Two-Way Tables Examples in Statistics

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 Statistics.

Concept Recap

A two-way table (contingency table) displays the frequency of data categorized by two different categorical variables simultaneously, with one variable in rows and the other in columns, allowing comparison of distributions across groups.

A two-way table is like a spreadsheet that shows how two questions relate. 'Do you like pizza?' and 'Are you a kid or adult?' becomes a $2 \times 2$ grid showing how many kid pizza-lovers, adult pizza-lovers, etc.

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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: Two-Way Tables asks whether the same cases connect two variables or groups in a pattern that can be described carefully.

Common stuck point: Students often know a procedure related to two-way tables but skip the recognition step: Am I studying a relationship between variables, and have I separated association from causation? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Am I studying a relationship between variables, and have I separated association from causation?

Common Mistakes to Watch For

Before you work through the examples, skim the mistake guide so you know which shortcuts and sign errors to avoid.

Conditional Probability Mistakes

Worked Examples

Example 1

medium

From a two-way table: Smokers (Yes/No) vs Disease (Yes/No). Smokers-Disease

= 30

; Smokers-NoDisease

= 70

; NonSmokers-Disease

= 10

; NonSmokers-NoDisease

= 90

. What is the proportion of smokers who get the disease?

What proportion of smokers get the disease?

Answer

30/100 = 0.30

First step

Conditional on Smokers, denominator

= 30 + 70 = 100

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Example 2

medium

Given: Boys

35

total, Girls

45

total, Soccer

45

total, Basketball

35

total, grand total

80

. Compute expected count of Boy-and-Soccer if Gender and Sport were independent.

Expected count of Boy-and-Soccer under independence = ?

Example 3

hard

In the treatment table (A:

90/30

; B:

60/120

), compute the expected count of (Treatment A, Cured) under independence.

Expected count of (Treatment A, Cured) under independence = ?

Example 4

hard

Conditional distributions of Sport given Gender are identical (

60\%

soccer,

40\%

basketball for both Boys and Girls). What does this say about association?

Example 5

challenge

A study finds Treatment A beats Treatment B overall, but within each severity subgroup, Treatment B is better. What is this phenomenon called and what does it teach us about two-way tables?

Example 6

medium

A survey of 80 people classifies Coffee (Yes/No) by Tea (Yes/No). 30 drink both; 20 coffee only; 15 tea only; the rest drink neither. Find each marginal total.

Find each marginal total.

Example 7

medium

100 students were surveyed: 70 play sports; among those, 49 also play music. Among non-sport students, 9 play music. Build the table and find the percent of music players overall.

Example 8

hard

A two-way table by Treatment (Drug/Placebo) and Outcome (Recover/Not): Drug 40/10; Placebo 25/25. Compare the recovery rates and decide which group did better.

Compare recovery rates: which group did better?

Example 9

easy

A survey of 60 students recorded their gender and whether they prefer cats or dogs. Boys who prefer cats: 8, boys who prefer dogs: 17, girls who prefer cats: 15, girls who prefer dogs: 20. Organise this into a two-way table with row and column totals.

Example 10

medium

From the two-way table: Passed exam — studied (40), didn't study (10); Failed exam — studied (5), didn't study (25). Calculate the percentage of students who passed among those who studied, and among those who didn't study.

Practice Problems

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

Example 1

easy

A survey records 30 students. The two-way table has rows Kid/Adult and columns Likes Pizza/Dislikes Pizza. Kids: 12 like, 3 dislike. Adults: 9 like, 6 dislike. How many people are in the survey total?

Grand total = ?

Example 2

easy

In the table, Kids who like pizza

=12

, Kids who dislike

=3

. What is the marginal total of Kids?

Example 3

easy

Adults: 9 like pizza, 6 dislike. What is the marginal total of Adults?

Example 4

easy

Using the table (Kids: 12 like, 3 dislike; Adults: 9 like, 6 dislike), how many people in total like pizza?

Example 5

easy

From the same table, how many people dislike pizza in total?

Example 6

easy

In the two-way table, what is the joint count of Kids who dislike pizza?

What is the joint count for Kids who dislike pizza?

Example 7

easy

2\times 2

table has cells

a=8

b=2

c=5

d=5

. What is the grand total?

Grand total = ?

Example 8

easy

A two-way table classifies people by Gender (row) and Owns a Bike (column). Which variable types make a two-way table the right tool?

Example 9

medium

Table: Sport played (Yes/No) by Grade (9th/10th). 9th: 20 Yes, 10 No. 10th: 15 Yes, 25 No. How many 10th graders are there, and what fraction of ALL students are 10th graders?

How many 10th graders are there, and what fraction are 10th graders?

Example 10

medium

Same table (9th: 20 Yes, 10 No; 10th: 15 Yes, 25 No). What fraction of students play a sport (Yes)?

What fraction of students play a sport?

Example 11

medium

Table cells:

a=18

b=12

(row 1),

c=6

d=24

(row 2). What proportion of the data is in cell

d

(joint relative frequency)?

What is the joint relative frequency of cell d?

Example 12

medium

A two-way table of Color preference by Age group has row totals

25

and

35

and a grand total of

60

. If one column total is

40

, what is the other column total?

Example 13

medium

Table: Pass/Fail by Studied (Yes/No). Studied-Yes: 28 Pass, 4 Fail. Studied-No: 6 Pass, 12 Fail. How many students total, and how many failed?

How many students total, and how many failed?

Example 14

medium

From the Pass/Fail table (Studied-Yes: 28 Pass, 4 Fail; Studied-No: 6 Pass, 12 Fail), what fraction of ALL students both studied and passed?

What fraction of ALL students both studied and passed?

Example 15

medium

A table has row totals

40

and

60

. A cell in the first row is

10

. What is the joint relative frequency of that cell if the grand total is

100

Example 16

medium

Survey of 80 people by Coffee (Yes/No) and Tea (Yes/No). Coffee-Yes/Tea-Yes

=30

, Coffee-Yes/Tea-No

=20

, Coffee-No/Tea-Yes

=18

, Coffee-No/Tea-No

=12

. How many drink coffee?

How many people drink coffee?

Example 17

medium

In a

2\times 3

table, the three column totals are

15

25

, and

20

, and one row total is

36

. What is the other row total?

Example 18

challenge

A two-way table (Treatment/Placebo by Recovered/Not) is consistent if all marginals match the grand total. Treatment:

a

recovered,

30-a

not (row total

30

). Placebo:

b

recovered,

40-b

not (row total

40

). If

52

total recovered, express

b

in terms of

a

, then find

b

when

a=28

Example 19

challenge

A table reports that

\frac{a}{a+b}=0.6

for row 1 and the row-1 total

a+b=50

. Cell

a

is also

\frac{1}{4}

of the grand total. Find the grand total.

Example 20

challenge

Two groups, A (size 100) and B (size 100), are merged. In A, 70 of 100 succeed; in B, 30 of 100 succeed. After merging into one

2\times 2

table (Group by Success), what is the overall success relative frequency, and why can it differ from each group's rate?

Example 21

easy

A two-way table classifies

80

students by Gender (Boy/Girl) and Sport (Soccer/Basketball). Boys:

20

soccer,

15

basketball. Girls:

25

soccer,

20

basketball. What is the total number of soccer players?

Total soccer players = ?

Example 22

easy

Using the same table (Boys:

20

soccer,

15

basketball; Girls:

25

soccer,

20

basketball), how many girls are in the survey?

Example 23

easy

Boys total

35

, Girls total

45

, soccer total

45

, basketball total

35

. What is the grand total?

What is the grand total?

Example 24

medium

Given (Boys:

20

soccer,

15

bb; Girls:

25

soccer,

20

bb), what proportion of boys play soccer (conditional)?

What proportion of boys play soccer?

Example 25

medium

Same data. What proportion of soccer players are boys?

What proportion of soccer players are boys?

Example 26

medium

Using the same smoker table, what proportion of non-smokers get the disease, and what does the comparison say about association?

What proportion of non-smokers get the disease?

Example 27

medium

A table has row marginal totals

40, 60

and column marginal totals

50, 50

. If variables are independent, what is the expected count in the (Row1, Col1) cell?

Example 28

medium

In a

2\times 2

table with grand total

200

, the cell (A, X) count is

50

. What is the joint relative frequency?

Example 29

medium

Joint distribution:

(P(A \cap X), P(A \cap Y), P(B \cap X), P(B \cap Y)) = (0.25, 0.15, 0.10, 0.50)

. Find

P(A)

, the marginal probability for row A.

Example 30

medium

A survey of

200

people:

60

like coffee,

140

don't;

80

are over

30

120

are under. If like-coffee and age-over-30 are independent, what is the expected count of 'likes coffee AND over

30

Example 31

hard

A two-way table cross-classifies

300

patients by Treatment (A/B) and Outcome (Cured/Not). Treatment A:

90

cured,

30

not. Treatment B:

60

cured,

120

not. Compute the cure rate for each treatment.

Compute the cure rate for each treatment.

Example 32

hard

A two-way table has expected count

40

and observed count

50

in one cell. Compute that cell's contribution to the

\chi^2

statistic.

Example 33

hard

2 \times 3

two-way table has

2

rows and

3

columns. How many degrees of freedom for the chi-square test of independence?

Example 34

medium

120

surveyed. Cell counts: Yes-Like

= 30

, Yes-Dislike

= 30

, No-Like

= 20

, No-Dislike

= 40

. What is

P(\text{Like} | \text{Yes})

Find P(Like | Yes)

Example 35

medium

Same table. Compute

P(\text{Yes}|\text{Like})

Find P(Yes | Like)

Example 36

medium

From a

2 \times 2

table with marginal proportions

P(\text{Row A}) = 0.4, P(\text{Col X}) = 0.5

, what would

P(A \cap X)

equal if rows and columns are independent?

Example 37

easy

A two-way table has row totals

18

and

22

. What is the grand total?

Example 38

easy

2\times 2

table has cells

a=7

b=3

c=4

d=6

. What is the column 1 total?

Example 39

easy

Pet/No Pet by Cat/Dog table: 20 cat owners, 30 dog owners. Of cat owners, 15 also own a dog. How many own only a cat?

Example 40

easy

A two-way table has

50

people. Row 1 total

= 30

. What fraction are in row 1?

Example 41

easy

A two-way table cell shows

8

students who are Female AND play Soccer out of

40

students. What is the joint relative frequency?

Example 42

easy

In a two-way table for Has Job (Yes/No) by Year (Junior/Senior), Juniors: 5 Yes, 15 No. Seniors: 10 Yes, 10 No. How many students total?

How many students total?

Example 43

medium

From the previous table (Juniors: 5 Yes, 15 No; Seniors: 10 Yes, 10 No), among Seniors, what fraction have a job (conditional relative frequency)?

Among Seniors, what fraction have a job?

Example 44

medium

2\times 2

table has

a=12

b=8

c=18

d=22

. What fraction of cases in column 1 are from row 1 (conditional)?

What fraction of column 1 is from row 1?

Example 45

medium

Movie Genre (Action/Comedy) by Age (Teen/Adult): Teens: 24 Action, 6 Comedy; Adults: 10 Action, 20 Comedy. Among Teens, what percent prefer Comedy?

Among Teens, what percent prefer Comedy?

Example 46

medium

In the same Action/Comedy table, among Adults, what percent prefer Action?

Among Adults, what percent prefer Action?

Example 47

medium

2\times 2

table with grand total

200

a=60

b=40

c=30

. Find

d

and the column 2 marginal total.

Find d and the column 2 marginal total.

Example 48

medium

In a

2\times 2

table the row 1 marginal is

40

, row 2 is

60

, column 1 is

50

. If cell

a

(row 1, col 1) is

25

, fill in the rest.

Fill in cells b, c, and d.

Example 49

medium

In a survey of 240 people,

\tfrac{3}{8}

are children and

\tfrac{1}{2}

of children own a bike. How many children own a bike?

Example 50

hard

2\times 2

table has cells

a, b, c, d

with

a+b=60

c+d=40

a+c=55

, and

a=30

. Find

b

c

d

Find cells b, c, and d.

Example 51

hard

A school of 400 has 60% boys; 25% of boys take art; 40% of girls take art. How many students take art total?

Example 52

hard

Two variables have joint relative frequency

0.3

in a cell and the row marginal proportion is

0.5

. What is the conditional relative frequency within that row?

Example 53

hard

In a

2\times 2

table, the four conditional row distributions read: Row 1 has

60\%

Yes,

40\%

No; Row 2 has

60\%

Yes,

40\%

No. Is there an association?

Example 54

hard

From a

2\times 2

table with cells

a=18,\ b=12,\ c=12,\ d=8

, is there an association between the row and column variables?

Is there an association between the row and column variables?

Example 55

challenge

2\times 2

table has marginal totals

40

and

60

(rows) and

50

and

50

(columns). If the two variables are independent (equal conditional distributions), find every cell.

Example 56

medium

A two-way table shows transport to school: Walk (Year 7: 30, Year 8: 20), Bus (Year 7: 15, Year 8: 25), Car (Year 7: 5, Year 8: 10). What percentage of Year 8 students take the bus?

Example 57

hard

A two-way table shows exercise frequency and health rating for 200 adults: Exercise regularly — Good health: 70, Fair: 20, Poor: 10. Don't exercise regularly — Good health: 30, Fair: 40, Poor: 30. Is there an association between exercise and health? Support your answer with conditional percentages.

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Related Concepts

Data Representation Categorical Data Conditional Probability

Background Knowledge

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

stat data representationcategorical data