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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn4024Paper 1 · Non-calculatorStatisticsAverages and range

Statistics · Subtopic 1 of 5

Averages and Range

The mean shares the total equally, the median is the middle value in order, and the mode is the most common. The range measures spread. For frequency tables, multiply each value by its frequency; for grouped data, use the class midpoints.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 5 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Find the mean, median, mode and range of a list.
Use totals to solve problems about the mean.
Find averages from a frequency table.
Estimate the mean of grouped data and find the modal class.

Pause and think

Nine people in a room each earn about $30\,000$ dollars a year. Someone earning $1$ billion dollars a year walks in. What happens to the mean income? And the median?

Section 2 of 14

Core idea

Start hereThe average salary trap

Nine people earn 30 000 and one earns 1 000 000. The mean salary is 127 000, which describes nobody. The median, 30 000, is far more honest. Choosing the right average matters.

The big idea

Mean $\frac{\sum fx}{\sum f}$ ; median: middle value, position $\frac{n+1}2$ ; mode: most common; range: spread. The median resists extremes.

1. The averages

Salaries in a small firm: one big value drags the mean

The boss's 100 pulls the mean up to 35.3, above six of the seven salaries. The median (25) is more typical here.

2. Frequency tables

Worked exampleGoals in 20 matches

0: 4, 1: 6, 2: 7, 3: 3.

Goal: Mean, median, mode.
Strategy: $\sum fx$ ; running totals.

$\frac{0+6+14+9}{20}=1.45$
Mean.
$4,10,17,20$
Running totals: 10th is 1, 11th is 2.
$\text{median}=1.5,\ \text{mode}=2$
Average of 10th and 11th; highest frequency.

Answer

1.45

1.5

2

✓ CheckAll between 0 and 3 ✓.

Worked exampleGrouped estimate

0–10: 5; 10–20: 8; 20–30: 7.

Goal: Estimated mean.
Strategy: Use midpoints.

$5(5)+15(8)+25(7)=320$
$\sum fx$ with midpoints.
$320\div20=16$
Divide.

Answer

16

(estimate).

✓ CheckIt's an estimate: exact values within classes are unknown ✓.

Common misconceptionDividing by the number of rows Divide by

\sum f

(20), not by 4 classes.

Pause and think

Mean of 5 numbers is 8; four are 6, 7, 9, 10. Fifth?

Summary

Mean $\sum fx/\sum f$ .
Median position $(n+1)/2$ .

Median: Middle value.
Mode: Most common value.

HistoryAveraging out errors. In the late 1500s, the astronomer Tycho Brahe took many measurements of the same star position and combined them to reduce errors. Averaging repeated measurements soon became standard scientific practice and is still the first step in most experiments.

Fun factAbove-average legs. Almost everyone has more than the mean number of legs. A few people have fewer than two, which pulls the mean slightly below

2

, while the median and mode are exactly

2

Section 3 of 14

Key formulae

Mean

\bar x=\frac{\sum x}{n}

Mean from a frequency table

\bar x=\frac{\sum fx}{\sum f}

Median position

\frac{n+1}{2}\text{th value}

TrickThink in totals. Any question about a mean changing ("one more value is added", "find the missing number") is easiest with totals: total

=

mean

\times

count. Work with totals, then divide at the very end.

Section 4 of 14

Worked examples

Example 1 · Adding a value0 / 2 steps

The mean of

8

numbers is

15

. A ninth number,

24

, is added. Find the new mean.

Example 2 · Median from a table0 / 2 steps

Scores:

0

(frequency

4

1

(

7

2

(

6

3

(

3

). Find the median.

Section 5 of 14

Explore the graph

An outlier pulls the mean

Loading the interactive graph…

Try this

The data are $2, 4, 5, 6$ and one more value $v$ . The rising line is the mean; the flat line is the median ( $5$ ).
Drag $v$ up to $60$ . How far does the mean move? Does the median move?
Which average better describes "a typical value" when $v=60$ ?

One extreme value moves the mean but leaves the median unchanged.

In the real worldAverage house prices. Official house-price reports usually quote the median rather than the mean. A few mansions selling for millions would drag the mean far above what a typical home costs.

Section 6 of 14

Common mistakes

IncorrectMedian of an unordered list

CorrectOrder the values first

The median is the middle in order.

Incorrect

\sum fx

divided by the number of rows

CorrectDivide by

\sum f

Divide by the total frequency.

IncorrectUsing upper class boundaries for grouped mean

CorrectUse midpoints

The midpoint represents the class.

IncorrectMode

=

the highest frequency

CorrectThe value with the highest frequency

The mode is a data value, not a count.

Mean

I'm the fair share

Median

I'm the middle one

Mode

I'm just the most popular. Don't overthink it.

Section 7 of 14

Exam tip

Exam tipAdd columns to the table. For a frequency-table mean, add a midpoint column and an

fx

column to the table in the question. Showing

\sum fx

and

\sum f

earns the method marks even if the final division slips.

Section 8 of 14

Guided practice

SA-4024-ST-G014 marksL1 FoundationNon-calculatorAverages

For the data

4,\ 7,\ 7,\ 9,\ 12,\ 15

, find the mean, the median, the mode and the range.

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-4024-ST1-0011 markL1 FoundationNon-calculatorAverages

Find the median of

3,\ 8,\ 1,\ 9,\ 5,\ 7,\ 2

Salam Academy Original Practice · Proposed mark scheme

SA-4024-ST-0062 marksL2 StandardNon-calculatorAverages

The mean of five numbers is

12

. Four of them are

10

8

15

and

11

. Find the fifth number.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-ST-0072 marksL2 StandardNon-calculatorAverages

The mean of

8

numbers is

15

. A ninth number,

24

, is added. Find the new mean.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-ST-0012 marksL2 StandardCalculatorFrequency tables

The scores of

20

students are: score

1

(frequency

3

), score

2

(frequency

5

), score

3

(frequency

8

), score

4

(frequency

4

). Calculate the mean score.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-ST1-0022 marksL2 StandardNon-calculatorFrequency tables

The scores of

20

students are: score

0

(frequency

4

), score

1

(frequency

7

), score

2

(frequency

6

), score

3

(frequency

3

). Find the median score.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-ST-0023 marksL2 StandardCalculatorFrequency tables

Journey times (minutes) of

20

people:

0<t\le10

4

;

10<t\le20

9

;

20<t\le30

5

;

30<t\le40

2

. Calculate an estimate of the mean time.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-ST-0031 markL1 FoundationCalculatorFrequency tables

For the journey times in the previous question (

0

–

10

4

;

10

–

20

9

;

20

–

30

5

;

30

–

40

2

), the modal class is

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-4024-ST-0164 marksL4 ChallengingNon-calculatorAverages

The four numbers

x

2x

3x

and

10

have a mean of

13

. (i) Find

x

. (ii) Find the median of the four numbers.

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Mean: total ÷ count. Median: middle in order. Mode: most common. Range: largest $-$ smallest.
Table mean $=\dfrac{\sum fx}{\sum f}$ ; grouped: use midpoints (an estimate).
Median position $\tfrac{n+1}2$ .
Median resists outliers; mode for "most popular".

Section 12 of 14

Quick quiz

Q1 · SA-4024-ST-Q011 markL1 FoundationCalculatorAverages

The range of a data set is

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-4024-ST-0151 markL2 StandardCalculatorAverages

House prices in a street include one very expensive house. Which average best represents a typical price?

Salam Academy Original Practice · Proposed mark scheme

Q3 · SA-4024-ST1-0031 markL1 FoundationCalculatorAverages

A shoe shop wants to know which size to order most of. Which average should it use?

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: 4024 O Level Mathematics
Where: Papers 1 and 2 · 9.3 Averages and range
Expected: Calculate mean, median, mode and range for individual and discrete data; estimate the mean of grouped data using midpoints; identify the modal class; choose an appropriate average.
Calculator: Paper 1 is non-calculator; Paper 2 allows a calculator.
Typical style: Calculate an estimate of the mean journey time from a grouped frequency table.
Open in: 4024 · Paper 1 4024 · Paper 2

Section 14 of 14

Past paper questions

5 questions (36 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

4024/22 · Oct/Nov 2025Question 23 marks
Median and range of quiz scores.
Question paper, page 3 Mark scheme
4024/22 · May/June 2025Question 187 marks
Estimate a mean from grouped data and draw a histogram.
Question paper, page 12 Mark scheme
4024/22 · Oct/Nov 2024Question 210 marks
Scatter diagram, line of best fit, and averages from a frequency table.
Question paper, page 4 Mark scheme
4024/12 · May/June 2024Question 33 marks
Median, mode, range and mean.
Question paper, page 4 Mark scheme
4024/22 · May/June 2024Question 913 marks
Cumulative frequency, median, percentiles and estimated mean.
Question paper, page 14 Mark scheme

All 4024 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Cambridge International 4024, Current syllabus (official specification).