Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for AP statistics sampling distributions? Then this AP Statistics lesson on Sampling Distributions of Proportions and Means explains each idea step by step, with worked examples.
Quick answer: A sampling distribution describes how a statistic such as x̄ or p̂ varies from sample to sample. Its center shows bias, its spread shows variability, and its shape is approximately normal under the Central Limit Theorem or the conditions np ≥ 10 and n(1 − p) ≥ 10.
Standards: AP Statistics Unit 2 (2026-27) · Level: Grade 12 AP Statistics
What is a sampling distribution?
A parameter describes a population (μ, p); a statistic describes a sample (x̄, p̂). A sampling distribution shows every value a statistic takes over all possible samples of the same size.
A statistic is unbiased if the center of its sampling distribution equals the parameter. Variability is the spread of that distribution; it shrinks as the sample size grows.
What is the sampling distribution of a sample proportion?
μp̂ = p σp̂ = √(p(1 − p)n)
Conditions: random sample, n less than 10% of the population, np ≥ 10 and n(1 − p) ≥ 10. Then the shape is approximately normal.
If 62% of voters support a measure and n = 150, then σ = √(0.62 × 0.38/150) = 0.0396. P(p̂ < 0.55): z = (0.55 − 0.62)/0.0396 = −1.77, so the probability is 0.0387.
What is the sampling distribution of a sample mean?
μx̄ = μ σx̄ = σ√n
If the population is normal, x̄ is normal for any n. Otherwise the Central Limit Theorem says x̄ is approximately normal when n is large (at least 30 is a common guide).
Waiting times have mean 15.2 and standard deviation 11 minutes, with a skewed population. For samples of 45, the center is 15.2 and σ = 11/√45 = 1.64, so the sampling distribution is nearly normal even though the population is not.
What about differences of two statistics?
| Statistic | Mean | Standard deviation |
|---|---|---|
| p̂1 − p̂2 | p1 − p2 | √(p1(1 − p1)/n1 + p2(1 − p2)/n2) |
| x̄1 − x̄2 | μ1 − μ2 | √(σ12/n1 + σ22/n2) |
Variances add for independent samples, so the spread of a difference is larger than the spread of either statistic alone.
Quadrupling the sample size cuts the standard deviation of the statistic in half, not to a quarter. Bigger samples reduce variability but never fix bias.
- p̂ has mean p and standard deviation √(p(1 − p)/n).
- x̄ has mean μ and standard deviation σ/√n.
- The Central Limit Theorem makes x̄ approximately normal for large n.
- Larger samples reduce variability but do not remove bias.
- For differences of independent statistics, subtract the means and add the variances.
Sampling Distributions of Proportions and Means: frequently asked questions
What is the difference between a population distribution and a sampling distribution?
A population distribution shows the values of individuals. A sampling distribution shows the values of a statistic, such as x̄, over all possible samples of the same size. The sampling distribution has the same center as the population for x̄ but much less spread.
What does the Central Limit Theorem say?
For a large random sample, the sampling distribution of x̄ is approximately normal, whatever the shape of the population, with mean μ and standard deviation σ/√n. Strongly skewed populations need larger samples than symmetric ones.
When can I use the formula σ = √(p(1 − p)/n) for the standard deviation of p̂?
When the sample is random and less than 10% of the population, so that observations are nearly independent. The normal shape also requires np ≥ 10 and n(1 − p) ≥ 10.
How do bias and variability differ?
Bias is a systematic miss: the sampling distribution is centered away from the parameter. Variability is how widely the statistic scatters from sample to sample. Good sampling methods give low bias, and large samples give low variability.
Why does the standard deviation of a difference use addition?
For independent samples the variances add, whether you subtract or add the statistics. The standard deviation is the square root of the sum of the two variances, so it is larger than either standard deviation alone.