11-15-2025, 04:35 PM
Chapter 9 — Frequency Tables
Frequency tables are one of the simplest but MOST powerful tools in probability and statistics.
If raw data looks messy or confusing, a frequency table instantly makes it clear.
Frequency = how many times something appears.
---
9.1 What Is a Frequency Table?
A frequency table is a simple chart that shows:
• each category or value
• how often it appears
Example:
People’s favourite fruit:
Apple: 7
Banana: 4
Grape: 3
Orange: 6
This is already a frequency table.
It helps you visualise patterns and calculate probabilities easily.
---
9.2 Why We Use Frequency Tables
Frequency tables help you:
• organise large amounts of data
• calculate probability
• spot trends
• prepare data for bar charts and pie charts
• calculate mean, median, and mode
They turn chaos into structure.
---
9.3 Building a Frequency Table — Step-by-Step
Example data:
5, 7, 5, 8, 9, 7, 5, 6, 8
Step 1 — List unique values
5, 6, 7, 8, 9
Step 2 — Count each value
5 → 3
6 → 1
7 → 2
8 → 2
9 → 1
Step 3 — Write the table:
Value | Frequency
5 | 3
6 | 1
7 | 2
8 | 2
9 | 1
Done.
---
9.4 Frequency Tables and Probability
Probability is simply:
P(event) = frequency / total
Using the previous table:
Total = 3 + 1 + 2 + 2 + 1 = 9
P(7) = 2/9
P(value ≤ 6) = (3 + 1) / 9 = 4/9
Frequency tables make probability EASY.
---
9.5 Grouped Frequency Tables
Used when the data set is large or continuous.
Example heights (cm):
142, 145, 151, 160, 167, 169, 150, 155, 162
Group into intervals:
140–149 | 2
150–159 | 4
160–169 | 3
Grouped tables help with:
• estimating mean
• creating histograms
• analysing large datasets
---
9.6 Relative Frequency
Relative frequency shows a proportion instead of a raw count.
Example:
If 15 out of 60 people prefer tea:
Relative frequency = 15/60 = 1/4 = 0.25
This helps turn data into probability:
P(prefers tea) = relative frequency = 0.25
---
9.7 Cumulative Frequency
Used for:
• medians
• quartiles
• large datasets
• grouped problems
Example (grouped lengths):
Length | Frequency | Cumulative
0–10 | 4 | 4
10–20 | 7 | 11
20–30 | 5 | 16
30–40 | 3 | 19
Cumulative frequency always increases as you move down the table.
---
9.8 Using Frequency Tables to Find Mode, Median, Mean
Mode → most common value (or group)
Median → middle value, use cumulative frequency
Mean → calculated using:
(sum of values × frequency) / total frequency
Example:
Value | Freq
3 | 2
4 | 5
6 | 3
Mean = (3×2 + 4×5 + 6×3) / (2+5+3)
= (6 + 20 + 18) / 10
= 44/10 = 4.4
---
9.9 Exam Example
A survey records how many books 20 students read last month:
Books | Frequency
0 | 3
1 | 6
2 | 5
3 | 4
4 | 2
Find:
(a) P(student read 2 books)
(b) P(student read at least 3 books)
© The mode
Solutions:
(a) 5/20 = 1/4
(b) At least 3 books → 3 or 4
= (4 + 2) / 20 = 6/20 = 3/10
© Mode → highest frequency = 1 book (freq 6)
---
9.10 Your Turn — Practice
1. A class records birthdays by month:
Jan 3, Feb 1, Mar 2, Apr 4, May 3, Jun 2, Jul 1, Aug 4
(a) Which month is the mode?
(b) Find P(birthday is in first 3 months).
---
2. Grouped weights:
Weight (kg) | Freq
40–50 | 3
50–60 | 8
60–70 | 9
(a) Estimate total number of students.
(b) Estimate probability weight > 60kg.
---
3. Scores on a test:
Score | Freq
2 | 1
3 | 3
4 | 5
5 | 6
6 | 5
(a) Find total number of students
(b) Find median score
---
4. A bag has sweets recorded:
Colour | Freq
Red | 7
Blue | 4
Green | 3
Yellow | 6
Find P(yellow), P(green or blue).
---
5. Heights in grouped form:
Height | Freq
120–130 | 2
130–140 | 5
140–150 | 8
150–160 | 5
Find cumulative frequency table.
---
Chapter Summary
• Frequency tables organise raw data
• They make probability simple and fast
• Grouped tables are used for larger datasets
• Relative frequency connects data to probability
• Cumulative frequency helps find medians and quartiles
Mastering frequency tables makes future statistics chapters much easier.
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive
Frequency tables are one of the simplest but MOST powerful tools in probability and statistics.
If raw data looks messy or confusing, a frequency table instantly makes it clear.
Frequency = how many times something appears.
---
9.1 What Is a Frequency Table?
A frequency table is a simple chart that shows:
• each category or value
• how often it appears
Example:
People’s favourite fruit:
Apple: 7
Banana: 4
Grape: 3
Orange: 6
This is already a frequency table.
It helps you visualise patterns and calculate probabilities easily.
---
9.2 Why We Use Frequency Tables
Frequency tables help you:
• organise large amounts of data
• calculate probability
• spot trends
• prepare data for bar charts and pie charts
• calculate mean, median, and mode
They turn chaos into structure.
---
9.3 Building a Frequency Table — Step-by-Step
Example data:
5, 7, 5, 8, 9, 7, 5, 6, 8
Step 1 — List unique values
5, 6, 7, 8, 9
Step 2 — Count each value
5 → 3
6 → 1
7 → 2
8 → 2
9 → 1
Step 3 — Write the table:
Value | Frequency
5 | 3
6 | 1
7 | 2
8 | 2
9 | 1
Done.
---
9.4 Frequency Tables and Probability
Probability is simply:
P(event) = frequency / total
Using the previous table:
Total = 3 + 1 + 2 + 2 + 1 = 9
P(7) = 2/9
P(value ≤ 6) = (3 + 1) / 9 = 4/9
Frequency tables make probability EASY.
---
9.5 Grouped Frequency Tables
Used when the data set is large or continuous.
Example heights (cm):
142, 145, 151, 160, 167, 169, 150, 155, 162
Group into intervals:
140–149 | 2
150–159 | 4
160–169 | 3
Grouped tables help with:
• estimating mean
• creating histograms
• analysing large datasets
---
9.6 Relative Frequency
Relative frequency shows a proportion instead of a raw count.
Example:
If 15 out of 60 people prefer tea:
Relative frequency = 15/60 = 1/4 = 0.25
This helps turn data into probability:
P(prefers tea) = relative frequency = 0.25
---
9.7 Cumulative Frequency
Used for:
• medians
• quartiles
• large datasets
• grouped problems
Example (grouped lengths):
Length | Frequency | Cumulative
0–10 | 4 | 4
10–20 | 7 | 11
20–30 | 5 | 16
30–40 | 3 | 19
Cumulative frequency always increases as you move down the table.
---
9.8 Using Frequency Tables to Find Mode, Median, Mean
Mode → most common value (or group)
Median → middle value, use cumulative frequency
Mean → calculated using:
(sum of values × frequency) / total frequency
Example:
Value | Freq
3 | 2
4 | 5
6 | 3
Mean = (3×2 + 4×5 + 6×3) / (2+5+3)
= (6 + 20 + 18) / 10
= 44/10 = 4.4
---
9.9 Exam Example
A survey records how many books 20 students read last month:
Books | Frequency
0 | 3
1 | 6
2 | 5
3 | 4
4 | 2
Find:
(a) P(student read 2 books)
(b) P(student read at least 3 books)
© The mode
Solutions:
(a) 5/20 = 1/4
(b) At least 3 books → 3 or 4
= (4 + 2) / 20 = 6/20 = 3/10
© Mode → highest frequency = 1 book (freq 6)
---
9.10 Your Turn — Practice
1. A class records birthdays by month:
Jan 3, Feb 1, Mar 2, Apr 4, May 3, Jun 2, Jul 1, Aug 4
(a) Which month is the mode?
(b) Find P(birthday is in first 3 months).
---
2. Grouped weights:
Weight (kg) | Freq
40–50 | 3
50–60 | 8
60–70 | 9
(a) Estimate total number of students.
(b) Estimate probability weight > 60kg.
---
3. Scores on a test:
Score | Freq
2 | 1
3 | 3
4 | 5
5 | 6
6 | 5
(a) Find total number of students
(b) Find median score
---
4. A bag has sweets recorded:
Colour | Freq
Red | 7
Blue | 4
Green | 3
Yellow | 6
Find P(yellow), P(green or blue).
---
5. Heights in grouped form:
Height | Freq
120–130 | 2
130–140 | 5
140–150 | 8
150–160 | 5
Find cumulative frequency table.
---
Chapter Summary
• Frequency tables organise raw data
• They make probability simple and fast
• Grouped tables are used for larger datasets
• Relative frequency connects data to probability
• Cumulative frequency helps find medians and quartiles
Mastering frequency tables makes future statistics chapters much easier.
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive