Two-Way Tables Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Two-Way Tables.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A table that displays frequencies for two categorical variables simultaneously, organized with one variable in rows and the other in columns. It shows joint frequencies (individual cells), marginal frequencies (row/column totals), and enables calculation of conditional frequencies.

Imagine surveying students about their favorite sport AND their grade level. A two-way table is like a grid: grades go down the side, sports go across the top, and each cell tells you how many students are in that specific combination. The totals on the edges (margins) tell you the overall counts for each category.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A two-way table arranges counts for two categorical variables in rows and columns, with margins as totals.

Common stuck point: The procedure for two-way tables is the easy part; the trap is confusing joint, marginal, and conditional frequencies. Asking "Are two categorical variables being crossed in a grid so each cell counts a combination of one category from each?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are two categorical variables being crossed in a grid so each cell counts a combination of one category from each?

Worked Examples

Example 1

medium
A two-way table shows: Smoker/Cancer=30, Smoker/No-Cancer=70, Non-smoker/Cancer=20, Non-smoker/No-Cancer=180. Calculate marginal and joint proportions, and find P(Cancer|Smoker) vs P(Cancer|Non-smoker).

Answer

P(Cancer∣Smoker)=0.30P(\text{Cancer}|\text{Smoker}) = 0.30 vs. P(Cancer∣Non-smoker)=0.10P(\text{Cancer}|\text{Non-smoker}) = 0.10. Smokers have 3× higher cancer rate.

First step

1
Total: 300 people; Smokers=100, Non-smokers=200

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Example 2

hard
From a 2×2 table: Group/Outcome frequencies: A-Success=40, A-Fail=10, B-Success=25, B-Fail=25. Test independence using the chi-square approach and calculate the relative risk.

Example 3

medium
A two-way table: 30 of 50 boys play a sport; 40 of 50 girls play a sport. Find P(plays sport∣boy)P(\text{plays sport} \mid \text{boy}) and P(plays sport∣girl)P(\text{plays sport} \mid \text{girl}).

Example 4

medium
A two-way table records 150 people: 90 like tea (60 of those are women) and 60 like coffee (20 of those are women). What fraction of women like tea?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A two-way table: Left-handed/Male=12, Left-handed/Female=8, Right-handed/Male=88, Right-handed/Female=92. Find the marginal proportion of left-handers and P(Left-handed|Male).

Example 2

hard
Construct a two-way table from this information: 200 students surveyed; 120 prefer online learning; 80 prefer in-person. Of online learners: 90 passed. Of in-person learners: 55 passed. Complete the table and find whether learning mode and passing are independent.

Example 3

easy
A two-way table shows 12 boys who like soccer and 8 girls who like soccer. How many students total like soccer?

Example 4

easy
In a two-way table, the cell for '6th grade AND likes art' shows 14. What kind of frequency is 14?

Example 5

easy
A two-way table's grand total is 50. The 'likes pizza' column total is 30. What fraction of all students like pizza?

Example 6

easy
A row total is 27 (students who chose soccer). Of these, 15 are 6th-graders. What fraction of soccer players are 6th-graders?

Example 7

easy
In a two-way table, what do the row and column totals represent?

Example 8

easy
A two-way table records two kinds of variables. What type of variables are they?

Example 9

easy
A table cell shows 0 students who are 'left-handed AND play violin.' What does this 0 mean?

Example 10

easy
Grand total is 100. If the row totals are 60 and 40, do they sum to the grand total?

Example 11

medium
A survey of 80 students: 30 are 6th-graders, of whom 18 walk to school; 50 are 7th-graders, of whom 20 walk. What fraction of 6th-graders walk?

Example 12

medium
Using the survey above (18 of 30 sixth-graders walk; 20 of 50 seventh-graders walk), which grade has a higher walking RATE?

Example 13

medium
A two-way table: Cats(owners) 24, Dogs(owners) 36, total 60. Of dog owners, 27 also have a yard. What fraction of dog owners have a yard?

Example 14

medium
A table shows 45 of 100 people prefer tea, and 28 of those tea-drinkers are adults. What fraction of ALL people are adult tea-drinkers?

Example 15

medium
A two-way table has a missing cell. Row total is 40; the other cell in that row is 15. What is the missing cell?

Example 16

medium
Of 200 surveyed, 120 own a smartphone. Among non-owners, 50 are over age 60. What fraction of non-owners are over 60?

Example 17

medium
A two-way table: of 90 students, 54 passed. Of those who passed, 36 studied. What fraction of passers did NOT study?

Example 18

medium
A two-way table shows 25 of 40 men and 30 of 60 women favor a policy. Is support associated with gender (do the rates differ)?

Example 19

challenge
A two-way table: Survey of 150. 90 like the app; of those, 54 are students. 60 dislike it; of those, 12 are students. What fraction of ALL students like the app?

Example 20

challenge
In a two-way table, P(A given B)=3040P(\text{A given B})=\frac{30}{40} and P(A given not B)=3040P(\text{A given not B})=\frac{30}{40}. Are events A and B independent? Explain.

Example 21

challenge
A two-way table is missing the grand total. Row totals: 70 and 30. Column totals: 55 and X. Find X and the grand total.

Example 22

medium
A two-way table: 60 students surveyed, 36 like math. Of the math-likers, 27 also like science. What fraction of math-likers do NOT like science?

Example 23

easy
A two-way table shows 22 sixth-graders and 18 seventh-graders prefer pizza. How many students total prefer pizza?

Example 24

easy
A row total is 50 and one cell in that row is 30. What is the other cell?

Example 25

easy
A grand total of 200 students were surveyed. The 'walks to school' column total is 80. What fraction of all students walk?

Example 26

easy
A column total is 60 and a row total is 40. The cell where these meet has 24 students. What is the joint frequency of that cell?

Example 27

medium
Of 80 students surveyed, 50 do homework daily; 32 of those who do daily homework earn A's. What fraction of daily-homework doers earn A's?

Example 28

medium
Two-way table has row totals 6060 and 4040; the cell '60-row, A-column' is 18. What is the conditional fraction P(A∣60-row)P(A \mid \text{60-row})?

Example 29

medium
Grand total 120. 'Owns a bike' column total is 72. Among bike-owners, 27 walk to school. What fraction of bike-owners walk?

Example 30

medium
Of 200 surveyed, 80 are adults; of those adults, 60 own a pet. What is the marginal proportion of adults who own a pet, divided by the grand total?

Example 31

medium
A two-way table: 100 people; 60 vegetarian, 40 non-vegetarian. 20 of the non-vegetarians chose salad. What is P(salad∣non-vegetarian)P(\text{salad} \mid \text{non-vegetarian})?

Example 32

medium
A 2x2 table has cells: (Male, Yes)=24, (Male, No)=16, (Female, Yes)=18, (Female, No)=42. Find P(Yes∣Male)P(\text{Yes} \mid \text{Male}).

Example 33

medium
Using the same table (Male,Yes)=24, (Male,No)=16, (Female,Yes)=18, (Female,No)=42, find P(Yes∣Female)P(\text{Yes} \mid \text{Female}).

Example 34

medium
Same table as before: do the rates P(Yes∣Male)=0.6P(\text{Yes} \mid \text{Male}) = 0.6 and P(Yes∣Female)=0.3P(\text{Yes} \mid \text{Female}) = 0.3 suggest the variables are associated or independent?

Example 35

medium
Two-way table has row totals 80 and 120, and column totals 90 and X. Find X and the grand total.

Example 36

medium
A study of 200 people: 80 use the app daily; 60 of those daily users are satisfied. 60 of the 120 non-daily users are satisfied. Are 'satisfaction' and 'daily use' independent?

Example 37

medium
A 2x2 table records 50 students: 30 in honors, 20 not. Of the honors students, 24 take a foreign language; of the non-honors, 12 take one. What fraction of foreign-language learners are honors students?

Example 38

hard
A two-way table: row totals 70, 30 (grand total 100). The (row1, columnA) cell is 28. What value in the (row2, columnA) cell makes the columns A and B INDEPENDENT of the row variable?

Example 39

hard
A 2x2 table: row totals 50 and 50; column totals 40 and 60. Compute the EXPECTED count in the (row1, column1) cell under independence.

Example 40

hard
A 2x2 table: (A, Yes)=30, (A, No)=20, (B, Yes)=10, (B, No)=40. Compute the relative risk P(Yes∣A)/P(Yes∣B)P(\text{Yes}\mid A)/P(\text{Yes}\mid B).

Example 41

hard
Using the same table (A,Yes)=30, (A,No)=20, (B,Yes)=10, (B,No)=40, compute the odds ratio of 'Yes' for A vs B.

Example 42

hard
A two-way table reports rounded percentages by ROW. Row 1: 60% Yes, 40% No. Row 2: 60% Yes, 40% No. Is the row variable a useful predictor of Yes/No?

Example 43

challenge
A two-way table: 240 people. 144 like soccer; 96 like basketball. Of soccer fans, 60 are kids. Of basketball fans, 48 are kids. Are 'sport choice' and 'kid vs adult' independent?

Example 44

challenge
A 2x2 table: (A,Yes)=20, (A,No)=30, (B,Yes)=10, (B,No)=40. Find the expected count for (A,Yes) under independence and compare to the observed.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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