A class $10<x\le25$ has frequency 36. Find the frequency density.

Class width $=15$. $36\div15=2.4$. Answer: 2.4

The cumulative frequency table gives the journey times, $ t $ minutes, of 80 commuters. $ t \le 10 $: 6; $ t \le 20 $: 22; $ t \le 30 $: 47; $ t \le 40 $: 68; $ t \le 50 $: 80. The cumulative frequency graph is drawn by joining the plotted points with straight lines. Use it to estimate the median journey time.

M1 median at cumulative frequency 40 M1 20 + (40 − 22)/(47 − 22) × 10 A1 27.2 minutes (accept 27 to 27.5 from a graph) Answer: 27.2 minutes

Edexcel IGCSE Maths (4MA1) · Unit 6: Statistics & Probability

Edexcel IGCSE 4MA1 Histograms and Cumulative Frequency: Exam Questions and Answers

Q: Which Edexcel IGCSE Maths papers test histograms and cumulative frequency?

Either Higher tier paper, 1H or 2H, can test histograms and cumulative frequency: both cover the whole Pearson Edexcel International GCSE Mathematics A (4MA1) specification and both allow a calculator. Show your method, because most marks are for working.

Q: Where can I practise more histograms and cumulative frequency questions?

In the IGCSE student hub: Unit 6 (Statistics & Probability) has more exam-style questions on this topic, each with a mark scheme. Practice starts with a 7-day free trial; the worked examples and questions on this page are free.

Histograms and Cumulative Frequency questions with full solutions for Pearson Edexcel International GCSE Mathematics A (4MA1), Higher tier. Read the 4 worked examples first, then try the 8 exam-style questions yourself and check your working against the mark scheme.

4 worked examples
8 questions with mark schemes
Higher tier · calculator allowed
Free, no sign-in

Practise histograms and cumulative frequency in Unit 6 →Histograms and Cumulative Frequency revision notes

What this topic covers

Representing Data 1: Choosing frequency tables, two-way tables, bar charts and pie charts for data (4MA1 does not examine stem-and-leaf diagrams, scatter graphs or box plots).
Representing Data 2: Frequency polygons and histograms.
Cumulative Frequency 1: Drawing a cumulative frequency curve from grouped data.
Cumulative Frequency 2: Reading median, quartiles from a c.f. curve.

Histograms and Cumulative Frequency worked examples

Worked example 1: Pie chart angle

45 out of 120 students chose football. Find the angle for football in a pie chart.

Solution

$\frac{45}{120} \times 360$ .
$= 135^{\circ}$ .

3 steps

Tip: Each person is worth 360 ÷ 120 = 3°.

Worked example 2: Frequency density

A class $10 < x \leq 25$ has frequency 36. Find the frequency density.

Solution

Class width $= 15$ .
$36 \div 15 = 2.4$ .

3 steps

Tip: Histogram bar height is frequency density, not frequency.

Worked example 3: Cumulative frequency table

Frequencies of four classes are 4, 9, 15, 12. Write the cumulative frequencies.

Solution

Running totals: 4, $4 + 9 = 13$ , $13 + 15 = 28$ , $28 + 12 = 40$ .

2 steps

Tip: Plot cumulative frequency at the upper class boundary.

Worked example 4: Quartiles from a list

Find the median and interquartile range of 3, 5, 6, 8, 9, 11, 12, 14, 15, 18, 20.

Solution

11 values: median is the 6th value, 11.
Lower quartile: 3rd value, 6. Upper quartile: 9th value, 15.
IQR $= 15 - 6 = 9$ .

4 steps · 1 to fill in

Tip: For n values use the (n + 1)/4 and 3(n + 1)/4 positions.

Where students lose marks on histograms and cumulative frequency

Examiner's noteCumulative frequency plotted at midpoints

Where marks are lost: Points are plotted at mid-interval or lower-interval values, bar charts or lines of best fit are drawn, and readings are not subtracted from the total when asked "how many more than".

Examiners flagged this in 4 exam series (November 2023, June 2024, November 2024, June 2025). Summarised in our own words from Pearson Edexcel examiner reports. Read the full note →

Histograms and Cumulative Frequency exam-style questions

Try each question before you open the answer (one has its full worked solution open). The mark schemes use Edexcel-style marks: M for method, A for accuracy (after the method mark), B for an independent result.

Question 1 · easy · 2 marks

The table gives the times, $t$ minutes, taken by 50 runners to finish a race. $20 < t \leq 25$ : 3; $25 < t \leq 30$ : 11; $30 < t \leq 35$ : 18; $35 < t \leq 40$ : 12; $40 < t \leq 45$ : 6. Write down the cumulative frequencies for the upper bounds 25, 30, 35, 40 and 45.

Show the answer and mark scheme

B2 3, 14, 32, 44, 50 (B1 for at least three correct)

Answer: 3, 14, 32, 44, 50

Question 2 · easy · 1 mark

In a cumulative frequency table for the masses of parcels, the cumulative frequency for $m \leq 60$ kg is 38 and the cumulative frequency for $m \leq 70$ kg is 71. How many parcels have a mass in the class $60 < m \leq 70$ ?

Show the answer and mark scheme

B1 33

Answer: 33

Question 3 · medium · 2 marks

The cumulative frequency table gives the heights, $h$ cm, of 120 students. $h \leq 150$ : 9; $h \leq 155$ : 27; $h \leq 160$ : 58; $h \leq 165$ : 91; $h \leq 170$ : 110; $h \leq 175$ : 120. Which class interval contains the median height?

Show the answer and mark scheme

M1 median is about the 60th value (120 ÷ 2)
A1 160 < h ≤ 165

Answer: $160 < h \leq 165$

Question 4 · medium · 3 marks

The cumulative frequency table gives the journey times, $t$ minutes, of 80 commuters. $t \leq 10$ : 6; $t \leq 20$ : 22; $t \leq 30$ : 47; $t \leq 40$ : 68; $t \leq 50$ : 80. The cumulative frequency graph is drawn by joining the plotted points with straight lines. Use it to estimate the median journey time.

Full worked solution Free, no sign-in

M1 median at cumulative frequency 40
M1 20 + (40 − 22)/(47 − 22) × 10
A1 27.2 minutes (accept 27 to 27.5 from a graph)

Answer: 27.2 minutes

Every question here has a full worked solution. See them all →

Question 5 · medium · 2 marks

In a test taken by 150 students, the cumulative frequency for marks $\leq 40$ is 96. What percentage of the students scored more than 40 marks?

Show the answer and mark scheme

M1 150 − 96 = 54
A1 36%

Answer: 36%

Question 6 · hard · 4 marks

The times, $t$ seconds, taken by 78 people to solve a puzzle are grouped: $0 < t \leq 10$ : 12; $10 < t \leq 15$ : 18; $15 < t \leq 20$ : 24; $20 < t \leq 30$ : 16; $30 < t \leq 50$ : 8. (a) Work out the frequency density for each class. (b) On a histogram of these data, the bar for $15 < t \leq 20$ is 9.6 cm tall. How tall is the bar for $30 < t \leq 50$ ?

Show the answer and mark scheme

M1 frequency ÷ class width for at least two classes
A1 1.2, 3.6, 4.8, 1.6, 0.4
M1 9.6 ÷ 4.8 = 2 cm per unit, or 0.4/4.8 × 9.6
A1 0.8 cm

Answer: (a) 1.2, 3.6, 4.8, 1.6, 0.4 (b) 0.8 cm

Question 7 · hard · 3 marks

The lifetimes, $t$ hours, of 200 batteries are summarised: $t \leq 10$ : 14; $t \leq 15$ : 52; $t \leq 20$ : 118; $t \leq 25$ : 173; $t \leq 30$ : 200. The cumulative frequency graph is drawn by joining the plotted points with straight lines. Use it to estimate how many batteries lasted more than 22 hours.

Show the answer and mark scheme

M1 cumulative frequency at 22 hours: 118 + (2/5) × 55 = 140
M1 200 − their 140
A1 60 (accept 57 to 63 from a graph)

Answer: 60

Question 8 · hard · 6 marks

The heights, $h$ cm, of 60 plants are grouped: $0 < h \leq 10$ : 5; $10 < h \leq 20$ : 13; $20 < h \leq 30$ : 20; $30 < h \leq 40$ : 15; $40 < h \leq 50$ : 7. (a) Work out the cumulative frequencies. (b) Joining the plotted points with straight lines, estimate the interquartile range. (c) Estimate the percentage of the plants that are taller than 35 cm.

Show the answer and mark scheme

B1 5, 18, 38, 53, 60
M1 lower quartile at the 15th value and upper quartile at the 45th value
M1 LQ = 10 + (10/13) × 10 = 17.69… or UQ = 30 + (7/15) × 10 = 34.66…
A1 16.9 to 17.1 (16.97…)
M1 cumulative frequency at 35 cm = 38 + 7.5 = 45.5, so 14.5 plants are taller
A1 24.2% (accept 24%)

Answer: (a) 5, 18, 38, 53, 60 (b) 16.97 cm (about 17 cm) (c) 24.2%

Every question and worked example here is original, written for IGCSE Math Revision rather than copied from Pearson papers, and each answer was re-solved independently before publishing.

Revise histograms and cumulative frequency for the exam

Unit 6 practice: Statistics & Probability in the student hub (more questions with mark schemes)
IGCSE Maths revision notes for this topic and the IGCSE Maths formula sheet
Edexcel IGCSE Maths 4MA1 past papers to practise under exam conditions
IGCSE Maths 4MA1 examiner report summary: where students lose marks

Questions students ask

Which Edexcel IGCSE Maths papers test histograms and cumulative frequency?

Either Higher tier paper, 1H or 2H, can test histograms and cumulative frequency: both cover the whole Pearson Edexcel International GCSE Mathematics A (4MA1) specification and both allow a calculator. Show your method, because most marks are for working.

Where can I practise more histograms and cumulative frequency questions?

In the IGCSE student hub: Unit 6 (Statistics & Probability) has more exam-style questions on this topic, each with a mark scheme. Practice starts with a 7-day free trial; the worked examples and questions on this page are free.

Practise histograms and cumulative frequency with your progress saved

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Edexcel IGCSE 4MA1 Histograms and Cumulative Frequency: Exam Questions and Answers

Try it first

What this topic covers

Histograms and Cumulative Frequency worked examples

Worked example 1: Pie chart angle

Solution

Worked example 2: Frequency density

Solution

Worked example 3: Cumulative frequency table

Solution

Worked example 4: Quartiles from a list

Solution

Where students lose marks on histograms and cumulative frequency

Histograms and Cumulative Frequency exam-style questions

Question 1 · easy · 2 marks

Question 2 · easy · 1 mark

Question 3 · medium · 2 marks

Question 4 · medium · 3 marks

Full worked solution Free, no sign-in

Question 5 · medium · 2 marks

Question 6 · hard · 4 marks

Question 7 · hard · 3 marks

Question 8 · hard · 6 marks

Revise histograms and cumulative frequency for the exam

Questions students ask

Practise histograms and cumulative frequency with your progress saved

More Unit 6 topics: Statistics & Probability