Lesson 10 / 13 · 13 min read

Sampling Distributions & the Central Limit Theorem

Why sample means are predictable — the engine behind all of inference.

What is a sampling distribution?

Take many samples of size $n$ from a population and compute the mean $\bar{x}$ of each one. Those means form their own distribution — the sampling distribution of the mean. It describes how much $\bar{x}$ bounces around from sample to sample.

The center and spread of $\bar{x}$

If the population has mean $\mu$ and standard deviation $\sigma$ , then across samples of size $n$ :

$\mu_{\bar{x}} = \mu \qquad \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

$\bar{x}$ is unbiased — it centers on the true $\mu$ .
Its spread, called the standard error, shrinks as $n$ grows. Bigger samples give more reliable means.

Notice the $\sqrt{n}$ : to halve the standard error you need four times the data.

The Central Limit Theorem

For a large enough sample size (a common threshold is $n \geq 30$ ), the sampling distribution of $\bar{x}$ is approximately normal — regardless of the population's shape.

This is the punchline of intro statistics. Even a skewed, lumpy population produces a bell-shaped distribution of sample means. It's the reason the normal distribution shows up everywhere in inference. (If the population is already normal, $\bar{x}$ is normal for any $n$ .)

Worked example

A population has $\mu = 70$ and $\sigma = 12$ . For samples of size $n = 36$ :

$\mu_{\bar{x}} = 70 \qquad \sigma_{\bar{x}} = \frac{12}{\sqrt{36}} = \frac{12}{6} = 2$

By the CLT, $\bar{x}$ is approximately normal with mean $70$ and standard error $2$ . So about $95\%$ of sample means fall within $70 \pm 2(2) = [66, 74]$ .

What's $P(\bar{x} > 74)$ ? That's 2 standard errors above the mean, so about $2.5\%$ .

Why it matters

Confidence intervals and hypothesis tests both depend on knowing how $\bar{x}$ behaves. The CLT hands that to us: approximately normal, centered at $\mu$ , with standard error $\sigma / \sqrt{n}$ .

Key takeaways