Suppose that we have a list of relevant numbers (e.g. maximum revenues for various years). If the leading digits are $1$ for all the years, can we say that trivially the maximum revenues satisfy Benford's Law?
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1$\begingroup$ AFAIK Benford's law says that in many real-world variables, 1 should be the first significant digit about 30% of the time, not 100% $\endgroup$– miuraCommented Jul 11, 2012 at 17:42
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$\begingroup$ @miura: I guess we can say that if the data does not follow Benford's law, then it was generated by some process? $\endgroup$– DamienCommented Jul 11, 2012 at 17:55
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$\begingroup$ You can't conclude much from noticing that the data does not follow Benford's law. The years don't follow Benford's law, either. $\endgroup$– Douglas ZareCommented Jul 11, 2012 at 19:15
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$\begingroup$ @Damien - we know the data was generated by some process no matter what. It might be a process with a random element, it might not. $\endgroup$– Patrick CaldonCommented Jul 11, 2012 at 23:20
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$\begingroup$ For tests of Benford's Law as applied to election data, you might check Walter Mebane's papers since 2006. $\endgroup$– CharlieCommented Jul 13, 2012 at 14:07
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1 Answer
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In short, no.
First, we expect around 30% of the numbers to be 1s.
Second, my guess is that you really want to know if the numbers were "generated by a natural process" or generated artifically, because that's what people use Benford's law for. Benford's law won't tell you much/anything on many kinds of time series data.
We have strong reason to believe the revenue of particular companies is autocorrelated year by year. If you look at first digits of an autocorrelated sequence, it oftentimes won't follow Benford's law despite being "generated by nature". I guess you could difference the sequence, but I don't know what a "(non?)-Benford's-law compliant first differenced sequence" means, I suspect not a lot.
How you might want to use Benford's law:
You should look at a chi-squared statistic for the expected (benford) distribution of digits and the real distribution, as a measure of the degree to which the real distribution satisfies Benford's law.
You can do statistical tests on this, but I don't think these add a lot of value - the real-world distribution might not correspond to Benford's law for a lot of reasons. Where the technique works ok/well is outlier detection - is when you have a population of distributions, say generated by different people. You can then get assurance from the population as a whole that the distribution "ought" to follow Benford's law to a reasonable degree, and look at outliers which satisfy Benford's law to a much lesser degree.
It works well on this intuitive/exploratory level. I've not seen anyone build more formal models which really worked effectively, but that might just be my ignorance.
