Data Booster 4 – Cumulative Frequency & Box Plots
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Cumulative Frequency
Clip 151
Check:
The heights
of 80 plants Clip
were measured and can be seen
© Mathswatch
Cumulative Frequency
© Mathswatch
Clip 151151
Cumulative
Frequency
in the table, below.
The heights of 80 plants were measured and can be seen
a) Complete the cumulative
The heights
of 80below.
plants were measured and can be seen
n the table,
Heighti(cm)
Frequency
frequency table for the plants.
in the table, below.
a) Complete the cumulative
a)frequency
Complete
the
table
forcumulative
the plants.
0 < h < 10
Height (cm) 2
Frequency
Height
(cm)
Frequency
0 < h < 10
2
10 < h < 20
5
0 < h < 10
2
10 < h < 20
5
20 < h < 30
19
10 <20h << h20
5 19
< 30
30 < h < 40
38
20 <30h << h30
19 38
< 40
40 < h < 50
13
< 50
30 <40h << h40
38 13
<
50
h < 60
3
< 60
40 <50h << h50
13 3
50 < h < 60
CF
CF
80
80
Height (cm)
Cumulative
(cm)
0 < hHeight
< 10
2 Frequency
Cumulative
Frequency
0 <Height
h < (cm)
10
2
0 < h < 20
2
0 0< <h h< <20 10
0 < h < 30
0 0< <h h< <30 20
0 < h < 40
0 0< <h h< <40 30
0 < 0h << h 50
0 < h< <50 40
0 < 0h << h 60
< 60
0 < h < 50
3
0 < h < 60
b) Draw
a cumulative
frequency
b) Draw
a cumulative
frequency
graph
for
your
table.
graph for your table.
CF
80
70
70
70
b) Draw a cumulative frequency
graph for your table.
c) Use your graph to find an
c) Use estimate
your graph
for to find an
estimate for
60
(i) the median height of a plant.
60
60
c) median
Use your
graph
find an
(i) the
height
ofto
a plant.
(ii)estimate
the interquartile
range
of the
for
heights
of
the
plants.
(ii) the interquartile range of the
(i) the
height of a plant.
heights
of median
the plants.
50
50
50
Cumulative
Frequency
frequency
table
for the plants.
thegraph
interquartile
range of the
d) Use(ii)
your
to estimate
heights
of had
the aplants.
how many
plants
height
d) Use your
graph
to estimate
that was
greater
than 45cm.
40
how many plants had a height
thatd)
was
greater
Use
your than
graph45cm.
to estimate
40
40
30
how many plants had a height
that was greater than 45cm.
30
20
30
20
10
20
10
0
0
10
20
30
40
50
60
Height (cm)
10
Page 144
0
0
10
0
20
30
Height (cm)
40
50
60
Page 144
© Mathswatch
Box Plots
Clip 152
1) The ages of 20 teachers are listed below.
22, 22, 24, 25, 27, 27, 28, 29, 29, 29, 34, 35, 41, 43, 44, 49, 55, 57, 58, 58
a) On the grid below, draw a boxplot to show the information about the teachers.
10
20
30
40
50
60
70
b) What is the interquartile range of the ages of the teachers?
Learn:
Maths Watch Reference – 151
The cumulative frequency graph has a characteristic “S” shaped curve. Cumulative frequency is
always
on the
You will
be given
2) Aplotted
warehouse
has y60axis.
employees
working
in it.a table of data and asked to find the cumulative
frequency, which is the running total from which you can plot the graph.
The age of the youngest employee is 16 years.
The age of the oldest employee is 55 years.
The median age is 37 years.
The lower quartile age is 29 years.
The upper quartile age is 43 years.
On the grid below, draw a boxplot to show information about the ages of the employees.
10
20
30
40
50
60
Page 145
Practice:
8T) The heights of 64 plants were measured and can be
8T) The heights of 64 plants were measured and can be
seen in this table.
Height (cm)
Frequency
seen in this table.
Height (cm)
Frequency
Grade
Grade B
B
Clip
Clip 151
151
0 <
0 <
10 <
10 <
20 <
20 <
30 <
30 <
40 <
40 <
50 <
50 <
h <
h <
h <
h <
h <
h <
h <
h <
h <
h <
h <
h <
10
10
20
20
30
30
40
40
50
50
60
60
3
3
7
7
23
23
24
24
5
5
2
2
a) Complete this cumulative frequency table.
a) Complete this cumulative frequency table.
Height (cm)
Height (cm)
0 < h < 10
0 < h < 10
0 < h < 20
0 < h < 20
0 < h < 30
0 < h < 30
0 < h < 40
0 < h < 40
0 < h < 50
0 < h < 50
0 < h < 60
0 < h < 60
a)
a)
Cumulative Frequency
Cumulative Frequency
Height (cm)
Height (cm)
0 < h < 10
0 < h < 10
0 < h < 20
0 < h < 20
0 < h < 30
0 < h < 30
0 < h < 40
0 < h < 40
0 < h < 50
0 < h < 50
0 < h < 60
0 < h < 60
3
3
Cumulative Frequency
Cumulative Frequency
3
3
10
10
33
33
57
57
62
62
64
64
b) Draw a cumulative frequency curve for your table.
b) Draw a cumulative frequency curve for your table.b)
c) Use your graph to find an estimate for the
b)80
c) Use your graph to find an estimate for the
80
interquartile range of the heights of the plants.
interquartile range of the heights of the plants. CF
CF
70
80
80
CF
CF
70
70
60
70
60
60
50
50
40
x
x
UQ
UQ
x
x
x
x
1< #7*+7-0 +*(
=&#-%&: #.' #*++#>- &%(). ' %#*?*5(
@h5. <A#
> - B#2C8
<7*'
60#
< .h5#
0 <#
60
b) these
Drawtwo
a cumulative
frequency
curve for your table.
1T) Solve
simultaneous
equations
b)
1T) Solve
these
two
simultaneous
equations
1T) Solve
these
simultaneous
c)2r +Use
to find anequations
estimate for the
3s your
= two
6 graph
80
r = 6 and s = –2
2r +3r
3s
=+2s
63s= =226 range of the heights of the plants.
2r–interquartile
r
=
6
and
= –2s = –2
r = 6s and
CF
3r – 2s3r=–22
2s
=
22
70
80
1S) Solve these two simultaneous equations
CF Solve
1S) Solve
these two
simultaneous
equations
1S)
two simultaneous
equations
60
h + these
3t = –10
h = 2 and t = –4
h70+ 3t
=
–10
h
+
3t
=
–10
h
=
2
= –4t = –4
hand
= 2t and
2h – t = 8
50
UQ
2h – t2h
= 8– t = 8
64
Grade B Clip 142
Grade
B BClip
142142
Grade
Clip
60
x
median
x
a) LQ = 152
a) a)
LQ = LQ
152= 152
b) UQ = 177
20
b) b)
UQ
177= 177
LQ =UQ
30
Grade B
Grade
B B
Grade
Clip
152
ClipClip
152 152
50
her class.
her class.
She put the heights in order.
She putShe
theput
heights
in order.
the heights
in order.
132
144 150 152 160 162 162 167
40
144
150
152
160
162
162
167
132
144172
150177
152181
160182
162182
162 167
167 170
170
167 172
170177
172181
177182
181182
182 182
x
10
x
0
0
30 the lower quartile
a) Find
a) Find
the
lower
quartile
a)
Find
theupper
lowerquartile
quartile
b) Find
the
b) Find
the
upper quartile
Find
upper
c)b) On
thethe
grid,
drawquartile
a boxplot for this data.
c) Onc)the
grid,
draw
a
boxplot
for thisfor
data.
On
the
grid,
draw
a boxplot
this data.
20
10
20
30
40
50
Height (cm)
c)
IR = UQ – LQ
= 34.5 – 23.5
= 11cm
10
0
0
10
20
30
40
50
60
Height (cm)
2S)45 Mary
recorded
the heights,
in centimetres,
the girls in
2S)
the heights,
in centimetres,
of theofgirls
PageMary
2S) recorded
Mary recorded
the heights,
inwww.mathswatch.com
centimetres,
of theingirls in [email protected]
©MathsWatch
LQ
= 154
her class.
a) a)a)
LQ =
154
her class.
LQ
= 154
her class.
b)
UQ
=
She
put
the
heights
in
order.
b) b)
UQ =UQ
181=181
She putShe
theput
heights
in order.
181
the heights
in order.
131
169
a)
b)
c)
x
40
2T) Mary recorded the heights, in centimetres, of the girls in
2T) Mary
recorded
the heights,
in centimetres,
of the girls
2T) her
Mary
recorded
the heights,
in centimetres,
of theingirls in
class.
132
167
x
Page 45
131
142
142
150
158
161
165
169
131142
142142
142150
150158
158161
161165
165169
169
169
169
173
179
183
185
186
188
169
173
179
183
185
186
188
169 169 173 179 183 185 186 188
a)a) Find
the
Find
the
lower
quartile
Find
thelower
lowerquartile
quartile
b)
Find
the
upper
quartile
Find
the
upper
quartile
b) Find the upper quartile
c)c)the
On
the
a boxplot
for
this
On
grid,
draw
adraw
boxplot
for this
data.
On
thegrid,
grid,draw
a boxplot
for
thisdata.
data.
PagePage
41
Model
Questions:www.mathswatch.com
4141 Exam
©MathsWatch
[email protected]
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www.mathswatch.com
[email protected]
Page
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[email protected]
PagePage
41
Page4141
60
Confirm:
8S) The weights of 72 boxes of recycled metals can be
8S) The weights of 72 boxes of recycled metals can be
seen in this table.
Weight (kg)
Frequency
seen in this table.
Weight (kg)
0 < w < 10
0 < w < 10
10 < w < 20
10 < w < 20
20 < w < 30
20 < w < 30
30 < w < 40
30 < w < 40
40 < w < 50
40 < w < 50
50 < w < 60
50 < w < 60
a)
a)
Frequency
8
8
12
12
18
18
24
24
7
7
3
3
Complete this cumulative frequency table.
Complete this cumulative frequency table.
Weight (kg)
Weight (kg)
0
w < 10
0 << w < 10
0 < w < 20
0 < w < 20
0 < w < 30
0 < w < 30
0 < w < 40
0 < w < 40
0 < w < 50
0 < w < 50
0 < w < 60
0 < w < 60
a)
a)
Cumulative Frequency
Cumulative Frequency
Weight (kg)
Weight (kg)
0
w < 10
0 << w < 10
0 < w < 20
0 < w < 20
0 < w < 30
0 < w < 30
0 < w < 40
0 < w < 40
0 < w < 50
0 < w < 50
0 < w < 60
0 < w < 60
8
8
Cumulative Frequency
Cumulative Frequency
8
8
20
20
38
38
62
62
69
69
72
72
b) Draw a cumulative frequency curve for your table.
b) Draw a cumulative frequency curve for your table. b)
c) Use your graph to find an estimate for the
b)80
c) Use your graph to find an estimate for the
80
interquartile range of the weights of the boxes.
interquartile range of the weights of the boxes. CF
CF70
80
80
x
x
70
CF
CF
x
x
60
60
70
70
x
x
UQ
UQ
50
50
60
60
40
40
median
median
x
x
30
30
50
50
20
20
40
40
10
10
0
0 0
0
30
30
c)
c)
20
20
10
10
0
00
0
10
10
20
20
30
30
40
40
50
50
60
60
LQ
LQ
x
x
x
x
10
10
20
20
30
30
40
40
Weight (kg)
Weight (kg)
IR = UQ – LQ
IR = UQ – LQ
= 36.2 – 18.5
= 36.2 – 18.5
= 17.7kg
= 17.7kg
50
50
60
60
0 < w < 60
b)
c)
0 < w < 60
72
Draw a cumulative frequency curve for your table.
b)
Use your graph to find an estimate for the
80
interquartile range of the weights of the boxes.
CF
80
70
CF
x
x
60
© Mathswatch
70
Box Plots
Clip 152
x
UQ
50
1) The ages of 20 teachers are listed below.
60
40
median
x
22, 22, 24, 25, 27, 27, 28, 29, 29, 29, 34, 35, 41, 43, 44,
30 49, 55, 57, 58, 58
50
a) On the grid below, draw a boxplot to show the information about the
teachers.
LQ
20
x
40
10
x
0
0
10
20
30
40
50
Weight (kg)
30
c)
20
10
20
30
40
50
IR = UQ – LQ
= 36.2 – 18.5
60
= 17.7kg
70
10
b) What is the interquartile range of the ages of the teachers?
0
0
10
20
30
40
50
60
Weight (kg)
Page 46
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[email protected]
Page 46
2) A warehouse has 60 employees working in it.
The age of the youngest employee is 16 years.
The age of the oldest employee is 55 years.
The median age is 37 years.
The lower quartile age is 29 years.
The upper quartile age is 43 years.
On the grid below, draw a boxplot to show information about the ages of the employees.
10
20
30
40
50
60
60
1
6
six
5
6
Grade B
not
Clip
154
six – Tree
Data Booster 5
Diagrams
What is the probability of rolling a six on
Check:
one dice and 'not a six' on the other dice?
10S) Two coins are flipped.
a) Complete the tree diagram to show the outcomes.
Coin 1
1
2
6
not
6
5
6
5
5
10
+
=
36
36
36
b)
b)
1
6
not
6
a)
Coin 2
1
2
1
2
H
H
1
2
head
1
2
T
1
2
H
T
1
2
T
tail
b)
What is the probability of flipping a head on
one coin and a tail on the other coin?
Page 47
©MathsWatch
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b)
1
1
+
=
4
4
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2
4
Page 47
Learn:
Maths Watch Reference – 154
Tree diagrams can be easy questions to spot on the exam paper. You will get two marks just for
completely the tree with probabilities so focus on these marks first. Calculating probabilities
involves multiplying fractions, which you will already know how to do. Make sure you read the
questions carefully to check if it is with or without replacement – this can change the second
branches of the tree.
Practice:
© Mathswatch
Tree Diagrams
Clip 154
1) Lucy throws a biased dice twice.
Complete the probability tree diagram to show the outcomes.
Label clearly the branches of the tree diagram.
1st Throw
2
6
2nd Throw
Six
.....
Not
Six
2) A bag contains 10 coloured balls.
7 of the balls are blue and 3 of the balls are green.
Nathan is going to take a ball, replace it, and then take a second ball.
a) Complete the tree diagram.
1st Ball
2nd Ball
.....
Blue
.....
.....
Green
.....
.....
Blue
.....
Green
Blue
Green
b) Work out the probability that Nathan will take two blue balls.
c) Work out the probability that Nathan will take one of each coloured balls.
d) Work out the probability that Nathan will take two balls of the same colour.
Model Exam Questions:
Page 147
200
172
134
101
78
25
a) 168.7 135.7 104.3 68.0
a)
Work out the 3-month moving averages for
this information.
b) Work out the 4-month moving averages for
Confirm:
this information.
b) 151.75 121.25 84.5
10T) Two fair six sided dice are rolled.
a) Complete the tree diagram to show the outcomes.
Red dice
1
6
Blue dice
6
5
6
not
6
Grade B
Clip 154
10S) Two coins are flipped.
a) Complete the tree diagram to show the outcomes.
Coin 1
1
6
6
5
6
not
6
1
6
6
not
6
5
6
5
5
10
+
=
36
36
36
b)
What is the probability of rolling a six on
one dice and 'not a six' on the other dice?
1
2
1
6
six
not
six
b)
a)
a)
Coin 2
1
2
1
2
H
H
1
2
head
1
2
T
1
2
H
T
1
2
T
tail
b)
Page 47
What is the probability of flipping a head on
one coin and a tail on the other coin?
©MathsWatch
www.mathswatch.com
b)
1
1
+
=
4
4
[email protected]
2
4
Page 47