why the central limit theorem is important in statistics
The central limit theorem implies that if the sample size \(n\) is large then the distribution of the partial sum \(Y_n\) is approximately normal with mean \(n \mu\) and variance \(n \sigma^2\). Equivalently the sample mean \(M_n\) is approximately normal with mean \(\mu\) and variance \(\sigma^2 / n\). The central limit theorem is of fundamental importance, because it means that we can approximate the distribution of certain statistics, even if we know very little about the underlying sampling distribution.