How much should a sample percentage vary?
You will be able to: Calculate and interpret the center, spread and approximate probabilities of p̂.
How much should a sample percentage vary?
Suppose 40% of a large population prefers digital notes. Random samples of 100 people will not all contain exactly 40 digital-note supporters.
A useful starting point: Can a good estimator miss in one sample? →
Words and symbols before equations
- Sampling distribution
- Distribution of a statistic over repeated samples of a fixed size.
- μp̂
- Mean of the sample-proportion distribution.
- σp̂
- Standard deviation of the sample-proportion distribution.
- Standardized value z
- Distance from the model mean measured in its standard deviations.
What this picture assumes
Synthetic study: a simple random sample without replacement from N=100000 independent units. The chosen sizes satisfy the 10% condition. Real studies require their own design checks. The model uses known p; normal tail probabilities are approximate without continuity correction.
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.
- Mean 0.4; SD 0.04899. Approximate P(p̂ ≥ 0.5) = 0.02061; no continuity correction.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
For independent observations, μp̂=p and σp̂=√[p(1−p)/n]. The variation is in sample shares; the population proportion is held fixed.
With p=0.40 and n=100, the standard deviation is √0.0024≈0.0490, or 4.90 percentage points. This is a typical scale of sampling fluctuation, not an absolute bound.
When expected successes and failures are at least 10, a normal approximation can estimate a region’s probability. A share of 0.50 is about 2.04 standard deviations above 0.40, so P(p̂≥0.50) is approximately 0.0206 without continuity correction. The exact discrete probability need not match.
A worked example, step by step
For independent sampling with p=0.50 and n=100, approximate P(p̂>0.60).
- The center is 0.50.
- The SD is √(0.5×0.5/100)=0.05.
- The cutoff has z=(0.60−0.50)/0.05=2.
- The normal upper-tail approximation is about 0.0228; it is an approximation to a discrete sampling distribution.
A standard deviation of 0.05 in a proportion means 5 percentage points, not 0.05%.
What happens to the SD when n is multiplied by four?
Compare with an explanation
It is divided by two, holding p fixed.
Predict. Change one thing. Explain.
Increase n while keeping p fixed. Read the predicted standard deviation and approximate upper-tail probability. Explain why the center stays fixed.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Mean 0.4; SD 0.04899. Approximate P(p̂ ≥ 0.5) = 0.02061; no continuity correction.
| Quantity | Value |
|---|---|
| Population p | 0.4 |
| Estimator mean | 0.4 |
| Sampling SD | 0.04899 |
| Expected successes / failures | 40 / 60 |
| Normal-count check | Pass |
Synthetic study: a simple random sample without replacement from N=100000 independent units. The chosen sizes satisfy the 10% condition. Real studies require their own design checks. The model uses known p; normal tail probabilities are approximate without continuity correction.
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 questionFor p=0.20 and n=400 independent observations, find the mean and SD and interpret the SD.
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Compare with the answer and four-point rubric
- 1 point: The mean is 0.20.
- 1 point: Variance is 0.2×0.8/400=0.0004.
- 1 point: SD is 0.02.
- 1 point: Repeated sample shares typically fluctuate on a scale of about 2 percentage points around 20%.
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 1Which quantity varies across repeated samples?
The statistic p̂.
RECALL 2Which p belongs in the theoretical SD?
The population/model p, when it is known.
RECALL 3Is a normal tail an exact binomial tail?
No; it is an approximation.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How much should a sample percentage vary?
- μp̂=p.
- σp̂=√[p(1−p)/n] for independent observations.
- z=(p̂−p)/σp̂.
Remember: A standard deviation of 0.05 in a proportion means 5 percentage points, not 0.05%.
Conditions: Synthetic study: a simple random sample without replacement from N=100000 independent units. The chosen sizes satisfy the 10% condition. Real studies require their own design checks. The model uses known p; normal tail probabilities are approximate without continuity correction.
Refresh Kid · AP Statistics Unit 3 · Objectives 3.2.A, 3.2.C · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 3.2, objectives 3.2.A, 3.2.C. Framework effective Fall 2026, checked September 17, 2026. Unit 3 includes inference for one and two population proportions, errors and power, and chi-square homogeneity/independence tests; it is part of the revised five-unit course.
Examples and datasets are synthetic, independently authored teaching material. Inference requires a justified design and appropriate counts. Intervals use observed proportions; null tests use their reference-model proportions. The revised CED states chi-square expected counts should be greater than 5; this unit follows that wording even though some companion texts use at least 5. Normal and chi-square inference are approximate. The chi-square goodness-of-fit test is not included in this unit’s official scope.
The Organic Chemistry Tutor companion title and destination were located; 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 categorical count grid uses self-hosted Three.js with its MIT license. Two rows and three columns organize synthetic people into categorical cells. Each stacked block represents one person. Camera rotation changes only the view; exact counts, expectations and contributions are always available in the 2D table. Inference curves and intervals remain 2D; use the exact table rather than apparent 3D size for comparisons. Complete labeled diagrams, count tables 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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