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In his 1995 paper, Hill points out that random samples from random samples will usually give rise to data that satisfy Benford's law. He mentions a newspaper frontpage as an example where data may come from economics, sports, or weather forecast; their mixed aggregate will follow Benford's law whereas individual distributions may not.

Durtschi et al. (2004), however, put the emphasis on individual variables being the outcome of several variables, e.g. accounts being a product of quantity sold and prices, for Benford's law to hold (among other conditions).

In the literature on Benford's law, the choice for analysing either all variables together (Hill) or variables separately (Durtschi) is rarely justified. Often, the two are combined. This extends to the treatment of longitudinal data where observations are sometimes analysed by period (e.g. a survey wave) and sometimes lumped together, effectively ignoring their longitudinal character.

My data contain several variables (communions, marriages, baptisms) from twenty parishes for ten successive years. Is it acceptable to

  1. lump the data together across parishes and years

  2. analyse all variables together instead of analysing them separately?

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I think it depends what you are trying to do. For example, people often use Benford's Law to try to detect whether data have been faked (personally I think they are on very shaky ground, but I know some who swear by it. Those same people never seem to be interested in experimenting with what happens when you apply the law to non-fraudulent data sets. But anyway... )

The general expectation is that the more unrelated data you throw together, the closer you should get to the ideal Benford's Law relationship. So if you throw all your data together and find that Benford's Law doesn't hold, then you ought to be in a stronger position when you claim that they have been faked. If you are applying a statistical test, this will also give you a bigger sample size and therefore more chance of getting a "significant" result.

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  • $\begingroup$ Thank you for your response. I share your scepticism regarding statistical tests and wanted to go more for a graphical comparison. And Benford's law will not be the only check. $\endgroup$ – joapo May 6 '15 at 13:49

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