Sometimes it’s amazing how deeply buried the useful nuggets of information are. Here is a power law that converts Amazon sales rank to units sold per week.
Sales/week = e to the power (10.526-(.87*sales rank)
As an Excel formula: =EXP(10.526-(0.87*LN(rank)))
This formula is buried deep inside an academic paper by MIT econometricians Brynjolfsson and Smith — equation (9), if you’re curious — referenced on Chris Anderson’s LONG TAIL web site.
The formula is backed up by four different studies conducted from 2001 to 2005 which have all found r, or measures of correlation, of .8 or higher for values of beta ranging from -.855 to .-91 (above, I used -.87) using sales data for samples containing as many as a thousand titles. In other words, although it obviously isn’t perfect, this is a pretty darned handy approximation of sales rank to units sold. Incredibly, although the formula is extremely simple and can be expressed in one line, I have never seen it reproduced anywhere else. I offer it here as a service to readers.
Tip: I did a retrospective calculation using my Ingram sales data and Amazon sales ranks for my best-selling title, which usually sells in the range between 10,000 and 40,000, and found that a beta of 0.936 accurately predicted total sales to date.
Related posts:
[...] estimates, you would be averaging a sales rank of 100. This is according to Nimble Book’s Power Law formula. That data is based on 2006 data. You may also have a sales rank around 300, just like “The [...]
[...] It’s a logarithmic function, so as sales rank decreases, unit sales rise exponentially. For a great explanation, see http://www.fonerbooks.com/surfing.html, and for the goriest possible details, see http://www.nimblebooks.com/wordpress/2006/06/power-law-converting-amazon-sales-ranks-to-units-sold/. [...]
[...] This model is based on Nimble Books sales data plus four different academic studies from 2001 to 2005 whose estimates of the relationship between Amazon unit sales and sales ranks all found r, or degrees of correlation, of .8 or higher; for details and citations, see this article on the Nimble Books website. The primary financial risks in this model are that any increase in printing costs or increase in the short discount rate would adversely affect both publisher and author net compensation. [...]
Or to express it in terms suitable for using Google rather than booting up a spreadsheet:
e^(10.526-(.87*ln(RANK)))
where RANK is replaced with a real number
The results understate sales at the bottom of the chart, those books that sell one or two a day while bouncing between 30,000 and 200,000
Unfortunately, I don’t yet have any experience at the opposite extreme.