Impact Factors and the Central Limit Theorem: Why citation averages are scale dependent

We apply the Central Limit Theorem to study how citation averages, and Impact Factors (IFs) in particular, depend on scale. For a journal of n papers randomly selected from a population, we expect from the Theorem that its IF fluctuates around the population average μ, and spans a range of values proportional to σ/√n, where σ² is the variance of the population's citation distribution. The 1/√n dependence has profound implications for IF rankings: The larger a journal, the narrower the range around μ where its IF lies. IF rankings therefore allocate an unfair advantage to smaller journals in the high IF ranks, and to larger journals in the low IF ranks. This implies a scale-dependent stratification of journals in IF rankings, whereby small journals occupy all ranks, mid-sized journals occupy the middle ranks, and very large journals have IFs that asymptotically approach μ. We confirm these predictions by analyzing 20 years of Impact-Factor and journal-size data, and the citation distributions of 11,000 journals. We propose the Φ index, a rescaled IF that accounts for size effects, and which can be readily generalized to account also for different citation practices across research fields [1].

[1] M. Antonoyiannakis, J. Informetrics 12, 1072 (2018).