The Practical Scope of the Central Limit Theorem

The $\mathit{\text{Central}}\mathit{\text{Limit}}\mathit{\text{Theorem}}\mathit{\text{(CLT)}}$ is at the heart of a great deal of applied problem-solving in statistics and data science, but the theorem is silent on an important implementation issue: \textit{how much data do you need for the CLT to give accurate answers to practical questions?} Here we examine several approaches to addressing this issue – along the way reviewing the history of this problem over the last 290 years – and we illustrate the calculations with case-studies from finite-population sampling and gambling. A variety of surprises emerge.