Sampling Distributions

What is a sampling distribution?

A sampling distribution is the distribution of a statistic (like the mean) computed from many random samples of the same size. It answers: "If I took many samples, how much would the sample mean vary?"

Key properties

For the sampling distribution of $\bar{X}$:

• Center: $\mu_{\bar{X}} = \mu$ (same as population mean)
• Spread: $\sigma_{\bar{X}} = \sigma / \sqrt{n}$ (standard error — gets smaller with larger $n$)
• Shape: Approximately normal for large $n$ (CLT)

Step-by-step

Population: $\mu = 50$, $\sigma = 10$. Sample size $n = 25$.

Standard error: $SE = 10 / \sqrt{25} = 10 / 5 = 2$

So the sampling distribution is approximately $N(50, 2)$: centered at 50, with SD = 2.

Most sample means will fall between $50 \pm 2(2) = [46, 54]$ (95% of them).

Why this matters

The sampling distribution is the foundation of all inference: confidence intervals, hypothesis tests, and p-values all use the sampling distribution to assess how unusual a sample result is.

Try it in the visualization

Watch 100 samples drawn from the population. Each sample mean is plotted. The histogram of means forms a bell shape centered at μ with spread SE.