📊 Two-Way Tables

Cambridge Lower Secondary · Grade 7 · Statistics & Probability

Reading a Cell

Find the row → find the column → read the value!
Each cell counts BOTH conditions at once

Row/Column Totals

Row total = sum of all cells in that row
Grand total = sum of all row totals = sum of all column totals

Probability from a Table

P(event) = ^{cell value}/_{grand total}
Always use the grand total for simple probability

A class was asked: Do you play sport? Do you own a pet? 🐶⚽

	Owns a Pet	No Pet	Total
Plays Sport	?	?	?
No Sport	?	?	?
Total	?	?	?

What you'll learn:

Reading values from two-way tables (rows, columns, totals)
Completing missing values using row & column totals
Calculating simple probabilities from a two-way table
Relative frequency and proportions from tables
Drawing your own two-way table from raw data
Using tables to compare two groups
Interpreting findings and drawing conclusions

Created by Mr Josh

📖 Learn: Two-Way Tables

Part 1: What Is a Two-Way Table?

A two-way table (also called a contingency table) organises data according to two categories at once — one along the rows and one along the columns.

	Column Category A	Column Category B	Row Total
Row Category 1	A & 1	B & 1	Row 1 Total
Row Category 2	A & 2	B & 2	Row 2 Total
Column Total	Col A Total	Col B Total	Grand Total

📌 Each cell in the middle counts items that satisfy both the row AND column condition.

📌 The row total = sum of all cells in that row.

📌 The column total = sum of all cells in that column.

📌 The grand total = sum of all row totals = sum of all column totals.

💡 Always check: grand total at the corner should equal both the row totals added up AND the column totals added up!

Part 2: Reading Values From a Table

80 students were asked if they prefer maths or English and whether they are in Year 7 or Year 8.

	Prefer Maths	Prefer English	Total
Year 7	18	22	40
Year 8	24	16	40
Total	42	38	80

How many Year 7 students prefer Maths? → Find Year 7 row, Prefer Maths column → 18

How many students prefer English in total? → Column total for Prefer English → 38

How many Year 8 students are there? → Row total for Year 8 → 40

💡 Think: Row = the "who", Column = the "what". Move across the row to the right column, then read.

Part 3: Completing Missing Values

If some cells are missing, use the fact that rows and columns must add up to their totals.

	Left-Handed	Right-Handed	Total
Male	?	27	30
Female	4	?	25
Total	7	?	55

Step 1: Male, Left-Handed = Row total − Right-Handed = 30 − 27 = 3

Step 2: Female, Right-Handed = Row total − Left-Handed = 25 − 4 = 21

Step 3: Right-Handed column total = 27 + 21 = 48 ✓ (Check: 7 + 48 = 55 ✓)

💡 Strategy: fill in any cell where you know all but one value in its row or column. Work outward!

Part 4: Probability From a Two-Way Table

Use the table to find probabilities. The grand total is the denominator for simple (unconditional) probability.

	Likes Football	Does Not Like Football	Total
Girl	12	18	30
Boy	20	10	30
Total	32	28	60

P(randomly selected student is a girl who likes football) = ¹²/₆₀ = 1/5

P(student likes football) = ³²/₆₀ = 8/15

P(student is a boy) = ³⁰/₆₀ = 1/2

💡 Always write the probability as a fraction: P(event) = specific cell ÷ grand total

Part 5: Relative Frequency & Comparing Groups

Relative frequency expresses each cell (or total) as a fraction or percentage of the grand total. This lets you compare groups of different sizes.

Relative frequency of girls who like football = 12 ÷ 60 = 0.2 = 20%

Relative frequency of boys who like football = 20 ÷ 60 ≈ 0.333 = 33.3%

Comparison: Boys are more likely to like football (33.3%) than girls (20%).

💡 To compare groups fairly, use proportions (relative frequency) rather than raw counts — especially when group sizes differ!

You can also find the proportion within a group: proportion of girls who like football = 12 ÷ 30 = 0.4 = 40%

Created by Mr Josh

💡 Worked Examples

Example 1: Reading a Two-Way Table

60 people at a bakery were asked their age group (Under 20 / 20 or over) and their favourite item (Cake / Bread / Pastry).

	Cake	Bread	Pastry	Total
Under 20	14	6	10	30
20 or Over	8	15	7	30
Total	22	21	17	60

Q: How many under-20s prefer Bread? → Under 20 row, Bread column → 6

Q: How many people prefer Pastry? → Pastry column total → 17

Q: How many 20-or-over people were surveyed? → Row total for 20 or Over → 30

📌 Always identify the row AND column needed before reading the cell.

Example 2: Completing Missing Values

Some cells are missing. Use row and column totals to find them.

	Has Siblings	No Siblings	Total
Has a Pet	15	?→ 9	24
No Pet	?→ 13	8	21
Total	?→ 28	17	45

Step 1: Has a Pet, No Siblings = 24 − 15 = 9

Step 2: No Pet, Has Siblings = 21 − 8 = 13

Step 3: Has Siblings column total = 15 + 13 = 28

Step 4: Check: 28 + 17 = 45 ✓ and 24 + 21 = 45 ✓

Example 3: Probability From a Table

Using the table from Example 1 (60 people). A person is chosen at random.

P(person is Under 20 AND prefers Cake) = ¹⁴/₆₀ = ⁷/₃₀

P(person prefers Bread) = ²¹/₆₀ = ⁷/₂₀

P(person is 20 or Over) = ³⁰/₆₀ = ¹/₂

P(person does NOT prefer Pastry) = 1 − ¹⁷/₆₀ = ⁴³/₆₀

📌 For "AND" probabilities from a table, use the cell that matches BOTH conditions. Denominator = grand total.

Example 4: Relative Frequency & Comparison

150 students were asked if they walk to school and if they are in Year 7 or Year 8.

	Walk to School	Do Not Walk	Total
Year 7	45	30	75
Year 8	35	40	75
Total	80	70	150

Proportion of Year 7 who walk = 45 ÷ 75 = 0.6 = 60%

Proportion of Year 8 who walk = 35 ÷ 75 ≈ 0.467 = 46.7%

Conclusion: Year 7 students are more likely to walk to school than Year 8 students (60% vs 46.7%).

💡 To compare fairly: divide each group's count by its own row total, not the grand total.

Created by Mr Josh

📊 Interactive Visualizers

🏗️ Table Builder

Enter the label names and data values below. The two-way table builds automatically with row and column totals!

Row 1 Label:

Row 2 Label:

Column 1 Label:

Column 2 Label:

Enter the 4 data values:

Row 1, Col 1:

Row 1, Col 2:

Row 2, Col 1:

Row 2, Col 2:

	Plays Sport ⚽	No Sport	Total
Has Dog 🐶	0	0	0
No Dog	0	0	0
Total	0	0	0

	Cat Owner 🐱	Not a Cat Owner	Total
Left-Handed ✋	8	16	24
Right-Handed	22	54	76
Total	30	70	100

Created by Mr Josh

✏️ Exercise 1: Reading a Two-Way Table

Use the table to answer the questions. Type the number answer in each box.

	Chocolate 🍫	Vanilla 🍦	Strawberry 🍓	Total
Boys	16	9	5	30
Girls	11	14	5	30
Total	27	23	10	60

Created by Mr Josh

✏️ Exercise 2: Complete the Missing Values

Work out the missing values in each table using row and column totals.

Created by Mr Josh

✏️ Exercise 3: Probability from Two-Way Tables

A student is picked at random. Give probabilities as simplified fractions.

	Walks to School 🚶	Gets a Bus 🚌	Cycles 🚲	Total
Year 7	18	9	3	30
Year 8	12	12	6	30
Total	30	21	9	60

Created by Mr Josh

📉 Exercise 4: Relative Frequency

Give answers as decimals (to 2 d.p. where needed) or percentages where stated.

	Passed 🟢	Failed 🔴	Total
Studied	36	9	45
Did Not Study	10	45	55
Total	46	54	100

Created by Mr Josh

🏗️ Exercise 5: Draw a Two-Way Table

Here are the results of a survey of 20 students. Complete the two-way table below.

Survey data (each student recorded as Sport/No Sport × Music/No Music):

Sport & Music, Sport & Music, No Sport & Music, Sport & No Music, No Sport & No Music,
Sport & Music, No Sport & Music, Sport & Music, Sport & No Music, No Sport & Music,
Sport & Music, Sport & No Music, No Sport & No Music, Sport & Music, No Sport & No Music,
No Sport & Music, Sport & Music, Sport & No Music, No Sport & Music, Sport & Music

💡 Tally up: Sport & Music = ?, Sport & No Music = ?, No Sport & Music = ?, No Sport & No Music = ?

Fill in every cell of the table below:

	Likes Music 🎵	No Music	Total
Plays Sport ⚽
No Sport
Total

Created by Mr Josh

📝 Practice Questions

Reference Table — use this for questions 1–12:

	Glasses 👓	No Glasses	Total
Year 7	15	25	40
Year 8	10	30	40
Total	25	55	80

Created by Mr Josh

🏆 Challenge Questions

These questions require more steps or careful reasoning. Give simplified fractions or percentages where asked.

Created by Mr Josh

📊 Two-Way Tables

📖 Learn: Two-Way Tables

Part 1: What Is a Two-Way Table?

Part 2: Reading Values From a Table

Part 3: Completing Missing Values

Part 4: Probability From a Two-Way Table

Part 5: Relative Frequency & Comparing Groups

💡 Worked Examples

Example 1: Reading a Two-Way Table

Example 2: Completing Missing Values

Example 3: Probability From a Table

Example 4: Relative Frequency & Comparison

📊 Interactive Visualizers

🏗️ Table Builder

🎲 Survey Simulator: Class of 30

🧩 Fill the Gaps Puzzle

🎯 Probability Extractor

✏️ Exercise 1: Reading a Two-Way Table

✏️ Exercise 2: Complete the Missing Values

✏️ Exercise 3: Probability from Two-Way Tables

📉 Exercise 4: Relative Frequency

🏗️ Exercise 5: Draw a Two-Way Table

📝 Practice Questions

🏆 Challenge Questions