AP Statistics · Topic 2.12 · Unit 2
AP Stats 2.12: Sampling Distributions & CLT
By Jude Wallis · Published
Topic 2.12 in the Fall 2026 AP Statistics course describes sampling distributions with simulation. A sampling distribution is the distribution of a statistic across all samples of a given size. The central limit theorem says the sampling distribution of a sample mean is approximately normal.
AP Statistics: Unit 2 (topics 2.12). In the Fall 2026 AP Statistics course, sampling distributions and the central limit theorem are Unit 2 Topic 2.12, aligned to skill 4.C.
What topic 2.12 covers
Topic 2.12 closes Unit 2 and sets up all the inference to come. The objective is to describe sampling distributions with simulations.
A sampling distribution of a statistic is the distribution of values of that statistic for all possible samples of a given size from a given population. It is not the distribution of the population and not the distribution of one sample; it is what you would see if you took every sample of that size and recorded the statistic each time, a distinction developed in sampling distributions explained.
Simulating a sampling distribution
You rarely take every possible sample, so you approximate a sampling distribution by simulation. You repeatedly generate a large number of random samples from the population, assuming known values for the parameters, then determine and record the statistic for each sample.
The resulting distribution of the recorded statistic values approximates the true sampling distribution. A related idea is the randomization distribution, built by repeatedly reassigning the response values to treatment groups and recording the statistic each time, which approximates the sampling distribution for a randomized experiment. You can build one interactively in the sampling distribution simulator.
The central limit theorem
The central limit theorem (CLT) states that the sampling distribution of a mean of a random sample has a shape that can be approximated by a normal distribution. The larger the sample, the better that approximation, and this holds even when the population itself is not normal.
That is why the normal model from topic 2.11 reappears throughout inference. Here in Unit 2 the focus is describing the shape and center of a sampling distribution from a simulation; the formulas that give its center and spread directly from population parameters appear on the formula sheet and are applied in later units. See central limit theorem for more.
Three distributions students confuse
Exam responses often blur three different distributions, so keep them labeled. The population distribution describes individual values across the whole group. The distribution of a single sample describes the values you actually collected on one draw.
The sampling distribution describes how a statistic, such as the sample mean, varies across many samples of the same size from that population. The central limit theorem is a claim about this third distribution for a mean, not about the population or any one sample. Stating which of the three you mean prevents the vague generalizations, such as 'larger samples have less variability,' that leave a reader unsure and cost points.
Reading a simulated sampling distribution
A population has mean 50. You simulate the sampling distribution of the sample mean for samples of size 40 by drawing 1000 random samples and recording each sample mean. Of the 1000 simulated means, 830 fall between 48 and 52. Estimate P(48 < sample mean < 52) and describe the distribution.
Estimate the probability as a relative frequency: .
The 1000 recorded sample means pile up in a bell shape.
Their center sits near the population mean of 50.
With samples of size 40, the central limit theorem makes the shape approximately normal even if the population is not.
The estimated probability is , and the simulated sampling distribution is roughly normal, centered near 50.
Frequently asked questions
What is a sampling distribution?
A sampling distribution of a statistic is the distribution of that statistic across all possible samples of a fixed size from a population. It differs from the population distribution and from a single sample; it describes how the statistic itself varies from sample to sample. You often approximate it by simulation.
What does the central limit theorem say?
The central limit theorem states that the sampling distribution of the mean of a random sample is approximately normal in shape, and the larger the sample, the better the approximation. It holds even when the underlying population is not normal, which is why the normal model appears throughout inference.