Central Limit Theorem

Statistics · Axiom Academy

Discover one of the most surprising and powerful results in statistics Imagine you have a population with ANY distribution - it could be skewed, uniform, bimodal, or completely bizarre. The Central Limit Theorem makes this astonishing claim: Watch as we take samples from a highly skewed population and see what happens to the distribution of sample means: 2. When Does the Magic Happen? For the Central Limit Theorem to work its magic, we need certain conditions: Independence: Each observation in the sample must be independent of the others Random Sampling: Samples must be randomly selected from the population Sample Size: The famous "n ≥ 30 rule" - for most distributions, a sample size of 30 or more is sufficient The animation below shows how the sampling distribution becomes more normal as sample size increases: The Central Limit Theorem works because of a beautiful mathematical principle: averaging reduces variability . When you take a sample mean, extreme values in your sample tend to cancel each other out. A few unusually high values might be balanced by a few unusually low values. The more values you average together (larger n), the more this balancing effect occurs. Center: μ (same as the population mean) Spread: σ/√n (standard error - gets smaller as n increases) Shape: Approximately normal (especially when n ≥ 30) Watch how individual samples may vary wildly, but their means cluster tightly around the population mean in a normal pattern:

This is the written version of the interactive lesson above. See the full Statistics course.