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adding an image which is referenced in the first link from this original answer; even though this image is static (and the link is a Java example), the static image does an excellent job illustrating the theory
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edit approved May 12, 2017 at 17:00
Nate Anderson
edit approved May 12, 2017 at 17:00
Nate Anderson
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The nicest animation I know: http://www.ms.uky.edu/~mai/java/stat/GaltonMachine.html

8 horizontal layers of equally spaced pins, each layer staggered, results in a "pachinko/pinball" style obstacle for balls dropped through these pins. Each ball falls at the bottom, and as the balls stack, their height approaches an outline of the Gaussian curve. This illustrates that the sum of many independent random events (the layers), will result in a Gaussian distribution of results (the stacked ball height)

The simplest words I have read: http://elonen.iki.fi/articles/centrallimit/index.en.html

If you sum the results of these ten throws, what you get is likely to be closer to 30-40 than the maximum, 60 (all sixes) or on the other hand, the minumum, 10 (all ones).

The reason for this is that you can get the middle values in many more different ways than the extremes. Example: when throwing two dice: 1+6 = 2+5 = 3+4 = 7, but only 1+1 = 2 and only 6+6 = 12.

That is: even though you get any of the six numbers equally likely when throwing one die, the extremes are less probable than middle values in sums of several dice.

The nicest animation I know: http://www.ms.uky.edu/~mai/java/stat/GaltonMachine.html

The simplest words I have read: http://elonen.iki.fi/articles/centrallimit/index.en.html

If you sum the results of these ten throws, what you get is likely to be closer to 30-40 than the maximum, 60 (all sixes) or on the other hand, the minumum, 10 (all ones).

The reason for this is that you can get the middle values in many more different ways than the extremes. Example: when throwing two dice: 1+6 = 2+5 = 3+4 = 7, but only 1+1 = 2 and only 6+6 = 12.

That is: even though you get any of the six numbers equally likely when throwing one die, the extremes are less probable than middle values in sums of several dice.

The nicest animation I know: http://www.ms.uky.edu/~mai/java/stat/GaltonMachine.html

8 horizontal layers of equally spaced pins, each layer staggered, results in a "pachinko/pinball" style obstacle for balls dropped through these pins. Each ball falls at the bottom, and as the balls stack, their height approaches an outline of the Gaussian curve. This illustrates that the sum of many independent random events (the layers), will result in a Gaussian distribution of results (the stacked ball height)

The simplest words I have read: http://elonen.iki.fi/articles/centrallimit/index.en.html

If you sum the results of these ten throws, what you get is likely to be closer to 30-40 than the maximum, 60 (all sixes) or on the other hand, the minumum, 10 (all ones).

The reason for this is that you can get the middle values in many more different ways than the extremes. Example: when throwing two dice: 1+6 = 2+5 = 3+4 = 7, but only 1+1 = 2 and only 6+6 = 12.

That is: even though you get any of the six numbers equally likely when throwing one die, the extremes are less probable than middle values in sums of several dice.

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glassy
glassy
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The nicest animation I know: http://www.ms.uky.edu/~mai/java/stat/GaltonMachine.html

The simplest words I have read: http://elonen.iki.fi/articles/centrallimit/index.en.html

If you sum the results of these ten throws, what you get is likely to be closer to 30-40 than the maximum, 60 (all sixes) or on the other hand, the minumum, 10 (all ones).

The reason for this is that you can get the middle values in many more different ways than the extremes. Example: when throwing two dice: 1+6 = 2+5 = 3+4 = 7, but only 1+1 = 2 and only 6+6 = 12.

That is: even though you get any of the six numbers equally likely when throwing one die, the extremes are less probable than middle values in sums of several dice.