Mathematical and Applied Statistics Lesson of the Day – The Central Limit Theorem Applies to the Sample Mean

Having taught and tutored introductory statistics numerous times, I often hear students misinterpret the Central Limit Theorem by saying that, as the sample size gets bigger, the distribution of the data approaches a normal distribution.  This is not true.  If your data come from a non-normal distribution, their distribution stays the same regardless of the sample size.

Remember: The Central Limit Theorem says that, if X_1, X_2, ..., X_n is an independent and identically distributed sample of random variables, then the distribution of their sample mean is approximately normal, and this approximation gets better as the sample size gets bigger.

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