Sampling distribution
Also known as: sampling distribution of a statistic
A sampling distribution is the distribution of a statistic — such as the sample mean or sample proportion — across all possible samples of a given size from a population. It describes how much a statistic varies from sample to sample.
A sampling distribution is not a distribution of individual data values. It is the distribution of a statistic computed from repeated samples of the same size. Imagine drawing every possible sample of 40 students from a school, computing the mean height in each, and plotting those means: that plot is the sampling distribution of the sample mean for n = 40.
Three properties define it. Its center equals the population parameter when the statistic is unbiased, so the mean of all sample means equals the population mean μ. Its spread, called the standard error, shrinks as sample size grows — for a sample mean it equals σ / √n, so quadrupling n halves the standard error. Its shape becomes approximately normal as n increases regardless of the population's shape, which is the central limit theorem; for proportions the usual rule of thumb is that np and n(1 − p) should both be at least 10.
This idea is the bridge between a single sample and a statement about the population. Because you know how much sample means vary, you can say how unusual your particular sample mean is, which is exactly what confidence intervals and significance tests do. Without the sampling distribution there would be no way to attach a margin of error or a p-value to a result. Sampling distributions also make clear why larger samples produce more precise estimates while a biased sampling method stays biased no matter how large n gets.
AP Statistics builds an entire unit around sampling distributions, using simulations to show how the distribution of sample means or proportions takes shape as sample size grows. Expect free-response questions asking you to check the normality conditions, compute a standard error, and use the sampling distribution to find a probability.
Key takeaways
- A sampling distribution shows how a statistic such as the sample mean varies across all possible samples of a fixed size.
- Its center equals the population parameter when the statistic is unbiased.
- Its spread is the standard error, which equals σ / √n for a sample mean and shrinks as sample size grows.
- The central limit theorem makes the sampling distribution approximately normal for large enough samples regardless of population shape.
- Sampling distributions underpin confidence intervals and significance tests, both heavily tested on AP Statistics.