K-12 Math (US): 9-12 Data Analysis K-12 Math Statistics Two-way tables

Study Guide: Two-Way Tables (Grade 9–12, Math – Statistics)

1. The Driving Question

"You survey 100 students about whether they prefer pizza or burgers and whether they play sports or not. How do you figure out if athletes are more likely to pick pizza—or if it’s just random? And why can’t you just compare the raw numbers without doing some math first?"

2. The Core Idea — Built, Not Listed

Imagine you’re the manager of Midtown High’s school store. You track two things: (1) whether a student buys a snack (yes/no) and (2) whether they’re in 9th grade or 12th grade. You dump the data into a table like this:

At first glance, more 9th graders buy snacks (45 vs. 25). But wait—there are equal numbers of 9th and 12th graders! The real story is in the percentages: 60% of 9th graders buy snacks vs. only 33% of 12th graders. Two-way tables force you to compare rates, not just counts, so you don’t get fooled by uneven group sizes.

Key Vocabulary:
- Joint frequency: A count in a single cell (e.g., "45 9th graders bought snacks").
Example: In a table tracking ice cream flavor (vanilla/chocolate) and toppings (sprinkles/nuts), the joint frequency for "chocolate + nuts" might be 18.
College shift: In advanced stats, joint frequencies become joint probabilities in probability distributions.

• Marginal frequency: A total in the margins (e.g., "75 9th graders total").
  Example: If a table tracks pet ownership (dog/cat) and apartment size (studio/1-bedroom), the marginal frequency for "dog owners" might be 60.
  College shift: Marginal frequencies become marginal probabilities in Bayesian statistics.

• Conditional relative frequency: A percentage within a subgroup (e.g., "60% of 9th graders buy snacks").
  Example: In a table of movie genres (action/comedy) and age groups (teens/adults), the conditional relative frequency for "teens who prefer action" might be 75%.
  College shift: This becomes conditional probability (P(A|B)), a cornerstone of machine learning.

• Simpson’s Paradox: When a trend appears in groups but reverses when the groups are combined.
  Example: A hospital’s overall survival rate might look better than another’s, but when you split by patient severity (mild/severe), the other hospital actually performs better for both groups.
  College shift: This paradox is a gateway to understanding confounding variables in causal inference.

3. Assessment Translation

How this appears on assessments:
- SAT/ACT: Multiple-choice questions testing interpretation of conditional relative frequencies (e.g., "What percent of cat owners are also dog owners?"). Distractors often mix up joint vs. marginal frequencies or miscalculate percentages.
- AP Statistics: Free-response questions requiring you to: 1. Construct a two-way table from raw data.
2. Calculate and interpret conditional relative frequencies.
3. Explain whether an association exists (e.g., "Is there evidence that 12th graders are less likely to buy snacks?").
Rubric priorities: Clear labeling of calculations, correct use of terminology, and justification of conclusions (e.g., "The conditional relative frequency of snack buyers is higher for 9th graders, suggesting an association").

Proficient vs. Developing Responses:
- Proficient: "60% of 9th graders buy snacks, compared to 33% of 12th graders. This suggests 9th graders are almost twice as likely to buy snacks, so there’s likely an association between grade level and snack purchases." - Developing: "More 9th graders buy snacks (45 vs. 25)." (Ignores group sizes and doesn’t calculate percentages.)

Model Student Response (AP Free-Response):
Prompt: "A survey of 200 students asks whether they prefer online or in-person learning and whether they have reliable internet at home. The results are shown below. Is there an association between internet reliability and learning preference? Justify your answer."

Response: "To determine if there’s an association, I’ll compare the conditional relative frequencies of learning preference given internet reliability.
- For students with reliable internet: 80/120 ≈ 66.7% prefer online, and 40/120 ≈ 33.3% prefer in-person.
- For students with unreliable internet: 30/80 = 37.5% prefer online, and 50/80 = 62.5% prefer in-person.
The percentages differ significantly (66.7% vs. 37.5% for online), suggesting an association: students with reliable internet are more likely to prefer online learning, while those with unreliable internet prefer in-person."

4. Mistake Taxonomy

Mistake 1: Confusing Joint and Marginal Frequencies
Prompt: "In a table of 150 students, 60 play sports and prefer math, while 90 don’t play sports. What percent of students who don’t play sports prefer math?" Wrong Response: "60/150 = 40%." (Uses joint frequency instead of marginal.) Why It Loses Credit: The question asks for a conditional percentage (math preference given no sports), but the student uses the total population.
Correct Approach: 1. Find the marginal frequency for "no sports": 90.
2. Find the joint frequency for "no sports and prefer math": (Total math students) – (Sports and math) = 100 – 60 = 40.
3. Calculate conditional relative frequency: 40/90 ≈ 44.4%.

Mistake 2: Ignoring Group Sizes
Prompt: "A table shows 50 boys and 30 girls prefer action movies, while 20 boys and 40 girls prefer comedies. Are boys more likely to prefer action movies?" Wrong Response: "Yes, because 50 boys prefer action vs. 30 girls." (Compares raw counts.) Why It Loses Credit: The groups (boys vs. girls) are different sizes, so raw counts are misleading.
Correct Approach: 1. Calculate conditional relative frequencies:
- Boys: 50/70 ≈ 71.4% prefer action.
- Girls: 30/70 ≈ 42.9% prefer action.
2. Conclude: Boys are more likely to prefer action movies.

Mistake 3: Misinterpreting Simpson’s Paradox
Prompt: "A drug trial shows Drug A has a 60% success rate overall, while Drug B has a 55% success rate. However, when split by patient age (under 40/over 40), Drug B has a higher success rate for both age groups. How is this possible?" Wrong Response: "The data must be wrong." (Denies the paradox exists.) Why It Loses Credit: The student doesn’t recognize that the distribution of patients across groups can reverse the trend.
Correct Approach: 1. Note that Drug A might have been given mostly to younger patients (who have higher natural recovery rates), while Drug B was given mostly to older patients.
2. Example:
- Drug A: 90/100 under 40 (90%), 10/100 over 40 (10%) → 100/200 = 50% overall.
- Drug B: 10/100 under 40 (10%), 90/100 over 40 (90%) → 100/200 = 50% overall.
- But if Drug A was given to 180 under 40 and 20 over 40, its overall rate would be (162 + 2)/200 = 82%, while Drug B’s would be (2 + 18)/200 = 10%.

5. Connection Layer

1. Within Math: Two-way tables → Probability (independent events)
   Why it matters: If the conditional relative frequencies for two groups are equal (e.g., 60% of boys and 60% of girls prefer pizza), the variables are independent. This is the foundation for understanding P(A|B) = P(A).

2. Across Subjects: Two-way tables → Biology (genetics Punnett squares)
   Why it matters: A Punnett square is a two-way table where the "joint frequencies" are the possible genotypes (e.g., AA, Aa, aa). Calculating conditional probabilities (e.g., "What’s the chance a child is a carrier given both parents are carriers?") uses the same logic as two-way tables.

3. Outside School: Two-way tables → Sports analytics (player performance by position)
   Why it matters: NBA teams use two-way tables to analyze player efficiency (e.g., "What percent of 3-point shots by guards are made vs. centers?"). The "surprising" part: A center might have a higher raw number of made 3s, but a guard’s percentage is higher—just like the snack example!

6. The Stretch Question

"A two-way table shows that 70% of students who study with music get A’s, while only 50% of students who study in silence get A’s. Can you conclude that music causes better grades? If not, what’s one way to redesign the study to get closer to causation?"

Pointer Toward the Answer: - Correlation ≠ causation: Maybe students who choose music are already higher achievers, or they study longer. To test causation, you’d need a randomized experiment—assign students randomly to music or silence and compare outcomes. Even then, you’d have to control for other variables (e.g., subject difficulty, prior grades). This is why real-world studies (like drug trials) use randomized controlled trials (RCTs).