When repeated samples of the same size $n$ are drawn from a population, the sample statistic (e.g., $\bar{x}$ or $\hat{p}$) varies from sample to sample. The sampling distribution is the distribution of this statistic over all possible samples.
For random samples of size $n$ from a population with mean $\mu$ and standard deviation $\sigma$:
$$\text{Mean of sampling distribution: } \mu_{\bar{x}} = \mu$$
$$\text{Standard deviation of sampling distribution (standard error): } SE = \frac{\sigma}{\sqrt{n}}$$
The standard error $SE$ decreases as $n$ increases — larger samples give more precise estimates.
For a population proportion $p$, the sample proportion $\hat{p}$ from samples of size $n$ has:
$$\mu_{\hat{p}} = p \qquad SE_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$$
| Sample size $n$ | Standard error | Precision |
|---|---|---|
| Small | Large | Low — estimates spread out |
| Large | Small | High — estimates clustered near $\mu$ or $p$ |
Doubling $n$ reduces $SE$ by a factor of $\sqrt{2} \approx 1.41$.
A population has mean $\mu = 40$ and standard deviation $\sigma = 10$. Samples of size $n = 25$ are drawn.
$$SE = \frac{10}{\sqrt{25}} = \frac{10}{5} = 2$$
The sample means will be centred at 40 with standard deviation 2. Most sample means will fall within \$40 \pm 2 \times 2 = 36$ to $44$.
The sampling distribution tells us:
- How much variation to expect in sample statistics by chance
- How to construct confidence intervals (using $SE$)
- How to decide if a sample result is unusual under a particular hypothesis
STUDY HINT: Think of the sampling distribution as answering: “If I took many samples and calculated $\bar{x}$ each time, what would that distribution look like?” It is a distribution of statistics, not raw data.
EXAM TIP: When given $\mu$ and $\sigma$ for a population, calculate $SE = \sigma / \sqrt{n}$ before answering questions about sample means. This is almost always the first required step.
