How can two groups be compared fairly?
You will be able to: Compare distributions on a common scale using center, spread, shape and context.
How can two groups be compared fairly?
Two bus routes have similar average travel times, but one has occasional very long trips. A commuter cares about reliability as well as a typical time.
A useful starting point: Where should the box and whiskers end? →
Words and symbols before equations
- Common scale
- Matching numerical axes so equal distances mean equal quantities.
- Comparative statement
- A sentence directly relating groups on the same feature.
- Variability
- How dispersed observations are.
- Overlap
- A range of values occurring in both distributions.
What this picture assumes
Group A: 4,6,8,10 minutes. B is shifted and stretched about A’s mean of 7. Both dotplots share a 0–24 minute axis. These are illustrative samples, not evidence of a cause.
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.
- A: median 7, IQR 4 minutes. B: median 11, IQR 4 minutes. B−A median difference 4 minutes. A data: 4, 6, 8, 10. B data: 8, 10, 12, 14.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Begin with the same variable and units. Use aligned dotplots, histograms or boxplots; different axis limits can manufacture apparent differences.
Compare center, spread, shape and unusual values with numbers. Say “Route B’s median is 3 minutes higher” rather than listing unrelated facts about each route.
A higher group center does not imply every observation is higher. Overlap may be substantial. Without a justified collection design, describe the observed groups and avoid claims about all future trips or why they differ.
A worked example, step by step
Route A times are 4,6,8,10 minutes. Route B times are 5,9,13,17. Compare median and IQR.
- A’s median is 7; Q₁=5 and Q₃=9, so IQR=4 minutes.
- B’s median is 11; Q₁=7 and Q₃=15, so IQR=8 minutes.
- B has a 4-minute higher median and twice the IQR in these samples.
- The ranges overlap from 5 to 10 minutes; not every B trip is longer than every A trip.
Compare the same feature in the same units. Group summaries do not rank every individual.
Can equal means guarantee equal distributions?
Compare with an explanation
No; groups can have equal means but different spreads, shapes or outliers.
Predict. Change one thing. Explain.
Change Group B’s shift and spread separately. Explain which changes move its center and which change its variability, then assess overlap.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
A: median 7, IQR 4 minutes. B: median 11, IQR 4 minutes. B−A median difference 4 minutes. A data: 4, 6, 8, 10. B data: 8, 10, 12, 14.
Group A: 4,6,8,10 minutes. B is shifted and stretched about A’s mean of 7. Both dotplots share a 0–24 minute axis. These are illustrative samples, not evidence of a cause.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the data values, graph scales, summary statistics or study-design conditions. 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 questionA has median 12 and IQR 4; B has median 15 and IQR 10, both in minutes. Give a comparison and two limits.
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Compare with the answer and four-point rubric
- 1 point: B’s median is 3 minutes higher.
- 1 point: B’s IQR is 6 minutes greater, indicating more spread in its middle half.
- 1 point: These summaries do not establish that every B value exceeds every A value.
- 1 point: They do not reveal full shape or establish a cause.
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 makes a statement comparative?
It directly relates the groups on a shared feature.
RECALL 2Why check overlap?
A difference in centers does not separate all observations.
RECALL 3What can equal medians conceal?
Different spreads, shapes and unusual values.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How can two groups be compared fairly?
- Use a shared axis.
- State differences and context.
- Check overlap, shape and unusual observations as well as center.
Remember: Compare the same feature in the same units. Group summaries do not rank every individual.
Conditions: Group A: 4,6,8,10 minutes. B is shifted and stretched about A’s mean of 7. Both dotplots share a 0–24 minute axis. These are illustrative samples, not evidence of a cause.
Refresh Kid · AP Statistics Unit 1 · Objectives 1.7.E, 1.9.A, 1.9.B, 1.9.C · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 1.9, objectives 1.7.E, 1.9.A, 1.9.B, 1.9.C. Framework effective Fall 2026, checked September 17, 2026. Unit 1 includes one-variable data and data collection; it is part of the revised five-unit course.
Examples and datasets are synthetic, independently authored teaching material. Quartile calculations state a median-of-halves convention. Outlier screens identify values to investigate, not data to discard. Random selection and random assignment have different inferential roles. These lessons introduce design and descriptive reasoning; 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 sampling model uses self-hosted Three.js with its MIT license. It shows labeled units in four groups; camera rotation does not change the sampling procedure. Quantitative graphs remain 2D to avoid perspective distortion. Complete labeled diagrams, selected IDs 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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