Editing Central limit theorem (section)

===Common misconceptions===
Studies have shown that the central limit theorem is subject to several common but serious misconceptions, some of which appear in widely used textbooks.<ref>{{cite journal |last=Brewer |first=J. K. |date=1985 |title=Behavioral statistics textbooks: Source of myths and misconceptions? |journal=Journal of Educational Statistics |volume=10 |issue=3 |pages=252–268|doi=10.3102/10769986010003252 |s2cid=119611584 }}</ref><ref>Yu, C.; Behrens, J.; Spencer, A. Identification of Misconception in the Central Limit Theorem and Related Concepts, ''American Educational Research Association'' lecture 19 April 1995</ref><ref>{{cite journal |last1=Sotos |first1=A. E. C. |last2=Vanhoof |first2=S. |last3=Van den Noortgate |first3=W. |last4=Onghena |first4=P. |date=2007 |title=Students' misconceptions of statistical inference: A review of the empirical evidence from research on statistics education |journal=Educational Research Review |volume=2 |issue=2 |pages=98–113|doi=10.1016/j.edurev.2007.04.001 |url=https://lirias.kuleuven.be/handle/123456789/136347 }}</ref> These include: 
* The misconceived belief that the theorem applies to random sampling of any variable, rather than to the mean values (or sums) of [[iid]] random variables extracted from a population by repeated sampling. That is, the theorem assumes the random sampling produces a sampling distribution formed from different values of means (or sums) of such random variables. 
* The misconceived belief that the theorem ensures that random sampling leads to the emergence of a normal distribution for sufficiently large samples of any random variable, regardless of the population distribution. In reality, such sampling asymptotically reproduces the properties of the population, an intuitive result underpinned by the [[Glivenko–Cantelli theorem]]. 
* The misconceived belief that the theorem leads to a good approximation of a normal distribution for sample sizes greater than around 30,<ref>{{Cite web |date=2023-06-02 |title=Sampling distribution of the sample mean |format=video |website=Khan Academy |url=https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/v/sampling-distribution-of-the-sample-mean |access-date=2023-10-08 |archive-url=https://web.archive.org/web/20230602200310/https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/v/sampling-distribution-of-the-sample-mean |archive-date=2 June 2023 }}</ref> allowing reliable inferences regardless of the nature of the population. In reality, this empirical rule of thumb has no valid justification, and can lead to seriously flawed inferences. See [[Z-test]] for where the approximation holds.