A statistic like a sample mean changes from sample to sample. Its sampling distribution shows every value it could take and how often. Pick a population, any shape, set the sample size, and draw thousands of samples to see the center, spread, and shape that the central limit theorem predicts.
Each sample is a simple random sample of size n. The mean of each sample becomes one dot in the sampling distribution.
The population mean μ is one fixed number. Every random sample gives a different x̄. The sampling distribution describes that sample-to-sample variability.
The sample means are centered at the population mean for every sample size. On average, x̄ neither overestimates nor underestimates μ.
Larger samples give less variable means. To cut the standard deviation in half, you need four times the sample size. This formula assumes the sample is no more than 10% of the population.
When n is large, the sampling distribution of x̄ is approximately normal, whatever the population's shape. A common rule of thumb is n ≥ 30. The more skewed the population, the larger n needs to be.
If the population itself is normal, the sampling distribution of x̄ is exactly normal for any sample size, even n = 2.
Code "yes" as 1 and "no" as 0, and p̂ is a sample mean. Its distribution is approximately normal when np ≥ 10 and n(1 − p) ≥ 10.
Every sampling-distribution question involves these three. Ask yourself which one the question is about.
| Distribution | What each value is | Center | Spread | Shape |
|---|---|---|---|---|
| Population | One individual | μ | σ | Whatever it is |
| One sample (data) | One individual in the sample | x̄ | s | Roughly resembles the population |
| Sampling distribution of x̄ | The mean of one whole sample | μx̄ = μ | σx̄ = σ/√n | Normal if population is normal; approximately normal if n is large (CLT) |
| Sampling distribution of p̂ | The proportion in one whole sample | μp̂ = p | σp̂ = √(p(1 − p)/n) | Approximately normal if np ≥ 10 and n(1 − p) ≥ 10 |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 2.12. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 2 terminology guide.