For a while now, I’ve been pondering a not entirely academic question — how, exactly, do you translate between amazon.com sales rank and the number of books moved. Since I have my own book finally out there, I thought it would be a good time to share my findings.

A note about method: All of this is based on amazon US sales only.

So here’s what I did. I programmed my computer to download my amazon book page once an hour for the last few months. I realize what this reveals about my psyche, and I’m comfortable with that. From there, I had it strip out all information except for the overall ranking. This predated the release of the book by quite a while, so I wouldn’t expect to sell more than a book every day or so — and I wasn’t disappointed.

In case you miss the essence of the plot, the idea is that between sales, my rank rises higher and higher (worse and worse) and then suddenly plunges once a sale is made. So long as sales are typically less than once an hour, I can even identify individual events, and figure out my model accordingly. In fact, for books with very little track record (not many sales in total), it seems like the decay time for ranks are about a day.

I was further curious as to the relationship between average rank and rate of sales. So as a next step, I took two random day and times, and counted the number of sales in between, and plotted them against the average rank in the interim. Again, this works fine if you assume that amazon has very limited memory. This is probably the case unless a book was a bestseller a few years ago, but now nearly forgotten. Here’s the relation between the average rank and rate of sales:

At the crap end of the distribution, it seems like the relationship is:

daily sales rate = 220,000/average rank

(Equations are why I marked this as “technical”).

I’m disinclined to believe that this trend holds all the way down to the good end. After all, with the exception of Harry Potter, Twilight, and the like, it’s unlikely that the #1 books are really selling a quarter of a million copies a day — on amazon alone.

But what about the long term effects?

Here’s where I had to make some assumptions about how amazon works. My model is that every book has a score:

y=Sum_i [ A*e^-(t-t_i)/tA+B*e^-(t-t_i)/tB+...]

In other words, you get more points for a sale if it was more recent. What’s more, it makes sense that amazon would have several relevant timescales. For instance, I found empirically that tA, the most highly weighted timescale, was about a day. I normalized that contribution to be A=1. There might be many timescales. Bestsellers in the last month, last year, and for all time. Due to my limited data, I only used 1 day, and 2 months. I found (empircally) a best fit for B=0.0003.

In case you’re curious, I basically did a sort of K-S test. My algorithm predicted a ranking of scores, and I saw whether my method put the various days and times into the right order. The final result gave a relationship between rank (x-axis) and score (y-axis) down to about 10,000.

I haven’t yet done any functional fitting, but if you take a look, a score of about 3 has a rank of about 20,000. This roughly, corresponds to about 10 sales a day on average (to get an average score of 20,000 with an exponential decay) — nicely fitting with our other estimate.

Interestingly — and this is noisy, so you’ll have to take this with a grain of salt — at the good end, the behavior is very flat. An increase of a factor of 3 in score (sales) corresponds to an increase in a factor of ~10 in rank. This suggests that on the good end, perhaps the rank/rate relationship is more like:

daily sales rate ~ C/sqrt(average rank)

But who can say? You, perhaps. I’d be interested in hearing about others’ experiences. For that matter, as my own book starts to get reviews it will (hopefully) rise enough in the ranks that I’ll be able to say something definitive from my own experience.

-Dave