Which total belongs in the denominator?
You will be able to: Distinguish joint, marginal and conditional relative frequencies.
Which total belongs in the denominator?
In a 40-student table, 12 are both morning attendees and digital-preferring. That is 30% of everyone, but 60% of the 20 morning attendees.
A useful starting point: How can two categorical variables share one picture? →
Words and symbols before equations
- Joint relative frequency
- One cell divided by the grand total.
- Marginal relative frequency
- A row or column total divided by the grand total.
- Conditional relative frequency
- A cell divided by the total of the specified subgroup.
- Percentage point
- The arithmetic difference between two percentages.
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
“Morning and digital” asks for a joint share: 12/40=0.30. “Morning” alone asks for a marginal share: 20/40=0.50.
“Digital among morning attendees” restricts the reference group before dividing: 12/20=0.60. Reversing the condition changes the denominator: morning among digital-preferring students is 12/18≈0.667.
Compare like conditional distributions. Digital shares of 60% and 30% differ by 30 percentage points; the first is also twice the second. Those are distinct descriptions, and neither is a causal statement.
| Feature | Joint or marginal | Conditional |
|---|---|---|
| Denominator | Grand total | Specified subgroup total |
| Numerator | Cell or category total | Cell inside the given subgroup |
| Question | What share of everyone? | What share within this group? |
A worked example, step by step
A table has A/yes=8, A/no=12, B/yes=18 and B/no=12. Find the joint A-and-yes share, marginal yes share and yes share within A.
- The grand total is 50; A total is 20 and yes total is 26.
- Joint A-and-yes is 8/50=0.16.
- Marginal yes is 26/50=0.52.
- Conditional yes within A is 8/20=0.40. State the reference group with each answer.
“Of all students” and “of those in A” describe different denominators, even if the numerator is the same.
Is P(digital given morning) always P(morning given digital)?
Compare with an explanation
No. The denominators are different subgroup totals.
Predict. Change one thing. Explain.
Read the joint, marginal and conditional values in the readout. Explain why changing the morning digital count affects both the digital total and the two directions of conditioning.
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 questionA/yes=5, A/no=5, B/yes=15, B/no=25. Calculate three shares and compare yes percentages across A and B.
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Compare with the answer and four-point rubric
- 1 point: Grand total 50, A total 10, B total 40, yes total 20.
- 1 point: Joint A-and-yes=5/50=0.10; marginal yes=20/50=0.40.
- 1 point: Yes given A=5/10=0.50; yes given B=15/40=0.375.
- 1 point: A is 12.5 percentage points higher in these data; association is not proof of 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 denominator does a joint share use?
The grand total.
RECALL 2Where are marginal counts found?
At row and column margins.
RECALL 3What changes when conditioning is reversed?
The given group and therefore the denominator.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Which total belongs in the denominator?
- Joint = cell / grand total.
- Marginal = category total / grand total.
- Conditional = cell / given-group total.
Remember: “Of all students” and “of those in A” describe different denominators, even if the numerator is the same.
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 2.2.A, 2.2.B, 2.2.C · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 2.2, objectives 2.2.A, 2.2.B, 2.2.C. 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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