Chances are, if you took anything away from that high school or college statistics class you were dragged into, you remember some vague notion about the Central Limit Theorem. It’s likely the most famous theorem in statistics, and the most widely used. Most introductory statistics textbooks state the theorem in broad terms, that as the sample size increases, the sample distribution of the sum of the sample elements will be approximately normally distributed, regardless of the underlying distribution. Many things used in statistical inference as justification in a broad variety of fields, such as the classical z-test, rely on this theorem. Many conclusions in science, economics, public policy, and social studies have been drawn with tests that rely on the Central Limit Theorem as justification. We’re going to dive into this theorem a bit more formally, and discuss some counterexamples to this theorem. Not every sequence of random variables will obey the conditions of theorem, and the assumptions are a bit more strict than are used in practice. …

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