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I do not want to know if some phenomena in nature have normal distribution, but whether we can somewhere see shape of normal curve as we can see it for example in Galton box. See this figure from Wikipedia.

enter image description here

Note that many mathematical shapes or curves are directly seen in nature, for example golden mean and logarithmic spiral can be found in snails.

First naive answer is whether nonskewed hills would often "fit" normal distribution :-).

sitems
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  • This has already been discussed elsewhere on the site: http://stats.stackexchange.com/questions/33776/real-life-examples-of-common-distributions – MånsT Oct 10 '12 at 09:38
  • But not about the shape of distribution, it is only question where we can find some types of distributions. – sitems Oct 10 '12 at 09:41
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    Right... that's a fun question then, but it seems to be [off-topic](http://stats.stackexchange.com/faq#questions). – MånsT Oct 10 '12 at 10:08
  • Often my students in statistics course ask me, where can they see it in nature. If I answer that normal curve is everywhere, but you cannot directly see it, then they are not satisfied. :-) – sitems Oct 10 '12 at 10:28
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    [This example](http://math.stackexchange.com/a/38828/7003) is a favorite of mine. – cardinal Oct 10 '12 at 12:23
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    @cardinal That's an intriguing example, but wear on steps is highly unlikely to be at all Normal. In fact, it would be a puzzle if it were. The CLT might possibly be invoked to describe horizontal variation in where people walk, but that will not lead to a Gaussian shape in the wear on the step. – whuber Oct 10 '12 at 14:40
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    Many years ago, the East Wing of the National Gallery of Art in Washington DC had a beautiful (and unintentional) example of a normal distribution showing on an exterior wall where two exterior walls met at a 45-degree angle instead of the usual 90-degree angle. People presumably had _touched_ the edge to see if it felt sharp, and the smudges from their fingers left a stain on the wall which showed as a bell curve (rotated 90 degrees clockwise) at about chest height. On a more recent visit, I found that the exterior walls had been cleaned and the smudges had disappeared. – Dilip Sarwate Oct 10 '12 at 15:02
  • @whuber: I agree it is still a bit of an idealization, but not altogether implausible, and I have seen some realistic examples. The material should be (originally) uniform in both construction and density, not prone to chipping and relatively unaffected by other environmental factors such as water runoff. Also, depending on the step height, some people will tend to step on the edge of the step and so, in these cases, the indentations will not be vertical, but somewhat diagonal. But, if the pressure exerted by each passerby is relatively constant, the wear could take the stated form. :) – cardinal Oct 10 '12 at 15:13
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    I have not seen wear like that, @cardinal. In your model the wear would be much flatter near the peak than a Gaussian and--of course--would be confined within natural limits, unlike any Gaussian. (The latter is not such a big deal because we cannot demand that the Gaussian fit *perfectly*.) The wear on steps I have examined is *far* flatter than a Gaussian. (Often it is bimodal, too, because most long-lived stairs have been used in both the up and down directions.) A better model would be a convolution of a fairly narrow Gaussian with a uniform distribution. – whuber Oct 10 '12 at 15:16
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    [This blog post](http://mathtourist.blogspot.ca/2010/05/irresistible-edge.html) shows the example that @Dilip mentions as well as one example of wearing patterns on stone steps (with links to other pictures of wear patterns). Some might find it interesting. – cardinal Oct 10 '12 at 17:20
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    @Cardinal Nice! It may only be in the eye of the beholder, but the cathedral steps image appears to validate my expectation of a flat curve and multimodality. Moreover, the handprint pattern clearly is non-Gaussian in outline. The density (of dirt) very well could be Gaussian, but that is hard to determine from this image. That blog exemplifies http://stats.stackexchange.com/a/4301: **"Everybody believes in the [the Normal distribution]: the experimenters, because they think it can be proved by mathematics; and the mathematicians, because they believe it has been established by observation."** – whuber Oct 10 '12 at 17:43
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    @whuber: I agree on all points. I really find that picture alluring; I believe the lighting actually *accentuates* the apparent troughs in the steps, i.e., they are flatter than they appear in the shot. The convolution model is certainly more faithful if for no other reason than that a point force is a rather poor approximation to that produced by a typical shoe. :-) – cardinal Oct 10 '12 at 18:00
  • Our (wooden) steps in the institute show distinct bimodal wear. I'd have said not because of up and down, but because people keep their legs beside each other (not like foxes). In addition, it is clearly visible that most people enter with the same foot. In terms of up/down I'd guess that there may be more down traffic than up traffic as we have elevators as well (there's a trade-off between climbing speed and waiting time for the elevator) but I don't know how one could see this. – cbeleites unhappy with SX Oct 11 '12 at 21:25
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    I'd argue that the Galton box is _not_ an example of the Normal distribution, but instead, an example of a binomial distribution (with $p \approx 0.5$). After all, "discrete" is a closer description of the distribution than "continuous", and the distribution is bounded. Instead, the Galton box demonstrates the appropriateness of Normal approximation to a binomial distribution (or, equivalently, the Central Limit Theorem applied to the total of a sample from a Bernoulli distribution). – Firefeather Oct 11 '12 at 22:45
  • @cardinal Thanks for the link to an actual picture! I guess the 19-degree angle at which the walls meet is far sharper than the 45-degree angle that I talked about in my comment, and more likely to create the urge to touch. – Dilip Sarwate Oct 11 '12 at 22:54

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I wouldn't think any pattern of erosion or deposition on Earth would fit because skewing factors including gravity and Coriolis are always involved (rivers meander more as they age, for example, and valley floors are sort of the average of rivers). Maybe the cross section of a stalagmite, assuming the drip remained in one fairly exact central location? I would think the drips would deposit the most precipitate right where they are moving slowest, which would be at the point of impact.

John Barnes
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I thought a lot about my question and probably I found something. U-shape of many valleys imitates "reversed" normal curve. Are there any reasons why this should not be gaussian (note that water makes the valleys smooth)?

Here is an example.

sitems
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    There seems to be a strong tendency for people to hope than any unimodal curve is normal. I see no reason why such a valley would be closely approximated by an inverted normal curve, and many factors such as the erosion from water which may be unimodal, but where any accurate physical model predicts something other than a normal curve. – Douglas Zare Oct 11 '12 at 22:26
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    This is an interesting empirical question - how closely the shape can be approximated as normal would depend on the age of the various features. A valley probably begins more poisson shaped, becomes normal-ish, and as the tops of the hills wear heads back in a poisson direction. – N Brouwer Oct 12 '12 at 13:46