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

March 20, 2014

(This article was originally published at The Chemical Statistician » Statistics, and syndicated at StatsBlogs.)

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.