Confusion about using box models to explain the central limit theorem

Question

A text I'm reading (Freedman, Pisani, Purves - Statistics (2007)) states the central limit theorem as: the probability histogram of sums of numbers from a box converges to the normal curve as the number of draws increases. (It notes that this only works with sums and not, for example, products.)

I don't really understand the application of the box model here. If we have a hypothetical box with a bunch of numbers in it, then the more we draw the larger our sum gets. How does this produce any sort of distribution? Here are two examples of this convergence the text includes: https://i.sstatic.net/GPGH5.png and https://i.sstatic.net/dBWHZ.png. The bottom axis on each plot is for the curve, the top axis is for the histogram.

I guess I don't really understand what is being plotted. Does anyone have a better idea of what is going on here?

Answer 1

Looking at the second example, it seems to me that the plot is of the probability of each possible sum. So for drawing from a box with nine 0's and one 1, your most likely sum would be 2 over 25 draws, but 3 is close behind. Increase that to 100 draws, and your most likely sum is 10, with 9 or 11 being close in likelihood, and so on. Although the possible sum values increase as the number of draws increases, the probability distribution still converges to normal.