A sample mean x̄ changes from sample to sample. The sampling distribution of x̄ describes all those possible values. Pick a population shape, choose a sample size, and draw samples to see its center, spread, and shape, then use it to find probabilities.
Each click takes simple random samples of size n from the population and records each sample's mean.
When the sampling distribution is approximately normal, standardize x̄ with σx̄ and use the normal model. The simulation gives a check.
The mean of the sampling distribution of x̄ equals the population mean: μx̄ = μ. Sample means do not systematically overshoot or undershoot μ, so x̄ is an unbiased estimator.
Averages vary less than single values. Larger samples give less variable sample means. To cut σx̄ in half you need four times the sample size, because of the square root.
σ/√n assumes the observations are independent. When sampling without replacement, check that n is no more than 10% of the population size.
If the population is normal, the sampling distribution of x̄ is exactly normal for any sample size, even n = 2.
If the population is not normal, the sampling distribution of x̄ becomes approximately normal as n grows. The usual guideline is n ≥ 30. Strongly skewed populations need larger samples.
The distribution of one sample looks like the population, whatever n is. The sampling distribution is the distribution of x̄ across many samples. Only the second one gets narrower and more normal.
Three distributions that students mix up, and what happens to each as n increases.
| Distribution | What it shows | Center | Spread | As n increases |
|---|---|---|---|---|
| Population | Every individual value x | μ | σ | Does not change |
| One sample (the data) | The n values you collected | x̄ | s | Looks more like the population; x̄ and s get closer to μ and σ |
| Sampling distribution of x̄ | The means of all possible samples of size n | μx̄ = μ | σx̄ = σ/√n | Narrower, and closer to normal (CLT) |
| Population shape | Sample size | Shape of the sampling distribution of x̄ |
|---|---|---|
| Normal | Any n | Normal |
| Not normal or unknown | n ≥ 30 | Approximately normal by the central limit theorem |
| Not normal or unknown | n < 30 | Cannot assume normal; it keeps some of the population's shape |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 4.1. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 4 terminology guide.