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Amazon sales rank, book sales and power law distributions

Discussion in 'Research' started by Forest, Jul 11, 2014.

  1. Forest

    Forest Beloved Grand Eagle

    In the following post, Dr. David Clarke brought up a point that I wanted to respond to:
    My response gets pretty technical and if people reply it could get even more technical, so I thought it would be a good idea to start a new thread for the new discussion.

    Anyway, here is what Dr. Clarke wrote:
    I haven't had a chance to read up on this, but I wonder if what you are saying is captured in the idea that book sales follow a power law. The following paper provides an introduction to this idea:
    The authors write, "In this paper, we analyze the real sales data of the book sales market in Japan. The data which we analyze is the data taken from POS (Point Of Sales) system of over 2,000 bookstores in all areas of Japan. The target term is from April 2005 to March 2006. As a result of analysis, it became clear that the relation between sales volume and sales rank follows power law in both of annual and monthly sales."

    In Appendix B of this article, economists use data to estimate a power-law formula that converts Amazon sales rank into weekly sales numbers:

    I believe the resulting formula is Sales/week = e to the power (10.526-(.87*sales rank)

    See also this link.

    Practically speaking, it seems inevitable that the formula will only provide a very rough estimate given how complex the market is, how the market can change over time, and how very rapidly Amazon sales rank fluctuates for individual books.

    More reference can be found on Google. But for anyone reading this thread later on, the bottom line seems to be that books with low Amazon sales ranks sell a LOT of books, so it gets harder and hard to move up the ranks, the farther you go.
  2. davidclarke

    davidclarke Author & Physician

    My reference to the relationship between Amazon Sales Rank and Book Sales per Week being exponential does mean the same thing as saying it follows a power law. The formula you quoted, however, should read:

    Sales/Week = e ^ (10.526 - (.87 * Ln(Sales Rank))) where

    e = 2.718282 (the base of the natural logarithms)
    ^ means "to the power of"
    Ln(Sales Rank) is the natural logarithm of the Sales Rank
    In Excel this would be: EXP(10.526-(0.87*LN(Sales Rank)))

    For the two days that my book averaged a Sales Rank of about 3600 we would have:

    Sales/Week = 2.718282 ^ (10.526 - (.87 * Ln(3600))) = 2.718282 ^ (10.526 - (.87 * 8.1887) = e^3.4 = 30

    But I only sustained that sales rank for two days implying I would have sold 2/7 x 30 books = 8-9 books during those two days.
    Eric "Herbie" Watson likes this.
  3. Forest

    Forest Beloved Grand Eagle

    Very interesting. Thanks for the correction, too.
    Eric "Herbie" Watson likes this.
  4. davidclarke

    davidclarke Author & Physician

    Amazon's statistics for the week in which I hit the 3000-4000 range in sales rank showed I sold 13 books which is consistent with selling 8-9 on that Monday-Tuesday and just a 4-5 more the rest of the week so the formula above was certainly in the ball park. Their author's web page also showed most of my sales that week were from the Portland, OR metro area. Not sure why that happened since I gave no lectures there that week but my book is popular there (my home town) and so it was likely just a temporary spike.
  5. Walt Oleksy (RIP 2021)

    Walt Oleksy (RIP 2021) Beloved Grand Eagle

    Good luck with the book sales, David. If you help even just one person with your book, it is a success.
    Forest likes this.
  6. mugwump

    mugwump Well known member

    Goodluck with the book sales!

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