1 · Observe

Gray: the population. Blue: the histogram of 5,000 sample means. Gold curve: the normal prediction.

2 · Manipulate

3 · Predict

Pick the bimodal population and n = 5. Before simulating: will the sample means look bimodal too, or something else? What changes when you raise n to 100?

4 · See

5 · Explain

The Central Limit Theorem: for large n, sample means are approximately normal with mean μ (the population mean) and standard deviation σ/√n (the standard error). Bigger samples → tighter bell → more precise estimates. This is why polls and experiments quote a “margin of error.”