When you're #253,287 you try harder

My new relativity textbook now has an Amazon sales rank -- I'm 253,287 as of this morning. I did my best to lord it over my wife, whose mathematics textbook is at a modest 607,187. Of course, she has one (very favorable) reader review, while mine us as yet uncommented-upon.

So, how many books does a sales rank like this represent? Though Amazon keeps its actual sales figures secret, various people have tried to reverse-engineer the ranking system and recover raw sales estimates. It turns out that Amazon sales follow a power law: the number of books sold per unit time is, very roughly speaking, proportional to the rank raised to a power, which empircally seems to be around -1/2. There are deviations from this law at the ends of the distribution -- that is, for books that sell very many or very few copies -- but it works pretty well overall.

So we have the relation between sales S and rank R:

S ∝ R^-1/2 .

Consider books A and B. Book A has a sales rank of 100, and sells (let us say) 30 copies per day. (This is consistent with the estimates here.) Book B has a sales rank of 10,000. The power law tells us that it should be selling around 3 copies per day -- 10 times fewer copies for a sales rank 100 times greater. It makes sense, of course, that the book with a lower rank number sells much better.

But notice that there are a lot of books with higher rank numbers. The books with ranks 100 through 200 might together sell about 2500 copies per day. (Figured how? I wrote a C++ program, naturally. But I needn't have bothered. If #100 sells 30/day, then #200 should sell around 21/day, and the average in the range between these two should be around 25/day. Since there are a hundred books in that range, that makes 2500/day.) The books with sales ranks between 10,000 and 20,000 sell ten times fewer copies, but there are a hundred times as many titles; together, they should sell around 25,000 copies per day.

Moral: Most of the books sold are not the best-sellers. The long "tail" of books, each one of which only has a few sales, contains most of the typical volume of Amazon sales.

I'm happy to do my part for Jeff Bezos. Just how many sales are implied by my rank of 253,287? Applying the power law, and assuming 30/day for the book with rank 100, I estimate ... about 0.6 copies per day. Four per week. Hey, that isn't as bad as I thought! (But it's going to be a while before I can afford that chateau in the Alps.)

Of course, I'm probably wrong, since 253,287 is getting close to a "bend" in the curve, where the sales figures start diminishing more rapidly with rank. At least I'm not out there with Howard R. Feldman's Brachiopods of the Onondaga Limestone in Central and Southeastern New York, which has a sales rank today of 3,559,913. (Though I must confess that my book is several times more expensive than his, and his book has a nice customer review: "Feldman is a genius ... Funnier than anything Jim Carey's ever done." You'd think that this is exactly the kind of word-of-mouth that really ought to move your paleontology book. Weird.)

Note: Amazon sales probably are not representative of overall sales for textbooks, which are commonly ordered by college bookstores directly from the publisher. A friend of mine has a best-selling textbook in another field with an Amazon rank up around 16,000, That is not very impressive-sounding, but he does pretty well. His chateau is in the Adirondacks.