Can a good estimator miss in one sample?
You will be able to: Distinguish a point estimate, estimator bias and sampling variability.
Can a good estimator miss in one sample?
A random sample finds that 24 of 40 students prefer a later club meeting. The sample share is 0.60, but another random sample can give a different share.
A useful starting point: Prerequisite: sampling distributions →
Words and symbols before equations
- Parameter p
- The fixed population proportion with the specified response.
- Statistic p̂
- The sample proportion x/n, where x counts successes and n counts observations.
- Estimator
- A rule used to estimate a parameter from sample data.
- Bias
- The estimator’s long-run mean minus the parameter.
What this picture assumes
Independent Bernoulli sampling. The curve is a normal approximation used only when expected counts pass. Adding .05 is a deliberately biased rule; shifted estimates can leave the parameter range.
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. Bias 0. A single estimate can miss even with zero bias.
- 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 estimate is 24/40=0.60. The rule p̂=X/n is an estimator; its realized numerical value is an estimate. Define the target population and response before interpreting either.
Under independent Bernoulli sampling with probability p, E(p̂)=p. An unbiased estimator centers at the true value across repeated samples; it need not equal p in any one sample.
Bias and variability differ. Adding 0.05 to every estimate moves its long-run center by 0.05. Increasing n narrows the sampling distribution but does not remove that systematic shift or fix a biased collection method.
A worked example, step by step
A population model has p=0.40. Compare p̂ with T=p̂+0.05 as estimators of p; one sample has 18 successes among 50.
- The observed p̂ is 18/50=0.36.
- Under the stated random model E(p̂)=0.40, so p̂ is unbiased.
- E(T)=0.45, so T has upward bias 0.05.
- The particular estimate 0.36 is below 0.40, but that single miss does not make the estimator biased.
Unbiased describes repeated-sampling behavior, not perfect accuracy in every sample.
Can an unbiased estimate be too high?
Compare with an explanation
Yes. Individual estimates vary around the true value.
Predict. Change one thing. Explain.
Compare the unbiased and shifted estimator. Change sample size while holding the population proportion fixed. Separate a change in center from a change in spread.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Mean 0.4; SD 0.04899. Bias 0. A single estimate can miss even with zero bias.
| Quantity | Value |
|---|---|
| Population p | 0.4 |
| Estimator mean | 0.4 |
| Sampling SD | 0.04899 |
| Expected successes / failures | 40 / 60 |
| Normal-count check | Pass |
Independent Bernoulli sampling. The curve is a normal approximation used only when expected counts pass. Adding .05 is a deliberately biased rule; shifted estimates can leave the parameter range.
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 questionIn a model with p=0.30, a rule has sampling mean 0.34. A random sample gives 21 successes out of 60. Describe the estimate and bias.
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Compare with the answer and four-point rubric
- 1 point: The sample estimate is 21/60=0.35.
- 1 point: The rule’s bias is 0.34−0.30=0.04.
- 1 point: Bias uses the rule’s repeated-sampling mean, not this one sample.
- 1 point: Larger samples alone do not guarantee removal of the rule’s systematic bias.
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 is a point estimate?
One sample-based number estimating a parameter.
RECALL 2What does unbiased mean?
The estimator’s long-run mean equals the parameter.
RECALL 3Does large n repair a voluntary-response sample?
No; sample size does not remove selection bias.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Can a good estimator miss in one sample?
- p̂=x/n.
- Bias=E(estimator)−parameter.
- A larger sample reduces random spread, not systematic bias.
Remember: Unbiased describes repeated-sampling behavior, not perfect accuracy in every sample.
Conditions: Independent Bernoulli sampling. The curve is a normal approximation used only when expected counts pass. Adding .05 is a deliberately biased rule; shifted estimates can leave the parameter range.
Refresh Kid · AP Statistics Unit 3 · Objectives 3.1.A, 3.1.B · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 3.1, objectives 3.1.A, 3.1.B. 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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