Power-law distribution and the right question

I really didn’t like this article at Forbes. I agree with the main line of the article, but not with one of the reasons to support the main argument. The central thesis is that it’s not enough to know math and statistics. You need to have contextual knowledge, sou you make the right questions. Fair enough.

However, one evidence that the article brings is that most of stuff measured by humans are power-law distributed, a realm where Central Limit Theorem doesn’t apply. The funny thing is, he’s technically correct, but probably asked the wrong question (which is the whole point of the article). Take a look at a paper linked on this blog post, if you’re interested about when power-law distribution do occur. But if you want to skip the details, I’ll summarize for you, and it’s pretty simple.

Roughly speaking, the Central Limit Theorem says that, if you add up several (independent) measures, the mean of this measures is Normally distributed (the Bell curve). Now, if some phenomenon is the result not of the sum of several components, but say, the multiplication of several components, the Central Limit Theorem will fail. However, in cases like that, we always have the logarithm. Remember that log (x*y) = log x + log y. So, using log, you get additivity out of multiplication! And we get back to the Central Limit Theorem.

And that’s the reason why some distributions, which seem to be a power-law distribution, are in fact log-normally distributed (yep, the log here is due to the logarithm, and the normal here is due to the central limit theorem).

So, the take-home point is: ask the right question, but sometimes, to ask the right question, you first need to know the technical details (aka, in this case, math and statistics).

Sobre Manoel Galdino

Corinthiano, Bayesiano e Doutor em ciência Política pela USP.
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