significant digit 1, 2, 3, ... 9?
Many people would. After all, if you generate numbers randomly, isn't that just the same thing as generating random digit sequences by rolling a 10-sided die over and over again?
No. It is not.
As it turns out, there are more 1s than 2s, 2s than 3s, etc. One way to see that this must be true is to begin with the idea that the "distribution" of measurements in the "significant digit histogram" should remain the same regardless of what unit is used. In other words, if there really were as many 2s as 1s, then it shouldn't matter if meters or inches are used to make the measurements. If you believe this, then consider what happens to your measurements when you switch from a unit of 1 cm to a unit of 0.5 cm : obviously all the length measurements would double. This means that all measurements which used to start with 1 will now start with 2 or 3, and all measurements which used to start with 2 will now start with 3, 4 or 5. But all measurements which used to start with 5, 6, 7, 8, or 9 will now all start with 1.
It turns out that the only distribution on the significant digits which is invariant under this sort of unit transformation is this:
prob(first digit = d) = log_base_10(1 + 1/d), for d=1,2,3,4,5,6,7,8,9This is called Benford's Law, and it was originally discovered in 1881 by astronomer & mathematician Simon Newcomb, observed that log tables in the public library were much dirtier near the beginning than near the end, and who published a paper on the topic in the
This is all written up quite nicely in Volume 86, Number 4 of the American Scientist, published in July of 1998, and has appeared in many other popular media since being recently applied with some success in fraud detection (naively cooked books don't satisfy Benford's law).