Several different sources (starting from Wikipedia) state that the Galton box is a (visual) demonstration of the Central Limit Theorem. This claim is actually bothering me a little because this result is only incidental.
Indeed, the Galton box simulates the outcomes of a binomial variable by dropping several balls across an interleaved grid of pegs, showing that when the number of balls becomes large their bottom arrangement yields a very good approximation of binomial distribution. In other words, it represents an empirical proof of the Law of Large Numbers.
On the other hand, my previous visualization shows that increasing the number of trials in a sequence of theoretical binomial distributions the normal one comes into view very quickly. This is the real sense of the Central Limit Theorem, in its simplest version. And the reason why the Galton machine works.
Let me rephrase the above distinction formally by writing that if Fm(Xn) is the sample distribution of a collection of n binomial variables which count the number of successfully events having constant probability p in m trials, then it holds, with some abuse of notation:
The first relation describes the Law of Large Numbers; the second one the Central Limit Theorem.
If you think about them statically, the Galton box points out that for a large value of n, even a moderate value for m is enough to obtain a sample distribution very close to the normal one.
But here I insist that the key word to really appreciate the meaning of the two laws is: convergence. From this point of view, in my opinion there is a misunderstanding about the Galton box: it refers to the former law but it is credited to the latter: increasing the number of the balls more and more, their distribution gets closer and closer to the binomial distribution, whose approximation to the normal one is good but it cannot get better because the number of peg levels doesn’t change.
Is it possible to imagine a different mechanism to highlight the distinction between the two convergence laws? …