For e-books on Amazon the relationship between the daily sales rate s and the rank r is approximately given by:
s = 100,000 / r
Such an inverse proportional relationship between a ranked quantity and the rank is called a Zipf distribution. So a book on rank r = 10,000 can be expected to sell s = 100,000 / 10,000 = 10 copies per day. As of November 2013, there are about 2.4 million e-books available on Amazon’s US store (talk about a tough competition). In this post we’ll answer two questions. The first one is: how many e-books are sold on Amazon each day? To answer that, we need to add the daily sales rate from r = 1 to r = 2,400,000.
s = 100,000 · ( 1/1 + 1/2 + … + 1/2,400,000 )
We can evaluate that using the approximation formula for harmonic sums:
1/1 + 1/2 + 1/3 + … + 1/r ≈ ln(r) + 0.58
Thus we get:
s ≈ 100,000 · ( ln(2,400,000) + 0.58 ) ≈ 1.5 million
That’s a lot of e-books! And a lot of saved trees for that matter. The second question: What percentage of the e-book sales come from the top 100 books? Have a guess before reading on. Let’s calculate the total daily sales for the top 100 e-books:
s ≈ 100,000 · ( ln(100) + 0.58 ) ≈ 0.5 million
So the top 100 e-books already make up one-third of all sales while the other 2,399,900 e-books have to share the remaining two-thirds. The cake is very unevenly distributed.