re: "Half the schools are below average" - not always true

quote:

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The central limit theorem only applies when the sample is randomly selected. In this case the sample is chosen by school

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If we are using the entire population then we aren't worrying about randomization of a sample because it's not a sample. Furthermore, it is often difficult to get a truly random sample of an entire population and instead use convenience sampling. Guess what? Most of the time the distribution is approximately normal. The generalization of results may be impacted though. Besides the variables themselves are still random and there are statistical methods that are used to account for the non-independence of participants (HLM) if that is necessary.