Classroom Companion: Business

 
Chapter 16 · The Long Tail

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Hence, the long tail makes up 26.6% of all sales of books by Amazon, provided 
that the demand for books follows Zipf ’s law. Note that empirical results based on 
Amazon sales yielded a long tail of 36.7%. This means that Zipf ’s law gives a rea-
sonable estimate of Amazon’s long tail sales under the rather arbitrary assumption 
that 40,000 titles are in the head of the distribution.
If the tail is twice as long (4.6 million books), then the sales from the tail increase 
to 29.8%; that is, the sales from the first 2.3 million books in the tail amounts to 
26.6% of the total sale, while the next 2.3 million books in the tail amounts to only 
3.2% of the total sales. If the tail is only half as long (1.1 million books), the sales 
from the tail are 22.9% of the total sales.
7
Box 
16.2
 contains a generalization of Ziff ’s law applied to infinitely long 
tails.
Box 16.2 Generalization of Zipf’s Law
The size of the tail can be adjusted by 
applying a general discrete dissemina-
tion instead of Zipf ’s law. In a general 
discrete power-law distribution, the fre-
quency of exactly k events is:
f k
k
 

 



 

,
,
1
in which 
 

 





k
k
1
is the Riemann 
zeta function of argument a. This dis-
tribution is called the zeta distribution.
The parameter a alludes to the size 
of the tail of the distribution. If the tail 
starts at k = K = 40,001 (as in the exam-
ple with Amazon above), then the rela-
tive number of books, R, sold in the tail 
will be:
R
k
k
 
 



1
1
40 000
,
.

 
The Zipfian distribution is the special 
case in which α = 1. In this case, there 
must be an upper cutoff N for which 
f(X > N) = 0, and the zeta function is 
replaced by 
k
N
k



1
1
. The same applies if 
α
< 1 since the zeta function diverges for 
α
≤ 1.
.
Figure 
16.3
shows the relative 
number of books sold in the tail (R) as 
a function of α. Here, the cutoff value 
is N = 2,300,000, corresponding to the 
number of books available on Amazon. 
There are 40,000 books in the head of 
the distribution. Observe that the rela-
tive number of books sold in the tail 
(R) decreases from 98.3% when α = 0 to 
about 0% when α > 1.6. For α = 0, all 
book titles (both in head and tail) sell 
in the same numbers. In this case, sales 
are uniformly distributed. Furthermore, 
for α > 1.6, the tail is too small to have 
any economic value since books in the 
tail collectively contribute to virtually 
no sales. Note that in the Zipfian distri-
bution α = 1, resulting in R = 26.6% as 
calculated above. Note that α ≈ 0.94 
will match the empirical data of 
Amazon’s sales R = 36.7% for the 
parameters 
N 
= 
2,300,000 and 
k = K = 40,001. In other words, 
Amazon book sales as presented here 
can be modeled accurately using gen-
eral discrete power-law distribution 
with α = 0.94.