The Central Limit Theorem

The Central Limit Theorem (CLT)

The distribution of sample means approaches a normal distribution as sample size increases, regardless of the original population's shape.

$\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right) \text{ as } n \to \infty$

Step-by-step demonstration

Step 1: Roll a die (uniform distribution, not normal at all). Single rolls: equally likely 1-6.

Step 2: Take samples of size $n = 30$ and compute the mean of each sample.

Step 3: Repeat thousands of times and plot the distribution of sample means.

Result: The histogram of sample means is bell-shaped (approximately normal), centered at $\mu = 3.5$, with $\text{SD} = \sigma/\sqrt{n} = 1.71/\sqrt{30} \approx 0.31$.

Why the CLT matters

• It justifies using the normal distribution for hypothesis tests and confidence intervals, even when the population isn't normal.
• Larger $n$ → better approximation → narrower spread.
• Rule of thumb: $n \geq 30$ is usually sufficient.

Try it in the visualization

Select a source distribution (uniform, skewed, bimodal). Watch sample means accumulate into a histogram that becomes increasingly bell-shaped, regardless of the source shape.