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I have tested the central limit theorem with 1000 samples and a sample size of 4. The resulting distribution was nowhere near normal, and when I used n = 30, it did start looking normal.

Is there a theoretical basis for this "rule of thumb"?

The resulting distribution looks like this:

enter image description here

Omry Atia
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  • Can you show us what you have done? – user2974951 Feb 12 '19 at 11:13
  • I took a distribution which was not normal, sampled from it 1000 times a sample of 4, and plotted the distribution. – Omry Atia Feb 12 '19 at 11:19
  • What was the distribution? – user2974951 Feb 12 '19 at 11:20
  • And what was the non-normal distribution that you used to sample from? – user2974951 Feb 12 '19 at 11:49
  • It was a bimodal distrubution quite similar to the one I posted – Omry Atia Feb 12 '19 at 11:52
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    There is no simple answer to this question. For some distributions the convergence is very fast, but likewise it is easy to make examples where millions of observations will not be enough. Si is often answered (maybe with simulation) on a case-wise basis. – kjetil b halvorsen Feb 12 '19 at 12:13
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    Some relevant Qs: https://stats.stackexchange.com/questions/61798/example-of-distribution-where-large-sample-size-is-necessary-for-central-limit-t, https://stats.stackexchange.com/questions/2541/what-references-should-be-cited-to-support-using-30-as-a-large-enough-sample-siz, https://stats.stackexchange.com/questions/228816/big-sample-size-n50-000-but-still-highly-skewed-is-central-limit-theorem-st, https://stats.stackexchange.com/questions/370445/question-on-central-limit-theorem – kjetil b halvorsen Feb 12 '19 at 12:54

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