The challenges of using ranks to estimate sales

Researchers have frequently used data on product ranks to estimate nonpublic sales quantities, believing that there is a power-law-induced linear relationship between logged sales ranks and logged sales. Using essentially complete data on book sales, the most commonly used product in this literature, we find that the (double-logged) relationship between sales ranks and quantity sold is not linear, but robustly concave. We demonstrate that this concavity is likely to cause very poor sales predictions in many instances. We provide two concrete examples where applying this linear method to the concave relationship has led to serious errors in sales estimates. First, in the claim that the Internet's greater product variety in books has a large positive impact on social welfare, and second, in a claim about relative sales levels in top 20 and top 50 music “charts.” We also explore the use of nonlinear specifications as an alternative method to predict sales from ranks and find a simple specification that provides much better sales estimates. Finally, we examine the possibility that a particular type of biased sample might allow reasonable linear estimates of industry sales and conclude that it is possible but quite unlikely.