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I know that the common explanation for this is the Central Limit Theorem being applicable to most probability distributions. However, I can't seem to apply the Central Limit Theorem to simple examples like: if I were to sample the height of everyone in the world and plotted them on a histogram, it would look close to a normal distribution. But the Central Limit Theorem tells us that $\sqrt{n}(\overline{X_n} - \mu) \sim \mathcal{N}(0, \sigma^2)$. How does the average $\overline{X_n}$ of iid random variables $X_1, \dots, X_i$ have anything to do with height? Or weight? There are more examples but you get my point...

timeinbaku
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    It's hard to get your point because it's based on misinterpretations of the CLT. Perhaps, then, some of the answers at https://stats.stackexchange.com/questions/3734 will help you. – whuber Jul 06 '23 at 19:21
  • @whuber Sorry, could you please let me know what exactly I am misunderstanding in regards to the CLT? I have read the thread you linked and still desire an explanation to why the normal distribution is so common. (and why it shows up when we look at heights and weights of people, for example) – timeinbaku Jul 06 '23 at 19:28
  • The normal distribution assigns positive probability to all real numbers, but it's physically impossible for a person to have negative weight or negative height, so this example does not help us understand the connection that you're trying to make between some real-world phenomenon and a mathematical result about the central limit theorem. – Sycorax Jul 06 '23 at 19:34
  • @Sycorax-OnStrike Thanks, does this mean that the examples (height/weight) I have mentioned in reality do not have any relation to the CLT and instead are just a special coincidence which I have made malformed deductions from? – timeinbaku Jul 06 '23 at 19:37
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    There's not enough detail in the question to understand the reasoning process that relates the example to the CLT. For instance, it's possible that you asking whether or not heights are random variables, and the CLT is not really essential to your question. – Sycorax Jul 06 '23 at 19:45
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    You can find some clear, accurate statements of the CLT (in fairly intuitive language) in our thread at https://stats.stackexchange.com/questions/3734/. – whuber Jul 06 '23 at 21:28

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