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The math that explains why bell curves are everywhere

quantamagazine.org

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Re: The math that explains why bell curves are everywhere

#121

Post by FabHK »

> Place a measuring cup in your backyard every time it rains and note the height of the water when it stops: Your data will conform to a bell curve.

That strikes me as unlikely, actually: that the amount of water to fall (per area) across rain showers ("when it stops") is normally distributed. Why would the author think that?

Also, not much of "the math that explains" the CLT in the article. The basic conditions are:

The samples you add together must be

- sufficiently independent

- sufficiently well-behaved in the sense of not having huge outliers (finite variance is good enough for this)

Not sure either condition holds for rainfall.

Re: The math that explains why bell curves are everywhere

#122

Post by thaumasiotes »

jibal wrote: post #71
thaumasiotes wrote: post #51

Earlier quoted context omitted.

As I'm sure tsunamifury would agree, it is incredibly common for people to label "bell curves" by eyeball, regardless of whether they are normal curves. To most people, "clumping" in a one-dimensional spectrum is all they mean by the phrase "bell curve".

This completely misses the point, which is that the central limit theorem says that it isn't just any old clumping, it's always the normal distribution. tsunamifury dismissed this strong finding as "tautology" because clumping is obvious ... but that it's always precisely a bell curve is far from obvious. Again, > your "aka" is incorrect --- there is all sorts of clumping that is not a normal distribution. That it's…

> the central limit theorem says that the distribution of the means is always a bell curve--a normal distribution--not merely a "bell curve"

It doesn't say that. And it shouldn't, because that isn't true.

Re: The math that explains why bell curves are everywhere

#123

Post by tsunamifury »

jibal wrote: post #71
thaumasiotes wrote: post #51

Earlier quoted context omitted.

As I'm sure tsunamifury would agree, it is incredibly common for people to label "bell curves" by eyeball, regardless of whether they are normal curves. To most people, "clumping" in a one-dimensional spectrum is all they mean by the phrase "bell curve".

This completely misses the point, which is that the central limit theorem says that it isn't just any old clumping, it's always the normal distribution. tsunamifury dismissed this strong finding as "tautology" because clumping is obvious ... but that it's always precisely a bell curve is far from obvious. Again, > your "aka" is incorrect --- there is all sorts of clumping that is not a normal distribution. That it's…

Wow aside from the fact that none of that support is in the article it still boils down to

Normal curves are everywhere normal curves are -- which are an observational tautology -- and a fundamental over our observation of "stuff". You're dismissive as if im some illiterate, but you'd be surprised at the contributions on math I've made to the world.

Re: The math that explains why bell curves are everywhere

#124

Post by abetusk »

jibal wrote: post #45

https://en.wikipedia.org/wiki/Central_limit_theorem > suppose that a large sample of observations is obtained, each observation being randomly produced in a way that does not depend on the values of the other observations, and the average (arithmetic mean) of the observed values is computed. If this procedure is performed many times, resulting in a collection of observed averages, the central limit theorem says that…

That sentence is flat out wrong.

If the probability distribution converges, it converges to a Levy stable distribution [0].

[0] https://en.wikipedia.org/wiki/Stable_distribution

Re: The math that explains why bell curves are everywhere

#125

Post by streptomycin »

bux93 wrote: post #108

The article doesn't share the actual math, but also not the relatively easy intuition. When you roll a pair of dice, there are more combinations that add up to 7 than any other number. Change the numbers on the dice (change the 1 to a 6, e.g.), there's again more combinations that add up to some numbers than to others. The histogram of the number of combinations that add up to different results is a bell curve. That'…

The distribution of the sum of two dice is actually triangular, not a bell curve https://math.stackexchange.com/a/1204492

Re: The math that explains why bell curves are everywhere

#126

Post by btilly »

srean wrote: post #76
btilly wrote: post #70

Earlier quoted context omitted.

The key principle is that you get CLT when a bunch of random factors add. Which happens in lots of places. In finance, the effects of random factors tend to multiply. So you get a log-normal curve. As Taleb points out, though, the underlying assumptions behind log-normal break in large market movements. Because in large movements, things that were uncorrelated, become correlated. Resulting in fat tails, where extreme…

Some correlations are fine though, there are versions of CLT that applies even when there are benign correlations. https://en.wikipedia.org/wiki/Central_limit_theorem#Dependen... I know you know that and were just simplifying. Just wanted this fact to be better known for practitioners. Your comment on multiplicative processes is spot on. I say more here https://news.ycombinator.com/item?id=47437152 It's bit of a sham…

Absolutely. The effect of straightforward correlations is a change in the variance, which can be measured in finance.

The effect of the nonlinear changing correlations is that future global behavior can't be predicted from local observations without a very sophisticated model.

Re: The math that explains why bell curves are everywhere

#127

Post by tsunamifury »

abetusk wrote: post #61
tsunamifury wrote: post #25

Bell curves are everywhere because all distributions of any properties clump in some way at some level. The basics of any probability shows this. The result is you “seeing” bell curves everywhere. Aka clumps. This is a tautology to the extreme.

No, that's not true. If sums of independent identically distributed random variables converge to a distribution, they converge to a Levy stable distribution [0]. Tails of the Levy stable distribution are power law, which makes them not Gaussian. [0] https://en.wikipedia.org/wiki/Stable_distribution

Yes but really what our brains do is use Gaussian Mixture model to cut up those distributions into more granular bell curves which we then call “normal”. Because we find what we are tuned to find.

Eg we find bell curves because we look for bell curves. And given infinite resolution we can find them at some granularity.

Re: The math that explains why bell curves are everywhere

#128

Post by srean »

Buttons840 wrote: post #44
bicepjai wrote: post #40

This is one of my favorite philosophical questions to ponder. I always ask it in interviews as a warmup to get their thoughts. I’ve noticed that interviewees often curl up, thinking it’s a technical question, so I’ve been modifying the question one after the other to make it less scary. The interviews are for data scientist roles.

I haven't read the article, but my understanding is that a normal curve results from summing several samples from most common probability distributions, and also a normal curve results from summing many normal curves. All summation roads lead to normal curves. (There might be an exception for weird probability distributions that do not have a mean; I was surprised when I learned these exist.) Life is full of sums. He…

> How did I do in my interview?

You did very well.

But if you haven't had exposure to this either through work experience or through course work it would be unfair to ask this question and use your answer to judge competence.

For a potential coworker role I would certainly be curious about your curiosity but a sharp ended question is not a way to explore that.

Re: The math that explains why bell curves are everywhere

#129

Post by roenxi »

fritzo wrote: post #119
roenxi wrote: post #78

Earlier quoted context omitted.

That isn't the case; mathematicians will do pages of calculations (particularly and especially the statisticians) if they can prove one approach is technically superior to another. These people, as a class, are the crazies who invented matrix multiplication. Something like MMSE is used because it provably optimum properties for estimating a posterior distribution. It is certainly possible that there are complex appro…

> optimum properties for estimating a posterior distribution Circular reasoning: that's true only if the posterior is normal, or if your "optimal" is defined by second moments. In infinite variance cases, the best estimator can be median or an alpha moment for alpha -- A mathematician who has indeed fallen into the beauty trap

> Circular reasoning: that's true only if the posterior is normal, or if your "optimal" is defined by second moments.

That doesn't sound right, it is an error minimising technique. Are we not talking about minimising mean square errors? Why would the posterior need to be normal? And why would optimal need to be defined by 2nd moments?

Re: The math that explains why bell curves are everywhere

#130

Post by mr_toad »

streptomycin wrote: post #125
bux93 wrote: post #108

The article doesn't share the actual math, but also not the relatively easy intuition. When you roll a pair of dice, there are more combinations that add up to 7 than any other number. Change the numbers on the dice (change the 1 to a 6, e.g.), there's again more combinations that add up to some numbers than to others. The histogram of the number of combinations that add up to different results is a bell curve. That'…

The distribution of the sum of two dice is actually triangular, not a bell curve https://math.stackexchange.com/a/1204492

The distribution of the sum of a finite number of dice only approximates a bell curve.

https://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distributio...

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