11-15-2025, 04:39 PM
Chapter 12 — Mean, Median, Mode & Range
These four key statistics describe the shape and behaviour of data.
If frequency tables organise information, these measures help you understand it.
Mastering them is essential for:
• exams
• data analysis
• probability
• real-world interpretation
Let’s break each one down clearly and intuitively.
---
12.1 Mean — The Average
Mean = total of all values ÷ number of values
Example:
Data: 4, 8, 6, 2
Mean = (4+8+6+2) / 4
Mean = 20 / 4
Mean = 5
When the mean is useful:
• when data is evenly distributed
• when no extreme values are present
When the mean is misleading:
• when data contains “outliers”
(e.g., salaries)
---
12.2 Median — The Middle Value
The median is the value in the middle when data is ordered from smallest to largest.
Example:
Data: 3, 9, 7
Ordered: 3, 7, 9
Median = 7
Example with even numbers:
Data: 6, 2, 10, 4
Ordered: 2, 4, 6, 10
Median = (4 + 6) / 2 = 5
When the median is useful:
• when data has outliers
• when the mean would be distorted
• when analysing typical values (e.g., house prices)
---
12.3 Mode — The Most Common Value
Mode = the value that appears most often
Example:
Data: 5, 4, 7, 5, 6
Mode = 5
There can be:
• one mode
• more than one mode
• no mode (if all values appear equally)
Mode is especially useful for:
• categorical data (favourite colour, etc.)
• most frequent behaviour
• identifying peaks or patterns
---
12.4 Range — The Spread of Data
Range = largest value − smallest value
Example:
Data: 3, 8, 2, 10
Range = 10 − 2 = 8
Range shows:
• how spread out the data is
• whether values are tightly grouped
• variation or inconsistency
Range is easy to calculate but sensitive to outliers.
---
12.5 Putting the Four Together
Let’s analyse some data:
Data: 12, 18, 10, 12, 20
Mean:
(12 + 18 + 10 + 12 + 20) / 5 = 72/5 = 14.4
Median:
Ordered: 10, 12, 12, 18, 20
Middle value = 12
Mode:
12 (appears twice)
Range:
20 − 10 = 10
What this tells us:
• typical (median) value is 12
• mean is slightly higher due to the 20
• range shows moderate variation
---
12.6 Using Frequency Tables to Calculate Mean
Example:
Value | Freq
2 | 3
4 | 5
6 | 2
Mean formula:
Mean = (value × freq total) ÷ total freq
= (2×3 + 4×5 + 6×2) / (3+5+2)
= (6 + 20 + 12) / 10
= 38 / 10
= 3.8
---
12.7 Using Cumulative Frequency for Median
Example:
Class interval | Frequency
0–10 | 4
10–20 | 8
20–30 | 5
30–40 | 3
Cumulative frequency:
4 → 12 → 17 → 20
If total = 20, median = the 10th value.
That sits in the 10–20 group.
---
12.8 Real-World Example
House prices:
• £80k
• £82k
• £85k
• £90k
• £900k
Mean = £1,237,000 / 5 = £247,400
(because of the outlier)
Median = £85k
(better representation)
This is why property websites use median, not mean.
---
12.9 Exam-Style Questions
1. Find the mean, median, mode, and range of:
7, 8, 4, 9, 7, 6
2. A frequency table records test scores:
Score | Freq
3 | 2
4 | 5
5 | 3
Find the mean score.
3. A class measured shoe sizes:
Sizes: 5, 6, 6, 7, 7, 7, 8
Find the mode.
4. The numbers below show weekly hours studied:
11, 9, 14, 12, 18
Find the range.
5. Grouped data:
Interval | Freq
0–5 | 4
5–10 | 6
10–15 | 5
Locate the median group.
---
12.10 Summary
• Mean uses all data
• Median shows the middle
• Mode shows the most common
• Range shows the spread
Knowing when to use each is the key skill:
• median for outliers
• mean for balanced data
• mode for categorical data
• range for variation
You now have the essential toolkit for describing any dataset.
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive
These four key statistics describe the shape and behaviour of data.
If frequency tables organise information, these measures help you understand it.
Mastering them is essential for:
• exams
• data analysis
• probability
• real-world interpretation
Let’s break each one down clearly and intuitively.
---
12.1 Mean — The Average
Mean = total of all values ÷ number of values
Example:
Data: 4, 8, 6, 2
Mean = (4+8+6+2) / 4
Mean = 20 / 4
Mean = 5
When the mean is useful:
• when data is evenly distributed
• when no extreme values are present
When the mean is misleading:
• when data contains “outliers”
(e.g., salaries)
---
12.2 Median — The Middle Value
The median is the value in the middle when data is ordered from smallest to largest.
Example:
Data: 3, 9, 7
Ordered: 3, 7, 9
Median = 7
Example with even numbers:
Data: 6, 2, 10, 4
Ordered: 2, 4, 6, 10
Median = (4 + 6) / 2 = 5
When the median is useful:
• when data has outliers
• when the mean would be distorted
• when analysing typical values (e.g., house prices)
---
12.3 Mode — The Most Common Value
Mode = the value that appears most often
Example:
Data: 5, 4, 7, 5, 6
Mode = 5
There can be:
• one mode
• more than one mode
• no mode (if all values appear equally)
Mode is especially useful for:
• categorical data (favourite colour, etc.)
• most frequent behaviour
• identifying peaks or patterns
---
12.4 Range — The Spread of Data
Range = largest value − smallest value
Example:
Data: 3, 8, 2, 10
Range = 10 − 2 = 8
Range shows:
• how spread out the data is
• whether values are tightly grouped
• variation or inconsistency
Range is easy to calculate but sensitive to outliers.
---
12.5 Putting the Four Together
Let’s analyse some data:
Data: 12, 18, 10, 12, 20
Mean:
(12 + 18 + 10 + 12 + 20) / 5 = 72/5 = 14.4
Median:
Ordered: 10, 12, 12, 18, 20
Middle value = 12
Mode:
12 (appears twice)
Range:
20 − 10 = 10
What this tells us:
• typical (median) value is 12
• mean is slightly higher due to the 20
• range shows moderate variation
---
12.6 Using Frequency Tables to Calculate Mean
Example:
Value | Freq
2 | 3
4 | 5
6 | 2
Mean formula:
Mean = (value × freq total) ÷ total freq
= (2×3 + 4×5 + 6×2) / (3+5+2)
= (6 + 20 + 12) / 10
= 38 / 10
= 3.8
---
12.7 Using Cumulative Frequency for Median
Example:
Class interval | Frequency
0–10 | 4
10–20 | 8
20–30 | 5
30–40 | 3
Cumulative frequency:
4 → 12 → 17 → 20
If total = 20, median = the 10th value.
That sits in the 10–20 group.
---
12.8 Real-World Example
House prices:
• £80k
• £82k
• £85k
• £90k
• £900k
Mean = £1,237,000 / 5 = £247,400
(because of the outlier)
Median = £85k
(better representation)
This is why property websites use median, not mean.
---
12.9 Exam-Style Questions
1. Find the mean, median, mode, and range of:
7, 8, 4, 9, 7, 6
2. A frequency table records test scores:
Score | Freq
3 | 2
4 | 5
5 | 3
Find the mean score.
3. A class measured shoe sizes:
Sizes: 5, 6, 6, 7, 7, 7, 8
Find the mode.
4. The numbers below show weekly hours studied:
11, 9, 14, 12, 18
Find the range.
5. Grouped data:
Interval | Freq
0–5 | 4
5–10 | 6
10–15 | 5
Locate the median group.
---
12.10 Summary
• Mean uses all data
• Median shows the middle
• Mode shows the most common
• Range shows the spread
Knowing when to use each is the key skill:
• median for outliers
• mean for balanced data
• mode for categorical data
• range for variation
You now have the essential toolkit for describing any dataset.
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive