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 \(\sigma\):
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:
| 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.
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.
