Demo: Random Samples and Sample Statistics
This demo repeats the same sampling experiment many times. Use it to connect iid sampling assumptions with the random behavior of familiar sample statistics.
Mathematical setup
Let $X_1,\ldots,X_n$ be iid observations from a population with mean $\mu$ and variance $\sigma^2<\infty$. The main statistics are
\[\bar X_n=\frac{1}{n}\sum_{i=1}^n X_i, \qquad S_n^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\bar X_n)^2.\]Then $E[\bar X_n]=\mu$, $\operatorname{Var}(\bar X_n)=\sigma^2/n$, and $E[S_n^2]=\sigma^2$ under iid sampling. The central limit theorem says that, for many population models,
\[\frac{\sqrt n(\bar X_n-\mu)}{\sigma}\Rightarrow N(0,1).\]For the heavy-tailed $t_3$ option, the variance is finite but large, so the normal approximation may need more samples to look convincing.
What to try
- Compare Normal, centered Exponential, Bernoulli, and $t_3$ populations at the same $n$. The sample mean centers correctly in all four cases, but the histogram shape changes.
- Increase $n$ while keeping the number of repeated samples fixed. The width of the sampling distribution should shrink at roughly the $1/\sqrt n$ rate.
- Increase the repeated samples slider when the histogram looks noisy. More repetitions improve the Monte Carlo picture, not the theoretical variance of one sample mean.
The plot is simulation-based. Changing the seed gives a new set of repeated samples with the same population model and sample size.