Editing Benford's law (section)

===Multiplicative fluctuations===
Many real-world examples of Benford's law arise from multiplicative fluctuations.<ref name=Pietronero>{{cite journal |title=Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf |author=L. Pietronero |author2=E. Tosatti |author3=V. Tosatti |author4=A. Vespignani |journal=Physica A |year=2001 |volume=293 |issue=1–2 |pages=297–304 |doi=10.1016/S0378-4371(00)00633-6 |bibcode = 2001PhyA..293..297P |arxiv=cond-mat/9808305}}</ref> For example, if a stock price starts at $100, and then each day it gets multiplied by a randomly chosen factor between 0.99 and 1.01, then over an extended period the probability distribution of its price satisfies Benford's law with higher and higher accuracy.

The reason is that the ''logarithm'' of the stock price is undergoing a [[random walk]], so over time its probability distribution will get more and more broad and smooth (see [[#Overview|above]]).<ref name=Pietronero/> (More technically, the [[central limit theorem]] says that multiplying more and more random variables will create a [[log-normal distribution]] with larger and larger variance, so eventually it covers many orders of magnitude almost uniformly.) To be sure of approximate agreement with Benford's law, the distribution has to be approximately invariant when scaled up by any factor up to 10; a [[log-normal]]ly distributed data set with wide dispersion would have this approximate property.

Unlike multiplicative fluctuations, ''additive'' fluctuations do not lead to Benford's law: They lead instead to [[normal probability distribution]]s (again by the [[central limit theorem]]), which do not satisfy Benford's law. By contrast, that hypothetical stock price described above can be written as the ''product'' of many random variables (i.e. the price change factor for each day), so is ''likely'' to follow Benford's law quite well.