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Say I flip a coin 80 times and I ask for the probability to get over 48 heads. I then flip a coin 800 times and ask for the probability to get over 480 heads.

Translating this into Central Limit Theorem concepts, we ask that the sample mean deviate from the expected value of the sample mean by a tenth. As N grows larger this same "tenth" chunk becomes smaller and so the answer to the second question is smaller than the first.

I have been trying to visualize the distribution, starting with a sum of N flips and dividing by N to get the mean of N flips. Yet I keep getting the same picture in my head of the distribution for any N.

In both questions above I look at the probability mass for the sum to go over 48 or 480 and this just seems to be the same probability. This gif has realized my confusion: http://en.wikipedia.org/wiki/File:De_moivre-laplace.gif

If I understand it properly, every frame in this gif was generated by taking a distribution of the sum of say, N coin flips, and normalizing it to the same scale - dividing by N. Where is that peak at the expected value the Central Limit Theorem is talking about?

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up vote 2 down vote accepted

The animation rescales the horizontal axis not by dividing by $N$ but by dividing by $\sqrt{N}$ such that the visual width of the bell stays constant.

It would probably have been less confusing if the animation had explicitly shown the vertical axis moving away towards the left in the first few steps.

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