Aberdare Community School
Mathematics Department
WJEC GCSE
Higher – Non Calculator
Data
Cumulative frequency
Name:
.............................................................
Set:
.............................................................
Date:
.............................................................
Teacher:
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18
Examiner
only
Arholwr
yn unig
15. The cumulative frequency diagram shows the length of time taken by two different groups of 50
students, A and B to complete a task successfully.
Cumulative frequency
50
Group A
Group B
40
30
20
10
0
0
(a)
5
10
15
20
Time in minutes
State which group of students, A or B, was quicker at successfully completing the task. Give
a reason for your decision.
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
[2]
(b)
State the median of the times taken for successfully completing the task for Group A.
....................................................................................................................................................................................................................................
[1]
(185-04)
16
Examiner
only
Arholwr
yn unig
14. The time taken by each of 50 students to sew a button on a shirt was recorded. A summary of the
results is shown in the cumulative frequency diagram.
Cumulative frequency
50
40
30
20
10
0
(a)
0
5
10
15
20
Time in minutes
Estimate the median of the times taken by these students to sew a button on to a shirt.
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
[1]
(b)
Estimate the interquartile range for the times to sew a button on a shirt.
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
[2]
(185-04)
18
Examiner
only
16. In order to raise money for charity, 120 children took part in a sponsored walk. The times taken by
the 120 children to complete the walk were recorded and the following cumulative frequency
diagram was drawn.
130
120
110
Cumulative frequency
100
90
80
70
60
50
40
30
20
10
0
10
20
Time (minutes)
30
40
Use the cumulative frequency diagram to find estimates for
(a)
the median,
....................................................................................................................................................................................................................................
[1]
(b)
the inter-quartile range.
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
(185-09)
[2]
17
Examiner
only
17. Each of 60 pupils is given a task and the time taken to complete the task is recorded. The results
are summarised in the cumulative frequency polygon below.
Cumulative
frequency
60
50
40
30
20
10
0
0
5
10
15
20
25
30
35
40
Time taken
(minutes)
Use the cumulative frequency polygon to answer the following questions.
(a)
Find an estimate for the interquartile range.
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
[2]
(b)
Giving full details, find an estimate for the number of pupils that take between 20 and 30
minutes to complete the task.
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
[2]
(185-04)
Turn over.
14
12. (a)
Examiner
only
A teacher recorded the time taken by each of 30 pupils in her class to complete a task. The
table below shows a summary of her results.
(i)
Time taken (t minutes)
Frequency
0 ! t X 10
14
10 ! t X 20
10
20 ! t X 30
4
30 ! t X 40
2
On the graph paper below draw a frequency polygon for this data.
Frequency
20
15
10
5
0
(ii)
10
20
30
40
Time, t minutes
Using the table, give the class interval which contains the median time taken.
....................................................................................................................................................................................................................
[3]
(185-13)
15
(b)
Examiner
only
Another teacher recorded the times taken to complete the same task for his class of 32
pupils and he drew the following cumulative frequency diagram.
Cumulative frequency
30
25
20
15
10
5
0
10
20
30
40
Time, t minutes
Use the cumulative frequency diagram to find an estimate for the interquartile range.
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
[2]
(c)
Is it possible for the median times of the two classes to be the same?
Give a reason for your answer.
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
(185-13)
[2]
Turn over.
12
Examiner
only
10. The times taken by one operator at a call centre to answer the telephone were recorded. The
cumulative frequency diagram of the results is shown below.
Cumulative frequency
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
(185-09)
0
5
10
15
20
25
30
Time in
seconds
13
Examiner
only
Use the cumulative frequency diagram to find an estimate for
(a)
the median,
....................................................................................................................................................................................................................................
[1]
(b)
the inter-quartile range,
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
[2]
(c)
the number of calls that took longer than 7 seconds to be answered.
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
[2]
(185-09)
Turn over.
18
Examiner
only
Arholwr
yn unig
14. The cumulative frequency diagram shows the length of time taken by two different groups of 50
students, A and B to complete a task successfully.
Cumulative frequency
50
Group A
Group B
40
30
20
10
0
0
(a)
5
10
15
20
Time in minutes
State which group of students, A or B, was quicker at successfully completing the task. Give
a reason for your decision.
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
[2]
(b)
State the median of the times taken for successfully completing the task for Group A.
....................................................................................................................................................................................................................................
[1]
(185-04)
16
Examiner
only
10. The arm lengths of 120 children were measured to the nearest cm.
The table gives the grouped frequency distribution for the arm lengths of these 120 children.
(a)
Arm length (cm)
41 - 45
46 - 50
51 - 55
56 - 60
61 - 65
Number of children
10
20
22
48
20
On the graph paper below, draw a frequency polygon to show this data.
[2]
Frequency
Arm length,
in cm
(b)
State which class interval contains the median.
........................................................................................................................................................................................................................
[1]
(0185-13)
17
(c)
Examiner
only
Complete the following cumulative frequency table.
Arm length (less than)
40·5
Cumulative frequency
0
45·5
50·5
55·5
60·5
65·5
[1]
(d)
On the graph paper provided, draw a cumulative frequency diagram to show this
information.
[2]
Cumulative
frequency
120
100
80
60
40
20
0
0
35
(e)
40
45
50
55
60
65
Arm length
70 (in cm)
Use your cumulative frequency diagram to find an estimate for the median.
........................................................................................................................................................................................................................
[1]
(0185-13)
Turn over.
16
Examiner
only
10. The frequency diagrams show the time taken by two computer company helplines to deal with
customer problems.
Frequency
Computer Tronics
Frequency
20
20
18
18
16
16
14
14
12
12
10
10
8
8
6
6
4
4
2
2
0
0
2
4
(a)
6
8
10 12 14 16 18 20
Time in minutes
0
0
2
Tech Solutions
4
6
8
10 12 14 16 18 20
Time in minutes
How many calls to Computer Tronics lasted between 4 minutes and 12 minutes?
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
[1]
(b)
One of the helplines won an award for their speedy response to customer questions.
Which company do you think won the award? Give a reason for your answer.
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................
[2]
(185-09)
17
(c)
Examiner
only
Complete the cumulative frequency table for Computer Tronics helpline times.
Time in
minutes, t
tX4
tX8
t X 12
t X 16
Cumulative
frequency
[1]
(d)
Use the graph paper below to draw a cumulative frequency diagram for the Computer
Tronics information.
[3]
Cumulative
Frequency
30
25
20
15
10
5
0
(e)
0
2
4
6
8
10
12
14
16
18
20
Time in
minutes, t
Use your cumulative frequency diagram to find
(i)
an estimate for the median helpline time for Computer Tronics,
....................................................................................................................................................................................................................
(ii)
an estimate for the inter-quartile range of the helpline times for Computer Tronics.
....................................................................................................................................................................................................................
....................................................................................................................................................................................................................
....................................................................................................................................................................................................................
[3]
(185-09)
Turn over.
12
9.
Examiner
only
The heights of 80 men were measured in centimetres.
The table below shows a grouped frequency distribution of the results.
Height, h cm 150 ! h X 160 160 ! h X 170 170 ! h X 180 180 ! h X 190 190 ! h X 200
Frequency
(a)
2
12
24
36
6
On the graph paper below, draw a frequency polygon to show this data.
[2]
Frequency
40
30
20
10
0
(185-13)
150
160
170
180
190
200
Height, h cm
13
(b)
Examiner
only
Complete the following cumulative frequency table.
Height, h cm
h X 150
h X 160
Cumulative
frequency
0
2
h X 170
h X 180
h X 190
h X 200
[1]
(c)
Use the graph paper below to draw a cumulative frequency diagram for these heights.
[4]
(d)
Use your cumulative frequency diagram to find estimates for
(i)
the median,
.........................................................................................................................................................................................................
(ii)
the interquartile range.
.........................................................................................................................................................................................................
.........................................................................................................................................................................................................
[3]
(185-13)
Turn over.
16
9.
(a)
Examiner
only
Write each of the following numbers in standard form.
(i)
5800
.........................................................................................................................................................................................................
(ii)
0·004
.........................................................................................................................................................................................................
(b)
[2]
5· 6 × 10
.
2 × 10− 3
6
Find, in standard form, the value of
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
(c)
Find the value of
( 8 × 103 ) − ( 2 × 103 ).
[2]
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
[1]
10. Customers collect reward points for shopping at a local supermarket.
The following table shows a grouped frequency distribution of the number of points collected
by 60 different customers.
Number of points collected
Number of customers
(a)
1 - 20
21 - 40
41 - 60
61 - 80
4
12
34
10
X60
X80
Complete the following cumulative frequency table.
Number of points collected
Cumulative frequency
X20
X40
4
[1]
(185-09)
17
Examiner
only
(b)
On the graph paper below, draw a cumulative frequency diagram to show this
information.
[3]
Cumulative
frequency
60
55
50
45
40
35
30
25
20
15
10
5
0
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
Number of points
(c)
Use your cumulative frequency diagram to find an estimate for
(i)
the median,
.........................................................................................................................................................................................................
(ii)
the interquartile range.
.........................................................................................................................................................................................................
.........................................................................................................................................................................................................
.........................................................................................................................................................................................................
[3]
(185-09)
Turn over.
18
Examiner
only
13. The table gives the grouped frequency distribution for the masses, each measured to the nearest
kilogram, of 60 dogs.
Mass, to the
nearest kg
16-20
21-25
26-30
31-35
36-40
Number of
dogs
8
26
10
14
2
(a) A dog with a mass of 30·52 kg was included in the grouped frequency distribution.
In which group would you expect this mass to be recorded?
[1]
(b)
Complete the following cumulative frequency table.
Mass, kg
<15·5
<20·5
<25·5
<30·5
<35·5
<40·5
Cumulative
0
8
frequency
[1]
(c)
On the graph paper below, draw a cumulative frequency diagram to show this
information.
[2]
Cumulative frequency
60
50
40
30
20
10
0
0
5
10
15
20
25
(0185-13)
30
35
40
45
Mass, in kg
19
Examiner
only
(d)
Use your cumulative frequency diagram to find
(i) the median,
(ii) the interquartile range.
[3]
(0185-13)
Turn over.
16
Examiner
only
10. The heart rate of each of 60 people at a gym was recorded.
The following table shows a grouped frequency distribution of the results.
Heart rate, r
(beats per
80 X r ! 90
minute)
Number of
people
90 X r ! 100 100 X r ! 110 110 X r ! 120 120 X r ! 130 130 X r ! 140
22
14
4
6
10
4
(a)
Complete the following cumulative frequency table.
Heart rate (less than)
80
90
100
Cumulative
frequency
0
4
18
110
120
130
140
[1]
(b)
On the graph paper below, draw a cumulative frequency diagram to show this
information.
[3]
Cumulative frequency
60
50
40
30
20
10
0
(0185-09)
80
90
100
110
120
130
140
Heart
rate
17
Examiner
only
(c)
Use your cumulative frequency diagram to find estimates for
(i) the median,
.........................................................................................................................................................................................................
.........................................................................................................................................................................................................
(ii) the interquartile range.
.........................................................................................................................................................................................................
.........................................................................................................................................................................................................
.........................................................................................................................................................................................................
[3]
(0185-09)
Turn over.
13
(b)
The total mass of tomatoes produced by each of 60 plants in a different greenhouse was
measured. The following cumulative frequency graph illustrates the results.
Cumulative frequency
60
50
40
30
20
10
0
(i)
0
5
10
15
20
25
Mass, in kg
Complete the grouped frequency table of the total mass of tomatoes on each plant.
Mass, x kg
0! xX 5
5 ! x X 10
Frequency
0
7
10 ! x X 15
15 ! x X 20 20 ! x X 25
[2]
(ii) Use the cumulative frequency diagram shown above to find estimates for each of
the following.
The median.
The inter-quartile range.
[3]
© WJEC CBAC Ltd.
(4370-05)
Turn over.
Examiner
only
16
12. The frequency diagrams show the lengths of telephone calls taken by two online shopping
companies one day in November.
Frequency
Cat Boots UK
Frequency
200
200
150
150
100
100
50
50
0
0
5
10
15
0
20
Time, in
minutes
0
Shoes 4 All
5
10
15
20
Time, in
minutes
(a) How many calls to Cat Boots UK lasted between 5 minutes and 15 minutes?
[1]
(b)
Which company had longer calls on average on this day?
Give a reason for your answer.
[1]
(c)
Complete the cumulative frequency table for Cat Boots UK times.
Time t, in
minutes
tX5
t X 10
t X 15
t X 20
Cumulative
frequency
[2]
© WJEC CBAC Ltd.
(4370-05)
Examiner
only
17
(d) Use the graph paper below to draw a cumulative frequency diagram for the Cat Boots
UK information.
Cumulative frequency
400
300
200
100
0
0
5
10
15
20
Time t, in minutes
[2]
(e) Use your cumulative frequency diagram to find
(i) an estimate for the median time of calls to Cat Boots UK,
[1]
(ii) an estimate for the inter-quartile range of the times for calls to Cat Boots UK.
[2]
© WJEC CBAC Ltd.
(4370-05)
Turn over.
Examiner
only
18
10. Help4U and First Call are two different telephone help lines.
Information was recorded for the first 800 callers of the day for each helpline.
The cumulative frequency diagram illustrates the time it took for these callers to get through to
each of these helplines.
Cumulative frequency
800
700
600
500
First Call
400
300
200
Help4U
100
0
0
20
40
60
Time, t seconds
The solid line represents the data for the telephone helpline Help4U.
The dotted line represents the data for the telephone helpline First Call.
(a) How many of the 800 callers to First Call took less than 15 seconds to get through? [1]
© WJEC CBAC Ltd.
(4370-05)
Examiner
only
19
(b)
Help4U were set a target to answer 40% of all calls in less than 25 seconds.
They actually met the target with these 800 calls.
By how many seconds did they beat their target?
Examiner
only
[3]
(c)
Which of the two companies had the lower median time and by how many seconds? [2]
(d)
Calculate the interquartile range of the times for each company.
[3]
First Call
Help4U
(e)
One of these companies was given an award for the speed of answering the telephone.
Which company do you think won the award?
You must give a reason for your answer.
[1]
© WJEC CBAC Ltd.
(4370-05)
Turn over.