What varies when you keep taking new samples?
You will be able to: Distinguish population data, one sample and a distribution of repeated sample statistics.
What varies when you keep taking new samples?
A tiny population has equally likely values 0 and 2 minutes. Draw two observations independently with replacement, compute their mean, and repeat.
A useful starting point: How do you find the value that leaves a chosen area? →
Words and symbols before equations
- Statistic
- A numerical summary computed from a sample.
- Sampling distribution
- Distribution of a statistic across all samples of a fixed size under a specified sampling process.
- Sampling variability
- How a statistic changes from one random sample to another.
- Simulation replication
- A fresh random sample and one recorded statistic.
What this picture assumes
Independent draws with replacement from a fixed finite-variance population. Each plotted observation in the sampling histogram is one sample mean. Changing the number of repeated samples changes Monte Carlo detail; changing n changes the theoretical spread.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- 200 independent samples, each of size 5. Population mean 1; theoretical SD of sample means 0.447. Histogram bins include the lower endpoint and exclude the upper, except the last includes both. Larger n reduces theoretical spread; more repetitions refine the simulation.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
The population contains individual values 0 and 2. One size-2 sample might be 0,2, with sample mean 1. A sampling distribution collects means from repeated size-2 samples, not all the individual values pooled together.
The ordered samples 00,02,20,22 are equally likely here. Their means are 0,1,1,2, so the sample-mean distribution has probabilities 1/4,1/2,1/4.
A simulation approximates this distribution by repeating the collection process. Keep sample size and statistic fixed while varying the random sample. More replications makes the approximation less noisy; changing sample size creates a different sampling distribution.
| Feature | Population distribution | Sampling distribution of a mean |
|---|---|---|
| One plotted value | An individual observation | A whole sample mean |
| Center | Population mean | Population mean under this model |
| Spread | Population SD | Population SD divided by square root of n |
A worked example, step by step
For independent draws of size 2 with replacement from equally likely values 0 and 4, list the sample means and probabilities.
- Ordered samples are (0,0),(0,4),(4,0),(4,4), each probability 1/4.
- Their means are 0,2,2,4.
- Thus P(x̄=0)=1/4, P(x̄=2)=1/2, P(x̄=4)=1/4.
- One dot in a simulated sampling distribution represents one sample mean, not one original observation.
A large number of simulation replications is not the same as a large sample size within each replication.
Does the sampling distribution contain the original individual measurements?
Compare with an explanation
It contains values of the chosen statistic, such as sample means; those may differ from individual measurements.
Predict. Change one thing. Explain.
Choose the two-value population and set n=2. Compare the simulated means with the exact 0,1,2 pattern. Then change n while keeping the replication count fixed.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
200 independent samples, each of size 5. Population mean 1; theoretical SD of sample means 0.447. Histogram bins include the lower endpoint and exclude the upper, except the last includes both. Larger n reduces theoretical spread; more repetitions refine the simulation.
| Population value | Probability |
|---|---|
| 0 | 0.5 |
| 2 | 0.5 |
| Quantity | Value |
|---|---|
| Population mean μ | 1 |
| Population SD σ | 1 |
| Theoretical mean of sample means | 1 |
| Theoretical SD σ/√n | 0.447 |
| Simulated average of sample means | 1.08 |
Independent draws with replacement from a fixed finite-variance population. Each plotted observation in the sampling histogram is one sample mean. Changing the number of repeated samples changes Monte Carlo detail; changing n changes the theoretical spread.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the probability values, reference groups, graph scales or model assumptions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionFrom equally likely values 1 and 3, take two independent draws with replacement. Give the exact distribution of the sample mean.
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Compare with the answer and four-point rubric
- 1 point: Possible ordered samples: (1,1),(1,3),(3,1),(3,3).
- 1 point: Means are 1,2,2,3.
- 1 point: Probabilities are 1/4 at 1,1/2 at 2,1/4 at 3.
- 1 point: This is a statistic distribution; the original population has values only 1 and 3.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What must stay fixed when defining a sampling distribution?
Population, sampling method, statistic and sample size.
RECALL 2What does more simulation replication improve?
The approximation to that fixed distribution.
RECALL 3Can a sample mean be absent from the population’s individual values?
Yes.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What varies when you keep taking new samples?
- Fix the population, sampling process, statistic and n.
- Collect one statistic per replication.
- Distinguish the population, sample and sampling distribution.
Remember: A large number of simulation replications is not the same as a large sample size within each replication.
Conditions: Independent draws with replacement from a fixed finite-variance population. Each plotted observation in the sampling histogram is one sample mean. Changing the number of repeated samples changes Monte Carlo detail; changing n changes the theoretical spread.
Refresh Kid · AP Statistics Unit 2 · Objectives 2.12.A · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 2.12, objectives 2.12.A. Framework effective Fall 2026, checked September 17, 2026. Unit 2 includes probability, random variables, probability models and introductory sampling distributions; it is part of the revised five-unit course.
Examples and datasets are synthetic, independently authored teaching material. Assumptions about independence, replacement and equal likelihood are stated before calculations. Simulation estimates fluctuate. Discrete probability is summed; continuous probability is area. Sampling distributions and randomization distributions use different repetition mechanisms. Formal inference comes in later units.
The Organic Chemistry Tutor companion title and destination were checked; the full video was not reviewed. Khan Academy’s destination was checked, but its lesson content was not fully readable by the research tool. OpenStax provides optional reference reading. No provider scripts, questions or graphics were copied. Refresh Kid is not affiliated with these providers.
GitHub’s 3D website collection informed optional spatial inspection. Our original three-toss outcome cube uses self-hosted Three.js with its MIT license. Eight corners represent eight equally likely sequences of three fair independent tosses. Conditioning removes ineligible sequences; camera rotation never changes probabilities. Quantitative graphs remain 2D to avoid perspective distortion. Complete labeled diagrams, outcome lists and explanations remain available without 3D.
Independent teacher review and observation of students remain pending. Technical checks do not certify statistical accuracy, accessibility or learning effectiveness. This is a review edition.
Released AP Statistics questions and scoring guides are optional. Older exams use the earlier framework, so check alignment before selecting parts. All practice on this page is original, not official AP material.
Learn → Explore → Practice → Review is informed by the IES learning guide. This implementation has not yet been evaluated with learners.
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