How can two categorical variables share one picture?
You will be able to: Read a two-way table and compare conditional distributions with honest displays.
How can two categorical variables share one picture?
A club asks 40 students which session they attend and whether they prefer a digital handout. Among 20 morning students, 12 choose digital; among 20 afternoon students, 6 do.
A useful starting point: Prerequisites: reading a number line, fractions and percentages →
Words and symbols before equations
- Two-way table
- Counts classified by two categorical variables at once.
- Cell
- The intersection of one row category and one column category.
- Segmented bar
- A whole group divided into category shares.
- Mosaic plot
- Rectangles whose areas encode joint shares; group widths encode group proportions.
What this picture assumes
Synthetic preferences: 20 morning and 20 afternoon students. Afternoon counts are fixed at 6 digital and 14 paper. Counts and percentages describe these 40 students, not a causal effect.
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.
- Morning digital: 12/20 = 60%; afternoon: 6/20 = 30%. Joint morning-and-digital: 12/40 = 30%. Marginal digital: 18/40 = 45%. Morning given digital: 12/18 = 66.7%. Teal = digital; gold = paper.
- 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 rows can be morning and afternoon; the columns digital and paper. Each student enters one cell. Add rows and columns to check the total before comparing.
Side-by-side bars can compare counts or percentages with a labeled zero-based axis. A 100% segmented bar gives every group equal total length so its internal category shares are easy to compare. A mosaic also shows unequal group sizes through different widths.
Here the digital shares are 12/20=60% and 6/20=30%. Those unequal conditional distributions indicate an association in the observed table. They do not establish that attending a session causes a preference, and sample differences alone do not prove a population association.
A worked example, step by step
A has 9 digital and 6 paper preferences; B has 6 digital and 9 paper. Describe the table and a 100% segmented display.
- Row totals are 15 each; column totals are 15 digital and 15 paper; grand total is 30.
- A’s digital share is 9/15=60%; B’s is 6/15=40%.
- Draw two equally long bars divided into 60/40 and 40/60 digital/paper shares, labeling categories and percentages.
- Digital preference is 20 percentage points higher in A in these data. The graph does not supply a causal explanation.
A taller count bar can reflect a larger group. Compare relevant proportions when group totals differ.
What does a wider group rectangle in a mosaic mean?
Compare with an explanation
That group contains a larger share of the observations, not necessarily a larger conditional digital preference.
Predict. Change one thing. Explain.
Switch between segmented, side-by-side and mosaic displays. Change the morning digital count. Read the table first, then explain which display makes within-group percentages easiest to compare.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Morning digital: 12/20 = 60%; afternoon: 6/20 = 30%. Joint morning-and-digital: 12/40 = 30%. Marginal digital: 18/40 = 45%. Morning given digital: 12/18 = 66.7%. Teal = digital; gold = paper.
| Session | Digital | Paper | Total |
|---|---|---|---|
| Morning | 12 | 8 | 20 |
| Afternoon | 6 | 14 | 20 |
| Total | 18 | 22 | 40 |
Synthetic preferences: 20 morning and 20 afternoon students. Afternoon counts are fixed at 6 digital and 14 paper. Counts and percentages describe these 40 students, not a causal effect.
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 questionGroup A has 8 yes and 12 no; Group B has 18 yes and 12 no. Compare yes shares and describe the group widths in a mosaic.
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Compare with the answer and four-point rubric
- 1 point: Totals are 20 and 30; combined total 50.
- 1 point: Yes shares are 40% in A and 60% in B.
- 1 point: Mosaic group widths are 20/50=40% and 30/50=60%.
- 1 point: B has a 20-percentage-point higher yes share; the display alone does not establish causation.
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 does a two-way cell count?
Units sharing both specified categories.
RECALL 2Why use conditional shares?
They allow comparison within groups of potentially different sizes.
RECALL 3What does a mosaic preserve?
Group proportions in widths and joint proportions in areas.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How can two categorical variables share one picture?
- Row and column totals must agree on the grand total.
- A segmented bar totals 100% within a group.
- Mosaic area is joint proportion; width is group proportion.
Remember: A taller count bar can reflect a larger group. Compare relevant proportions when group totals differ.
Conditions: Synthetic preferences: 20 morning and 20 afternoon students. Afternoon counts are fixed at 6 digital and 14 paper. Counts and percentages describe these 40 students, not a causal effect.
Refresh Kid · AP Statistics Unit 2 · Objectives 2.1.A, 2.1.B · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 2.1, objectives 2.1.A, 2.1.B. 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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