Opening
Take a deeply skewed distribution, average many independent observations, and something rather beautiful happens. The individual observations are still just as skewed as before.
Teaching narrative
Probability density is not probability: probability is the area under a density over an interval. A normal curve is merely one possible density shape, alongside uniform and exponential models with very different behavior.
The central limit theorem concerns standardized sums or averages under suitable independence and finite-variance assumptions. The distribution of sample means can become approximately bell-shaped even when the original population is strongly skewed.
Another change happens as sample size grows: the sample mean's standard error generally decreases in proportion to one over the square root of sample size. Quadrupling observations roughly halves that error under the standard model.
No universal sample size of thirty guarantees normality. The original distribution's skewness, tail behavior, and dependence matter. The simulation is an illustration of assumptions, not a proof that every dataset behaves nicely.
Before the experiment
Does the exponential population become symmetric when the sample size grows, or does the distribution of averages change?
Live demonstration
Generate sample means from an exponential distribution with sample sizes 1, 5, 30, and 80. Record the mean and spread of each simulated sampling distribution.
Exit question
Write one correct version and one incorrect version of the central limit theorem.