By Popular Demand: A Non-Rant About Statistics

[melindadansky]

and something that I consider so beautiful it makes me want to cry (This one's for you markmc03Allow me to introduce you to Benford's law: a lovely mathematical fact that proves our intuition is often wrong, that "common sense" isn't always either of those things, and that funky mathematical facts can often be turned to practical use

Comments

[markmc03]

Hmmmm. Huh? If each digit is purely randomly selected from numbers 1 through 9, a 1 is more likely to be selected than a 9? But if each number has a 1 in 9 chance of being selected, how is it that 1 would come up with greater frequency than 9? I'm missing something. But then I'm not a mathematician. So I must defer to greater knowledge. I still place more faith in Murphy's law than Bender's, but what do I know? :)

[athenian_abroad]

The trick is to notice that each digit isn't randomly selected; only the whole number is. Consider some measurement that is uniformly distributed on the interval [1, 200]. There are two hundred possible "draws", but more than half have leading ones. (Specifically 1, 10-19 and 100-199). Now, that's an extreme case; on the interval [1, 300], only about a third of the results has a leading one -- but that's still way ahead of one in nine. If you go all the way to [1, 999], the odds of getting a leading one fall to just about one in nine. And then, as you continue to raise the upper bound, the probability of getting a leading one begin rising again.

[markmc03]

I think my brain just exploded. I'll stick to acting and leave the statistical measurements to the real mathematicians. Cheers.

[kuroshii]

even the entire numbers aren't precisely randomly selected...they're defined as being the "result of a natural process." meaning, measurements of something. or results of something.

[melindadansky]

Look! A thing!

[badger]

I love the bit in the wikipedia entry describing an empirical observation that led to an earlier recognition of Benford's Law:

"...the American astronomer Simon Newcomb noticed that in logarithm books (used at that time to perform calculations), the earlier pages (which contained numbers that started with 1) were much more worn than the other pages."

[ricevermicelli]

I love that bit too.

[melindadansky]

That is awesome.

[eoyount]

Very nice explanation. I prefer Bender's law though - kill all humans.

[melindadansky]

I also prefer Bender's law, but it's more difficult to apply.

[bruceb]

Fascinating stuff. Thanks. :)

[melindadansky]

Thank you!