The single most important idea in the course: even when a population is lumpy or skewed, the distribution of sample means turns bell-shaped as you take more samples — and gets skinnier as n grows.
1 · Pick a population
Sample size n5
Population (μ and σ shown below)
2 · Draw samples
The latest sample (each dot = one score)Its mean x̄
3 · The sampling distribution of the mean
Sample means collected so farCLT prediction: Normal(μ, σ/√n)
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Population μ
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Population σ
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Samples drawn
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Mean of the means
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SD of the means
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σ/√n (theory)
Try this:
Pick Right-skewed with n = 2 and draw 5,000 samples. Skewed or bell-shaped? Now set n = 30 and draw 5,000 again. That change is the Central Limit Theorem.
Watch the last two readouts: the SD of your collected means should home in on σ/√n. Double n from 25 to 100 — the standard error should cut in half. Does it?
Set n = 1. What is the "sampling distribution" now, and why does it look exactly like the population?
Sigma says: The population almost never looks normal — and it doesn't have to. It's the means that behave. That's why we can use the normal curve for inference about averages.