Definition: The central limit theorem says the distribution of sample means becomes approximately normal as the sample size grows, whatever the shape of the population.
What does it say?
Take random samples of size n from any population with mean mu and standard deviation sigma. When n is large enough (a common guideline is 30 or more), the sample mean has an approximately normal distribution with mean mu and standard deviation sigma / square root of n.
Why is it so useful?
It lets us use normal probabilities for means even when the population itself is skewed.
Worked example: a population has mean 50 and standard deviation 12. For samples of n = 36, the standard deviation of the mean is 12/6 = 2. The chance that a sample mean exceeds 54 corresponds to z = (54 – 50)/2 = 2, which is about 2.5%.
A single value above 54 would be much more common, since the individual spread is 12.
What mistakes occur?
Dividing by n instead of its square root, and applying the theorem to a single observation. Mathbot reminds you that the theorem is about averages of samples.
Where to learn it
- Sampling Distributions of Proportions and Means – lesson
- Inference for Means with t Procedures – lesson