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The Complete Idiot's Guide to Statistics, 2nd Edition by Robert Donnelly

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Why Does the Central Limit Theorem Work?

Let me explain why the central limit theorem behaves the way it does. If this concept does not interest you, feel free to skip this section. I promise you won’t hurt my feelings.
 
Going back to our original experiment with a sample size of two, the following table shows all the two-ball combinations that are possible along with the sample mean. This represents the theoretical sampling distribution of the mean because it represents all the possible combinations of samples along with their respective probabilities.
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We can convert this table into a relative frequency distribution, which is shown ...

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