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I am trying to understand the central limit theorem with simple example. As written in some texts, the central limit theorem can be stated as (I rephrase): for a population P, randomly pick n independent samples, X1, X2, ... , Xn. the average of (X1, X2, ..., Xn) approaches normal distribution as n is large enough.

Here is my questions

1) Central limit theorem is telling that it is the set of averages of every possible samples set approaching the normal distribution not the samples themself, right?

2) to obtain the normal distribution, we should take many sets of samples (says k sets) from the population, for each set, we have n samples X1, X2, ..., Xn, which gives one average. So k sets will give k averages, all these averages will show a normal distribution if n is large enough?

3) Let's consider one die, 6 faces. The population is {1, 2, 3, 4, 5, 6}. Randomly toss the die, which gives X1, and average is also X1 (only one die). Repeat this again for k times, we get a set of k averages, but these set of data will definitely be uniform. So small n won't reveal the bell shape (normal) distribution. Do I understand this correctly?

4) Let's consider the same problem with 3 dice, if we take the sum of dice into account, the population will be {3, 4, 5, ..., 18}. This time, each set of sample has three elements X1 X2 X3 and which will give one average. Repeat this process k times will give k averages, which more or less show a normal distribution but not exactly. However, the more dice you use, the closer to normal distribution will be.

5) Ok. Now look at different example about a warped roulette, found in a text. The roulette used in the experiment is not an ordinary one such that 17's appears at the probability of 2/38 instead of 1/38. To tell if this roulette is really a warped one or not, people spin it for many times and compute the related standard deviation. They find that the distribution for the warped one does not overlap with that for ordinary one; hence, it is easy to tell if the one in question is warped or not. My question is, if the author uses the normal distribution and standard deviation to tell the fact, he silently take central limit theorem into account. However, if my understanding in question 3) is correct, that is, only one die is not enough to reveal the property of normal distribution. So in this case, in each spin, we only get one number, it is no difference from one die. In other word, no matter how many times you spin the roulette, you will not get the normal distribution for the averages and it is meaningless to talk about the central limit theorem. Hence, the way mentioned above to tell if the roulette is warped is not correct. Right?