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Gaussian Distribution being questioned
Robert Wilde writes "The Financial Times is reporting in two stories that a group of scientists have discovered that any scale-independent system does not follow the traditional Gaussian Bell Curve but a new curve. " Interesting implications-for above systems. For what I can gather from the article, for those systems in which this curve is more appropriate, rare events will occur more often then predicted by the Gaussian distribution. Anyone have more comments on this?
First, let me say that the graph in the article is poorly labeled (or at least their example poorly chosen), IMHO, since "rarity" is related to the number of standard deviations you are from the mean (whether or not the distribution is symmetrical), whereas their graph has rarity monotonically decreasing from left to right. I guess in this sense ("rarity of a species"), rarity != probability.
This new graph stikes me as a bit odd, since it's not symmetrical. With the bell curve, you only need to know how many standard deviations you are from the mean. With this curve, "above the mean" and "below the mean" are vastly different territories.
This curve brings up two questions for me:
I guess this new curve is just another way of saying that "Hey, there's a class of 'random' events out there that share a common non-uniform distribution!" While that's useful to know, I don't see it as the ultimate refutation of the Gaussian distribution.
--Joe--
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The use of the gausian curve is based on the assumption that the random variable we are considering is actually gereated as an average of many many independent random variables. It has been shown for all 'reasonable' independent random variables in the limit their average will be a gausian distribution. This is straightforwad mathematics no arguing with this.
As such from a mathematical point of view this has nothing to do with replacing the gausian curve...it is still clearly the most 'natural' mathematical curve. However, what I understand the authors to be claiming is that certain types of real world events are not actually gaussian and are described better by this model. This shouldn't be that surprising as often the 'extreme' cases are not caused by a mere sum of the independent random variables mentioned earlier.
For instance intelligence might be regarded as the influence of a great deal of small random variables (how some genes got arranged upbring etc..) but the truly tale end cases such as mental retardation do not occur because all of these factors go bad, (someone who is retarded is the result of some genetic defect usually not a combination of bad upbringing poor nutrition etc..). This is probably not the kind of thing the distribution describes but it shows that the gaussian really never has been the end all and be all.
So while this is undoubtly a very interesting subbject it really isn't that exciting. Ohh and the claim that the greater incidence of natural disasters disproves the gaussian was really BS, while they may not be gaussian this doesn't appear to be a large enough sample size to make such definitive claims
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The problem here is how you define and measure a rare occurence. Let me give you an example.
Let's say one night you watch the results of the lottery on TV, and the numbers '1-2-3-4-5-6' come up. Is that a rare occurence? No. That sequence is as likely to occur than your birthday and your girlfriend's birthday combined into esoteric equations.
Example number 2: I'm with this girl one night. I say my astrological sign is Scorpio. "Really!" she exclaims, "I'm Scorpio too!" What are the probabilities of that happening? 1/144? No, just 1/12. At one point (and cryptos will be familiar with this) if you add people, it becomes a rare event that you do not find people with the same sign.
All that graph is showing me is that the guys (I'm hesitating to call them scientists - I mean, they published in "serious papers"? Come on. Names, please) looked purposefully for freak occurences, discarding other "rare" occurences that were perfectly normal. That's why the left side of the graph is wider.
Thing is, the Gaussian curve doesn't come out of nowhere; it's not arbitrary. For instance, in statistical mechanics and quantum mechanics, you get bell curve distributions precisely because of the distribution of particle states.
All these guys are saying is, "rare events are not as rare as we think they are". That's not because the bell curve is wrong, it's because we seem to forget how huge the Earth provides a sample.
What are the odds of being struck by lightning twice? One in a billion? We're 6 billion on this Earth. It's bound to happen to someone. Same thing with winning the grand prize lottery once or twice.
And, again, same thing with floods or tornadoes. Yes, in themselves they're rare. When taken alone they seem improbable. But on the scale of the planet, that's the kind of thing that happens.
Alright, anyone got another article on cold fusion lying around?
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