GCSE Cumulative Frequency, Box Plots &
Quartiles
Exercise 1 – Quartiles & Median
1. Determine the Lower Quartile, Median, Upper
Quartile and Interquartile Range of each list of
numbers.
a. 1, 7, 9, 10, 14, 15, 20
b. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15
c. -3, 0, 4, 5, 6, 7, 9, 9, 10, 10.5, 11
d. 8, 10, 14, 15, 20, 22, 25
e.
Age (years)
Freq
18
6
19
7
20
1
21
2
22
3
2. Determine the median class interval and modal
class interval:
Weight of cat (kg)
Freq
2≤𝑤<6
6
6≤𝑤<8
8
8 ≤ 𝑤 < 20
5
20 ≤ 𝑤 < 30
3
30 ≤ 𝑤 < 32
1
3. For the following marks:
3 9 2 10 1 11 4
Determine:
(a) The median
(b) Lower Quartile
(c) Upper quartile
(d) Interquartile range
4. [Edexcel IGCSE Jan2015(R)-3H Q12a]
Here are the marks scored in a test by the girls in
class 8C.
2 8 10 12 15 16 16 17 18 19 20
Work out the interquartile range of the girls’
marks.
5. [Edexcel IGCSE May2015(R)-4H Q5b]
Jordan’s school awards certificates for
outstanding work.
The table shows information about the numbers
of certificates awarded in Jordan’s class during a
term. Work out the interquartile range of the
numbers of certificates awarded.
6. [Edexcel GCSE June 2017 2H Q12a]
The table gives information about the heights of
35 girls. Find the class interval that contains the
median.
7. [June2003-5H Q14a] Mary recorded the heights,
in centimetres, of the girls in her class.
She put the heights in order.
132 144 150 152 160 162 162
167 167 170 172 177 181 182 182
Determine the lower and upper quartiles.
8. [Nov 2017 3H Q1a] The table shows information
about the heights of 80 children. Find the class
interval that contains the median.
9. [Edexcel IGCSE Jan2015(R)-3H Q12b]
Here are the marks scored in a test by the girls in
class 8C.
2 8 10 12 15 16 16
17 18 19 20
(a) Find the interquartile range of this data.
(b) The boys in class 8C did the same test.
The boys’ marks had a range of 19 and an
interquartile range of 11 marks. Gareth says
that the girls’ marks are more spread out than
the boys’ marks. Is Gareth right?
10. [JMC 2017 Q20] The range of a list of integers is
20, and the median is 17. What is the smallest
possible number of integers in the list?
11. [Hamilton 2016 Q5] James chooses five different
positive integers, each at most eight, so that their
mean is equal to their median.
In how many different ways can he do this?
Exercise 2 – Box Plots
1. Above the number line provided, draw a box
plot to represent the following information:
Minimum value:
Maximum value:
Lower quartile:
Median:
Upper quartile:
5
23
10
12
18
2. Use the following weights to draw a box plot
above the provided number line.
3.5kg 4.2kg 5.1kg 5.2kg 5.5kg 8.0kg 11.9kg
3. [Edexcel GCSE Nov2015-2H Q18]
The box plot shows information about the
number of countries competing in each
Winter Olympic Games since 1948.
a. Work out the median.
b. Work out the interquartile range.
4. [Edexcel GCSE
Nov2011-4H
Q16a]
Barry drew an
angle of 60°.
He asked some
children to
estimate the size of the angle he had drawn.
He recorded their estimates. The box plot
gives some information about these
estimates.
a. Write down the median of the
children’s estimates.
b. Find the interquartile range of the
children’s estimates.
5. [Edexcel GCSE June 2017 1H Q19 Edited]
Some students were asked how many times
they each used their mobile phones last week.
The box plots give information about the male
students’ answers and about the female
students’ answers.
6. [Edexcel GCSE Jun2016-1H Q15b]
The box plots give information about the
wages of a group of 16 year old workers and a
group of 18 year old workers.
There are 200 workers who are 16 years old.
Work out an estimate for the number of these
workers whose wages are £130 or more.
7. Edexcel GCSE(9-1) Nov 2017 1H Q12b]
The table shows information about the
heights, in cm, of a group of Year 11 girls.
The box plot below shows information about
the heights, in cm, of a group of Year 7 girls.
Compare the distribution of heights of the
Year 7 girls with the distribution of heights of
the Year 11 girls.
Exercise 3 – Cumulative Frequency Graphs
Test Your Understanding
[Edexcel IGCSE(9-1) SAM 2H Q12]
The table shows information about the amount of
money that 120 people spent in a shop.
(a) Complete the cumulative frequency table.
(b) On the grid, draw a cumulative frequency graph for the table.
(c) Use the graph to find an estimate for the median amount of money spent in the shop by these people.
……………………………..
Question 1: The table shows information about the heights of 40 bushes.
(a)
Height (h cm)
Frequency
170 ≤ h < 175
5
175 ≤ h < 180
18
180 ≤ h < 185
12
185 ≤ h < 190
4
190 ≤ h < 195
1
Complete the cumulative frequency table.
Height
(h cm)
Cumulative Frequency
170 ≤ h < 175
170 ≤ h < 180
170 ≤ h < 185
170 ≤ h < 190
170 ≤ h < 195
(b)
On the grid, draw a cumulative frequency graph for your table.
40
Cumulative
frequency
30
20
10
0
170
(c)
175
180
185
190
195
Height (h cm)
Use the graph to find an estimate for the median height of the bushes.
……………………… cm
Question 2.
40 boys each completed a puzzle.
The cumulative frequency graph below gives information about the times it took them to complete the puzzle.
(a)
Use the graph to find an estimate for the median time
............... seconds
(1)
50
60
40
Cumulative 30
frequency
20
10
O
10
20
30
Time in seconds
40
For the boys
the minimum time to complete the puzzle was 9 seconds
and
the maximum time to complete the puzzle was 57 seconds.
(b)
Use this information and the cumulative frequency graph to draw a box plot showing information
about the boy’s times.
0
10
20
30
Time in seconds
40
50
60
(3)
The box plot below shows information about the times taken by 40 girls to complete the same
puzzle.
0
(c)
10
20
30
Time in seconds
40
50
60
Make two comparisons between the boys’ times and the girls’ times.
...............................................................................................................................................
............................................................................................................................................... (2)
Question 3.
(a)
The table gives information about the ages of 160 employees of an IT company.
Age (A) in years
Frequency
15 < A ≤ 25
44
25 < A ≤ 35
56
35 < A ≤ 45
34
45 < A ≤ 55
19
55 < A ≤ 65
7
Write down the modal class interval.
.......................................
(1)
(b)
Complete the cumulative frequency table.
Age (A) in years
Cumulative Frequency
15 < A ≤ 25
15 < A ≤ 35
15 < A ≤ 45
15 < A ≤ 55
15 < A ≤ 65
(1)
(c)
On the grid on the next page, draw a cumulative frequency graph for your table.
(2)
(d)
Use your graph to find an estimate for
(i)
the median age of the employees,
................................... years
(i)
the interquartile range of the ages of the employees.
................................... years
(3)
Another IT company has 80 employees.
The age of the youngest employee is 24 years.
The age of the oldest employee is 54 years.
The median age is 38 years.
The lower quartile age is 30 years.
The upper quartile age is 44 years.
(e)
On the grid at the bottom of the page, draw a box plot to show information about the ages of the
employees.
(2)
180
Cumulative
frequency
160
140
120
100
80
60
40
20
0
0
10
20
30
40
Age (A) in years
50
60
70
10
20
30
40
Age (A) in years
50
60
70
Diagram for part (e).
0
(Total 9 marks)
Question 4.
A company tested 100 batteries. The table shows information about the number of hours that
the batteries lasted.
(a)
Time (t hours)
Frequency
50 ≤ t < 55
12
55 ≤ t < 60
21
60 ≤ t < 65
36
65 ≤ t < 70
23
70 ≤ t < 75
8
Complete the cumulative frequency table for this information.
(1)
Time
(t hours)
Cumulative
frequency
50 ≤ t < 55
12
50 ≤ t < 60
50 ≤ t < 65
50 ≤ t < 70
50 ≤ t < 75
(b)
On the grid, draw a cumulative frequency graph for your completed table.
(c)
Use your completed graph to find an estimate for the median time.
You must state the units of your answer.
(2)
100
Cumulative
frequency
80
60
40
20
0
50
55
60
65
70
75
Time (t hours)
.....................................
(2)
(Total 5 marks)
Question 5.
[Edexcel GCSE Nov2007-3I Q20ii, Nov2007-5H Q11ii]
Here is the cumulative frequency curve of the weights of 120 girls at Mayfield
Secondary School.
140
120
100
80
Cumulative
frequency
60
40
20
0
40
50
60
70
80
90
100
Weight (kg)
Use the cumulative frequency curve to find an estimate for the
(i)
median weight,
..................................... kg
(ii)
interquartile range of the weights.
..................................... kg
(Total 3 marks)
Question 6.
Here are four cumulative frequency diagrams.
A
B
C
D
Here are four box plots.
P
Q
R
S
For each box plot, write down the letter of the appropriate cumulative
frequency diagram.
P and .....................................
Q and .....................................
R and .....................................
S and .....................................
(Total 2 marks)