Sampling Distributions (CLT) – AP Statistics

The Central Limit Theorem

Regardless of the parent population's shape, the sampling distribution of the sample mean ($\bar{x}$) approaches a Normal Distribution as the sample size $n$ increases.

Theoretical Properties

Center: $\mu_{\bar{x}} = \mu_{\text{pop}}$ (Unbiased Estimator)

Spread: $\sigma_{\bar{x}} = \frac{\sigma_{\text{pop}}}{\sqrt{n}}$ (Standard Error drops!)

Shape: Normal if $n \ge 30$, or if parent is Normal.

Instruction

Pick a highly skewed parent distribution. Set $n=30$. Spam "Draw 10000 Samples" and watch the bottom graph perfectly construct a bell curve.