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Sir Francis Galton has described the Central Limit Theorem as follows.

I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the "Law of Frequency of Error". The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshaled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.

I have since become fascinated with the CLT, and have looked for cool examples in real life where this holds and wherein one can actually "see" the Bell curve. The only example I could find till now was the Galton Box.

Are there any more examples of this sort?

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1 Answer 1

Find an old stone step or lintel in front of a doorway (one that is old enough that it has been worn down by generations of people walking over it). If you look at how it is worn down, the wearing down won't be uniform over the surface of the step: rather, it will be in a bell-curve. (People tend to walk through the middle of the doorway.)

If the door doesn't open all the way, the bell-curve won't be perfect; there will a bias towards the side away from the hinges.

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This just blows my mind. Excellent answer. –  kodyv May 13 '11 at 4:45
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@Matt: Dear Matt, This is a very interesting answer. –  Akhil Mathew May 13 '11 at 4:49
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@Akhil: Dear Akhil, Thanks. At Harvard you should be able to find some examples to test its correctness! Best wishes, –  Matt E May 13 '11 at 4:53
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@Rahul: Dear Rahul, This is how I think of it, but perhaps I am confused: each person walks on the step, and wears it down some amount. This wearing is distributed over the step. As people cross the step, the wearing accumulates, i.e. the wearing down by different people is added. Thus, when we look at the long-term pattern of wearing, we are seeing the sum of a large number of independent variables. (Of course it is a sum, not a mean, since the total amount of wearing increases with each step; but that is just a matter of scaling.) Thus the central limit theorem applies, and explains ... –  Matt E May 13 '11 at 6:06
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... why the pattern of wearing forms a bell curve. Regards, –  Matt E May 13 '11 at 6:07

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