Late last year, Matthew Beckler was nice enough to make a sales rank tracker for How to Teach Physics to Your Dog. Changes in the Amazon page format made it stop working a while ago, though, and now Amazon reports roughly equivalent data via its AuthorCentral feature, with the added bonus of BookScan sales figures. So I’ve got a new source for my book sales related cat-vacuuming.

Still, there’s this great big data file sitting there with thousands of hourly sales rank numbers, and I thought to myself “I ought to be able to do something else amusing with this…” And then Corky at the Virtuosi did a post about Benford’s Law, and I said “Ah-ha!”

Benford’s Law, if you’re not familiar with it, says that in a large assortment of numbers generated by some process, you expect the first non-zero digits of all the numbers to be distributed in a logarithmic fashion. About 30% of the first digits should be “1,” and only about 4% of the first digits should be “9.” This goes against the naive expectation that the numbers ought to be evenly distributed, and is actually used by “forensic accountants” to catch people who are cooking their books– someone who is making up numbers to fill a phony set of books is fairly likely to pick numbers that don’t follow a Benford’s Law distribution.

So, I’ve got 6,818 hourly values of the Amazon sales rank for my book, spanning almost three orders of magnitude. How do those digits match up with Benford’s Law? Well:

That’s… pretty good, really. The blue diamonds are the actual frequency of the digit, the red squares are the prediction of Benford’s Law. There’s a slight shortage of 1’s and a surplus of 5’s and 6’s, but all the actual frequencies are within about 5% of the expected values. The most basic assumption about the statistics of this sort of data set would lead you to expect an uncertainty of about 1% (that is, 1 over the square root of 6818), but that’s pretty crude.

What does this tell us? Not a whole lot, really. if Amazon is somehow fudging their sales rank data (which I have no reason to suspect them of doing), they’re clever enough not to get caught by this really crude analysis of one book’s figures.

Making this graph has, however, given me a way to put off some tedious and annoying work for another hour or so, so let’s hear it for Benford’s Law!

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