Benford's law
Take a long list of numbers and look only at the first digit of each. You would expect 1 through 9 to show up equally, about 11% each. For many real-world lists they do not. Pick a list, and for the random ones note which generator made it. The bars are the first-digit counts; the outlines are what Benford's law predicts.
What's going on
Benford's law says that in many lists the first digit is 1 about 30.1% of the time, 2 about 17.6%, down to 9 at 4.6%. The formula is log10(1 + 1/d). The orange outlines show that prediction; the blue bars are what you counted.
Why? Think about something that grows by multiplying, like a number that keeps doubling. To get from 1,000 to 2,000 it has to double once, but to get from 9,000 to 10,000 it needs only a small nudge. So the number spends a long time with a leading 1 and hurries through the leading 9s. On a logarithmic scale the numbers starting with 1 fill 30.1% of each decade and the numbers starting with 9 fill 4.6%, so a list whose entries land evenly along that scale obeys the law. Powers of 2, Fibonacci numbers and factorials happen to do this.
The condition is where each number lands inside its decade, not growth and not range. The powers of 10 grow by multiplying and span thirty orders of magnitude, yet every one starts with 1. Whole dollars drawn uniformly from $5 to $4,000 span a wide range, yet 1, 2 and 3 each get about 27.8% and the other six digits about 2.8% each. Switch between the uniform and log-uniform generators with the same bounds: the generator decides the tally, not the range. Squares and uniform draws from 1 to 1000 miss for the same reason.
Accountants use the law as a screening tool. People inventing figures often spread the first digits evenly, so a tally far from the outlines is a reason to look closer. It is only a reason to look. Honest data fails the test whenever there was no reason to expect the law, as the uniform dollars show, and anyone who knows the law can invent numbers that pass it. Before reading anything into a tally, ask where the numbers came from and whether you expected the pattern at all.
Sheet: The law of the first digit.