Sampling Distribution

The Distribution of Sample Statistics

Download PNG Save to Pinterest

Related Tools

Related Labs

Related Worksheets

Related Cheat Sheets

A sampling distribution describes how a statistic changes from sample to sample when many random samples of the same size are taken from a population. It matters because in real studies we usually observe only one sample, but we still want to make conclusions about the whole population. The sampling distribution tells us how much natural variation to expect in statistics like the sample mean. That idea is the foundation of confidence intervals, hypothesis tests, and margin of error.

For the sample mean, the center of the sampling distribution is the population mean, so $\bar{x}$ is an unbiased estimator of $\mu$ . Its spread is measured by the standard error, given by $\frac{\sigma}{\sqrt{n}}$ when the population standard deviation is known. As sample size increases, the sampling distribution becomes narrower, meaning sample means cluster more tightly around the true mean. By the Central Limit Theorem, the sampling distribution of $\bar{x}$ becomes approximately normal for large enough $n$ , even if the population itself is not normal.

Key Facts

Sampling distribution: the distribution of a statistic over many repeated random samples of the same size n.
For the sample mean, mean of the sampling distribution: $\mu_{\bar{x}} = \mu$ .
Standard error of the sample mean: $SE = \frac{\sigma}{\sqrt{n}}$ .
As n increases, SE decreases because SE is inversely proportional to √n.
If the population is normal, then x̄ is normally distributed for any sample size n.
Central Limit Theorem: for large n, the distribution of x̄ is approximately normal, even when the population distribution is not normal.

Vocabulary

Population: The full set of individuals or values that a study wants to describe.
Sample: A subset of the population selected for observation or measurement.
Statistic: A numerical summary computed from a sample, such as the sample mean x̄.
Sampling distribution: The distribution formed by a statistic from many repeated random samples of the same size.
Standard error: The standard deviation of a sampling distribution, showing how much a statistic typically varies from sample to sample.

Common Mistakes to Avoid

Confusing the population distribution with the sampling distribution, because they describe different things. The population distribution shows individual data values, while the sampling distribution shows values of a statistic such as x̄.
Thinking the standard error is the same as the sample standard deviation, which is wrong because they measure different spreads. Sample standard deviation describes variability among individuals in one sample, while standard error describes variability of the statistic across many samples.
Assuming a larger sample size makes the sampling distribution wider, which is wrong because SE = σ/√n decreases as n increases. Larger samples make the sample mean more stable.
Believing the Central Limit Theorem says the population becomes normal, which is incorrect. The theorem applies to the distribution of the sample mean, not to the original population data.

Practice Questions

1 A population has mean $\mu = 50$ and standard deviation $\sigma = 12$ . If samples of size $n = 36$ are taken, find the mean and standard error of the sampling distribution of $\bar{x}$ .
2 A population has σ = 20. Compare the standard error of x̄ for sample sizes n = 25 and n = 100.
3 Explain why the sampling distribution of the sample mean can be approximately normal even when the population distribution is strongly skewed, and state what condition makes this approximation more reliable.