NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
Basic
9
Measures of Dispersion
Introduction to Measures of Dispersion
Range and Inter-quartile Range
Worksheet 9.1
(Refer to Book 5B Ch9 p. 9.6 – 9.19)
Name: _________________________
Class: ___________
Key Points
1. Range
Ungrouped data:
Grouped data:
range = largest datum – smallest datum
range = upper class boundary of the last class interval
– lower class boundary of the first class interval
2. Inter-quartile range
Ungrouped data:
A set of data arranged in ascending order
inter-quartile range = upper quartile (Q3) – lower quartile (Q1)
Grouped data:
The inter-quartile range of grouped data is also defined as Q3 − Q1 . The quartiles ( Q1 , Q2 and Q3 )
of grouped data can be read from the corresponding cumulative frequency polygon or curve.
Complete the following tables.
1.
(1 – 2)
Data
Minimum
Maximum
(a)
1, 1, 1, 3, 4, 5, 6
1
6
(b)
4, 5, 5, 7, 8, 9, 9, 9
(c)
6, 17, 41, 53, 55, 82
(d)
‒ 1.2, ‒ 0.3, 0, 0.3, 0.9, 1
2.
Data
Q1
Q2
Q3
(a)
3, 5, 7, 8, 10, 15
5
7.5
10
(b)
1, 3, 6, 11, 17, 25
8.5
(c)
4, 6, 7, 12, 18, 19, 22
12
(d)
0, 2, 10, 14, 22, 29, 31, 42
40
Range
(Maximum – minimum)
IQR
(Q3 – Q1)
NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
9
Measures of Dispersion
3. The heights of 10 students in a class are given below.
106 cm
108 cm
110 cm
111 cm
112 cm
112 cm
114 cm
116 cm
120 cm
121 cm
Find the range of the students’ heights.
Solution
Range of the students’ heights
= [(
)–(
)] cm
=
4. The following table shows the amounts in the savings accounts of 100 teenagers.
Amount ($)
1 – 2000
2001 – 4000
4001 – 6000
6001 – 8000
Frequency
18
25
35
22
Find the range of the amounts of the 100 teenagers.
Solution
Upper class boundary of the last class interval = $(
Lower class boundary of the first class interval = $(
∴ Range of the amounts of the 100 teenagers =
5. The histogram shows the distribution of the weights of a
group of students. Find the range of the weights of the
students.
Solution
41
)
)
NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
9
Measures of Dispersion
6. The heights jumped by 10 students in a high jump competition are recorded below:
0.4 m, 0.6 m, 1 m, 1 m, 1.1 m, 1.2 m, 1.2 m, 1.3 m, 1.4 m, 1.6 m
(a) Find the lower quartile and the upper quartile of the heights jumped.
(b) Find the inter-quartile range of the heights jumped.
Solution
(a)
(b)
7. The following are the IQ scores of 12 students in a class.
120
112
141
132
100
88
90
78
105
115
Find the range and the inter-quartile range of the IQ scores of the students.
Solution
42
120
108
NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
9
8. The cumulative frequency curve on the right shows
the distribution of the average daily working hours
of 40 employees of a company.
(a) Find the lower quartile and the upper quartile
of the working hours of the employees.
(b) Find the range and the inter-quartile range of
the working hours of the employees.
Solution
(a)
(b)
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Measures of Dispersion
NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
Enhanced
9
Measures of Dispersion
Introduction to Measures of Dispersion
Range and Inter-quartile Range
Worksheet 9.1
(Refer to Book 5B Ch9 p. 9.6 – 9.19)
Name: _________________________
Class: ___________
Key Points
1. Range
Ungrouped data:
Grouped data:
range = largest datum – smallest datum
range = upper class boundary of the last class interval
– lower class boundary of the first class interval
2. Inter-quartile range
Ungrouped data:
A set of data arranged in ascending order
inter-quartile range = upper quartile (Q3) – lower quartile (Q1)
Grouped data:
The inter-quartile range of grouped data is also defined as Q3 − Q1 . The quartiles ( Q1 , Q2 and Q3 )
of grouped data can be read from the corresponding cumulative frequency polygon or curve.
1. Complete the following table.
Data
(a)
Range
Q1
Q3
IQR
5, 4, 9, 14, 7, 2, 0
− 1, 5 ,
(b)
13
2
, 1, − 4 ,
5
2
(c)
14 kg, 10 kg, 21 kg, 23 kg, 31 kg, 18 kg, 30 kg
(d)
a + 13, a, a + 8, a + 24, a + 26, a + 1, a – 10
2. The following table shows the numbers of family members in a household of a group of people.
Number of family members
2
3
4
5
6
Frequency
8
18
23
15
6
(a) Find the median, the lower quartile and the upper quartile of the above data.
(b) Find the inter-quartile range of the above data.
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NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
9
Measures of Dispersion
Solution
(a)
(b)
3. The following stem-and-leaf diagram shows the distribution of the weights of 20 fruits sold in a
supermarket.
Weights of 20 fruits sold
Stem (10 g)
Leaf (1 g)
10
2
5
9
11
4
5
7
12
0
6
8
8
13
3
4
5
5
14
1
3
7
8
5
9
(a) Find the range of the weights of the fruits sold.
(b) (i) Find the lower quartile and the upper quartile of the weights of the fruits sold.
(ii) Hence, find the inter-quartile range of the weights of the fruits sold.
Solution
(a)
(b) (i)
(ii)
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NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
9
Measures of Dispersion
4. The following back-to-back stem-and-leaf diagram shows the distribution of marks of students
of classes A and B in a Mathematics examination.
Marks of students of classes A and B in a Mathematics examination
Class A
Class B
Stem (10
Leaf (1 mark)
Leaf (1 mark)
marks)
8 4 3 1 0
3
7 0 0
4
0 1 2 7
6 3
5
1 4 4 5 7 7
5 4 2 1 0
6
0 2 3
8 6 0 0
7
4 5 8 8 9
6 4 1
8
2 4 6 7 8 9 9
9 7 7 3 2 0
9
1 3 3 4 4
0 0
10
1
(a) Find the range and the inter-quartile range of the marks of each class.
(b) Hence, which class of students has a greater dispersion in marks?
Solution
(a)
(b)
5. Consider the following data sets.
X = {2.3, 4.1, 0.5, –0.6, 1.2, 3.3}
Y = {–3, 6, 1.5, 7, –2, 6.3, 0.1}
Z = {2, 1.5, 4.9, 8, 5.5, 2.1, 0, 1.1}
(a) Find the inter-quartile range of each data set.
(b) Considering the inter-quartile ranges found in (a), determine which data set has the greatest
dispersion. Explain your answer.
Solution
(a)
(b)
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NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
9
Measures of Dispersion
6. The cumulative frequency polygons on the right
show the waiting time of buses A and B recorded in the
past 30 days.
(a) Find the median, the lower quartile and the upper
quartile of the waiting time of each bus.
(b) Find the range and the inter-quartile range of the
waiting time of each bus.
(c) By comparing the range and the inter-quartile
ranges of the waiting time of the two buses, which
bus has a smaller variation in waiting time?
Solution
(a)
(b)
(c)
7. The following table shows the speeds of 80 vehicles passing through a checkpoint of a highway.
Speed of vehicles (km/h)
Frequency
1 – 40
41 – 80
81 – 120
121 – 160
161 – 200
4
16
32
18
10
(a) Construct a cumulative frequency table for the above data.
(b) Draw a cumulative frequency polygon from the table in (a).
(c) Find the median and the inter-quartile range of the speeds of vehicles.
Solution
(a)
Speed of vehicles less than (km/h)
0.5
40.5
Cumulative frequency
(b)
(c)
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NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
9
Measures of Dispersion
Basic
Worksheet 9.2
Box-and-Whisker Diagram
(Refer to Book 5B Ch9 p. 9.20 – 9.30)
Name: _________________________
Class: ___________
Key Points
A box-and-whisker diagram shows the lower quartile, the upper quartile, the median, the maximum and
the minimum values of a data set.
1. In each of the following box-and-whisker diagrams, find the lower quartile Q1 , the median Q2 ,
the upper quartile Q3 , the maximum and the minimum values of the data set.
(a)
(b)
Minimum value = (
)
Minimum value = (
)
Maximum value = (
)
Maximum value = (
)
Lower quartile Q1 = (
Median Q2 = (
Lower quartile Q1 = (
)
Median Q2 = (
)
Upper quartile Q3 = (
Upper quartile Q3 = (
)
)
)
)
2. The table below shows the information on two data sets. Draw a box-and-whisker diagram for
each data set.
Minimum
Lower quartile
Median
Upper quartile
Maximum
(a)
0
1
6
8
10
(b)
3
5
7
8
12
Solution
(a)
(b)
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NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
9
Measures of Dispersion
Draw a box-and-whisker diagram for each of the following data set. (3 – 4)
3. {1, 4, 12, 13, 18, 20, 21}
4. {4, 5, 7, 9, 15, 22, 26, 30}
Solution
Solution
∵ Minimum value =
Maximum value =
Median =
Lower quartile =
Upper quartile =
∴ The required box-and-whisker diagram is:
5. In each of the following pairs of box-and-whisker diagrams, compare the medians, the ranges
and the inter-quartile ranges of data sets A and B by circling the correct answers.
(a)
Median of A ( > / = / < ) median of B
Range of A ( > / = / < ) range of B
IQR of A ( > / = / < ) IQR of B
(b)
Median of A ( > / = / < ) median of B
Range of A ( > / = / < ) range of B
IQR of A ( > / = / < ) IQR of B
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NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
9
Measures of Dispersion
6. The box-and-whisker diagram shows the distribution of the time taken (in s) for
a group of 80 athletes to run 100 m.
(a) Find the range and the inter-quartile range of the time taken.
(b) What is the percentage of athletes who can run 100 m within 14 s?
(c) According to the diagram, can we find an athlete, in the group, who takes
12 s to run 100 m? Explain your answer.
Solution
(a)
(b)
(c)
7. The following box-and-whisker diagram shows the marks of 40 students in an English test.
(a) What percentage of students have marks below 70?
(b) If the passing mark is 50, how many students pass the English test?
(c) State which part of the data, the lower half or the upper half, is more dispersed. Explain
your answer.
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NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
9
Measures of Dispersion
Solution
(a)
(b)
(c) From the diagram, the median is closer to the (minimum / maximum) than the
(minimum / maximum). Therefore, the distribution of the data in the
(lower half / upper half) is more dispersed.
8. The cumulative frequency polygon on the right
shows the distribution of the numbers of sit-ups
performed by 60 students in a fitness test.
(a) Find the lower quartile, the median and the
upper quartile of the numbers of sit-ups
performed.
(b) Draw a box-and-whisker diagram to present
the information in (a).
Solution
(a)
(b)
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NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
9
Measures of Dispersion
Enhanced
Worksheet 9.2
Box-and-Whisker Diagram
(Refer to Book 5B Ch9 p. 9.20 – 9.30)
Name: _________________________
Class: ___________
Key Points
A box-and-whisker diagram shows the lower quartile, the upper quartile, the median, the maximum and
the minimum values of a data set.
1. The heights of the commercial buildings in city A are measured and are shown in the following
box-and-whisker diagram.
(a) Find the range and the inter-quartile range of the heights of the buildings in city A.
(b) What is the percentage of buildings with height above 60 m?
Solution
(a)
(b)
2. The stem-and-leaf diagram below shows the distribution of the weights of 20 primary students.
Weights of 20 primary students
Stem (10 kg)
Leaf (1 kg)
3
9
4
0
1
1
2
4
7
9
5
0
2
4
6
7
8
8
6
2
2
3
4
5
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NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
9
Measures of Dispersion
(a) Find the lower quartile, the median, the upper quartile, the maximum and the minimum
values of the weights of the primary students.
(b) Draw a box-and-whisker diagram to represent the above data.
(c) State which part of the data, the lower half or the upper half, is more dispersed. Explain
briefly.
Solution
(a)
(b)
(c)
3. The following box-and-whisker diagrams show the numbers of handbags sold each day in last
month in two accessory shops.
(a) There is no left whisker in the box-and-whisker diagram for shop B. What does this mean?
(b) Which shop’s sale performance is more consistent?
Solution
(a)
(b)
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NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
9
Measures of Dispersion
4. The table below shows the distribution of the daily average temperatures of two cities.
Day
Temperature
(ºC) of city A
Temperature
(ºC) of city B
1
2
3
4
5
6
7
15
14
16
22
20
23
25
18
19
15
24
21
22
23
(a) Present the information by box-and-whisker diagrams on the same number line.
(b) Which city has a higher median temperature?
(c) Based on the box-and-whisker diagrams, can we say that the daily average temperature in
city B is more stable than that of city A? Explain your answer.
Solution
(a)
(b)
(c)
5. (a) The box-and-whisker diagram below shows distribution of Julia’s weights in the past 12
months.
Find the median, the range and the inter-quartile range of Julia’s weights.
(b) The following table shows the distribution of Christy’s weights in the past 12 months.
Month
Jan
Feb
Weight (kg)
52
50
Mar Apr May Jun
50
52
53
54
Jul
56
Aug Sep
57
54
Oct
53
Nov Dec
53
52
(i) On the diagram in (a), draw a box-and-whisker diagram to present Christy’s weights.
(ii) Based on the box-and-whisker diagrams in (a) and (b)(i), can we say that Julia is
heavier than Christy in at least 3 months? Explain your answer.
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NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
9
Measures of Dispersion
Solution
(a)
(b) (ii)
6. The following box-and-whisker diagrams show the distribution of the monthly transportation
expenses of students in S6A and S6B.
(a) (i) Which class has a larger range of monthly transportation expenses?
(ii) Which class has a smaller inter-quartile range of monthly transportation expenses?
(b) It is known that the numbers of students in the two classes are the same. If the monthly
transportation expenses of the two classes of students are considered together by
combining two classes as a group, what percentage of students in the group have monthly
transportation expenses higher than or equal to $40?
Solution
(a) (i)
(ii)
(b)
55
NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
Basic
9
Measures of Dispersion
Standard Deviation
Comparing Dispersions Using Appropriate
Measures
(Refer to Book 5B Ch9 p. 9.31 – 9.45)
Worksheet 9.3
Name: _________________________
Class: ___________
Key Points
Standard deviation
Standard deviation  =
( x1 − x ) 2 + ( x2 − x ) 2 + ... + ( xN − x ) 2
for ungrouped data x1, x2 , ... , xN with
N
mean x .
Standard deviation  =
( x1 − x )2 f1 + ( x2 − x )2 f 2 + ... + ( xN − x )2 f N
for grouped data with class marks
f1 + f 2 + ... + f N
x1, x2 , ... , xN , and corresponding frequencies f1 , f 2 , ... , f N , and x is the mean.
Note: The variance  2 of a data set is equal to the square of its standard deviation.
(In this worksheet, unless otherwise specified, numerical answers should be either exact or correct
to 3 significant figures.)
1. Complete the following table.
Data
(a)
2, 3, 4, 7
(b)
–4, 0, 3, 9
(c)
0, 4, 6, 8, 14
(d)
1, 3, 4, 5, 8
Standard deviation σ
Mean x
4
(2 − 4) 2 + (3 − 4) 2 + (4 − 4) 2 + (7 − 4) 2
=(
4
)
2. Find the standard deviations of the following sets of data by using a calculator.
(a) {11, 11, 12, 14, 15, 15, 16}
(b) {20, 20, 22, 22, 24, 24, 26, 26}
Standard deviation = _______________
Standard deviation = _______________
(c) {42 kg, 54 kg, 48 kg, 53 kg, 45 kg }
(d) {22 m, 27 m, 25 m, 24 m, 23 m, 28 m}
Standard deviation = _______________
Standard deviation = _______________
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NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
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Measures of Dispersion
3. Choose the suitable measure(s) of dispersion among the range, the inter-quartile range and the
standard deviation for each of the following situations.
(a) A researcher wants to study the dispersion of lifetime of batteries.
________________
(b) A technician wants to monitor the consistency of the weights of cans. ________________
(c) A weather forecaster wants to reports the daily variation of
air quality health index.
________________
4. The following table shows the ages of children in a tutorial centre.
Age
11
12
13
14
15
Frequency
4
7
18
10
1
Find the mean, the variance and the standard deviation of the ages of the children.
Solution
5. The following table shows the distribution of the volumes of 50 bottles of shampoo from
different brands.
Volume (mL)
1 – 200
201 – 400
401 – 600
601 – 800
801 – 1000
4
7
11
22
6
Class mark (mL)
Frequency
(a) Complete the above table.
(b) Find the mean and the standard deviation of the volumes of the shampoo.
Solution
(b)
57
NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
9
6. The histogram on the right shows the age distribution of the
volunteers in an event.
(a) Find the mean age of the volunteers.
(b) Find the standard deviation of the ages of the volunteers.
Solution
(a)
(b)
7. It is given that the mean of x, 2x, 5, 7, 8 is 7.
(a) Find the value of x.
(b) Hence, find the variance and the standard deviation of the data set.
Solution
(a)
(b)
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Measures of Dispersion
NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
Enhanced
Worksheet 9.3
9
Measures of Dispersion
Standard Deviation
Comparing Dispersions Using Appropriate
Measures
(Refer to Book 5B Ch9 p. 9.31 – 9.45)
Name: _________________________
Class: ___________
Key Points
Standard deviation
Standard deviation  =
( x1 − x ) 2 + ( x2 − x ) 2 + ... + ( xN − x ) 2
for ungrouped data x1, x2 , ... , xN with
N
mean x .
Standard deviation  =
( x1 − x )2 f1 + ( x2 − x )2 f 2 + ... + ( xN − x )2 f N
for grouped data with class marks
f1 + f 2 + ... + f N
x1, x2 , ... , xN , and corresponding frequencies f1 , f 2 , ... , f N , and x is the mean.
Note: The variance  2 of a data set is equal to the square of its standard deviation.
(In this worksheet, unless otherwise specified, numerical answers should be either exact or correct
to 3 significant figures.)
1.
The scores obtained by 7 participants in a game show are as follows:
0, 2, 7, 9, 11, 18, 23
(a)
Find the mean score obtained by the participants.
(b)
Find the variance and the standard deviation of the scores obtained by the participants.
Solution
(a)
(b)
2. The performances of two long jump athletes in 6 trials are recorded as follows:
Athlete A: 7.5 m, 8 m, 7.8 m, 8.1 m, 7.8 m, 7.4 m
Athlete B: 8 m, 8.1 m, 8.5 m, 7 m, 7 m, 7.4 m
(a) Find the standard deviation of the records of each athlete.
(b) Comparing the standard deviations of the records, which athlete has a more stable
performance? Explain your answer.
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NSS Mathematics in Action (2nd Edition)
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9
Measures of Dispersion
Solution
(a)
(b)
3. The following table shows the age distribution of the members in debate club.
Age
11
12
13
14
15
16
Frequency
2
3
7
5
4
1
(a) Find the mean and the standard deviation of the ages of the members.
(b) It is known that the standard deviation of the ages of the members in poem club is 1.54.
Which club, debate club or poem club, whose members’ ages are less dispersed? Explain
your answer.
Solution
(a)
(b)
4. The following table shows the water expenses of 150 families in March.
Expense ($)
0 – 99
100 – 199
200 – 299
300 – 399
400 – 499
Frequency
2
27
86
29
6
(a) Find the mean and the standard deviation of the water expenses.
(b) The standard deviation of the water expenses of those families in April is $76.6. In which
month, March or April, the variation of the water expenses of these families is smaller?
Solution
(a)
(b)
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NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
9
Measures of Dispersion
5. An engineer compares the volumes of two brands of lemon tea. The following back-to-back
stem-and-leaf diagram shows the distribution of the volumes of 20 boxes of lemon tea, 10 boxes
of brand X and 10 boxes of brand Y.
Volumes of 20 boxes of lemon tea of brand X and brand Y
Brand X
Brand Y
Leaf (0.1 ml)
Stem (1
Leaf (0.1 ml)
ml)
3
5
371
372
373
374
375
376
2 1 0
0
8 5 3
9
9
3
2 5 6 6 7
0 1 4
(a) Which measure of dispersion would you suggest to compare the volume variation of lemon
tea of two brands? Explain your answer.
(b) Using the measure of dispersion you suggest in (a), determine which brand has a greater
variation in volume.
Solution
(a)
(b)
6. Jam and Michael decide to join an archery competition. In a practice, they take turns to shoot an
arrow at the target 10 times. Their scores in 10 trials are recorded as follows:
Jam:
3 6 4 5 7 8 x
Michael: 2 3 5 7
2
3 6
8 5 2 10 9 4
The mean scores of Jam and Michael are 5.4 and y respectively.
(a) Find the values of x and y.
(b) Find the variance of the scores of Jam and Michael respectively.
(c) If a participant with mean score more than 5 can join the archery competition, who should
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9
Measures of Dispersion
be chosen to participate in the competition? Explain your answer.
Solution
(a)
(b)
(c)
7. Two farms A and B provide eggs to the market. 6 eggs from each farm are selected for quality
check. The weights of the selected eggs are shown as follows:
Eggs from farm A: 13.3 g, 9.8 g, 7.9 g, 10.8 g, 8.8 g, 12.4 g
Eggs from farm B: 9.3 g, 10.8 g, 8.9 g, 11.4 g, 8.1 g, 10.0 g
(a) Find the mean weight of eggs from farm A and farm B respectively.
(b) Find the standard deviation of the weights of eggs from each farm.
(c) Mary claims that if the selected eggs are combined, the standard deviation of the weights of
all eggs will be greater than that from either farm A or farm B. Do you agree? Explain your
answer.
Solution
(a)
(b)
(c)
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5B Section Worksheets
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Measures of Dispersion
NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
9
Measures of Dispersion
Basic
Worksheet 9.4
NF
Applications of Standard Deviation
(Refer to Book 5B Ch9 p. 9.45 – 9.59)
Name: _________________________
Class: ___________
Key Points
1. Standard score
For a set of data with mean x and standard deviation , the standard score z of a given datum x is
defined as z =
x−x

.
It measures how far away a datum lies from the mean in units of the standard deviation.
2. Normal distribution
The frequency curve of a normal distribution is represented
by a normal curve.
The characteristics of a normal curve include the following.
I. It is bell-shaped.
II. It has reflectional symmetry about x = x .
III. The mean, the median and the mode are all equal to x .
(In this worksheet, unless otherwise specified, assume that 68%, 95% and 99.7% of the data of a
normal distribution lie within one, two and three standard deviations from the mean respectively.)
1. Refer to the following table. Find the standard scores for Olivia’s results in 3 tests.
Mean of the class
(x)
Standard deviation
of the class
(σ )
48
2
(a)
Test 1
Marks of
Olivia
(x)
50
(b)
Test 2
45
46
0.5
(c)
Test 3
35
38
3
Standard score
of Olivia
(z)
2. Complete the following table for Wilson’s test results in 4 subjects.
Subject
Marks of
Wilson
(x)
Mean of
the class
(x)
Standard
deviation of the
class
(σ )
(a)
Chinese
70
62
4
(b)
English
86
82
(c)
Mathematics
40
1
8
64
Standard score
of Wilson
(z)
–2.5
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5B Section Worksheets
9
Measures of Dispersion
(d)
Liberal Studies
51
6
–1.5
3. In each of the following normal curves, find the percentage of data lying in the shaded regions.
(a)
Percentage (b)
of data
Percentage
of data
(c)
Percentage (d)
of data
Percentage
of data
4. The mean and the standard deviation of the Air Quality Health Index in Hong Kong at a certain
day are 5 and 1.25 respectively. The Air Quality Health Index in Kwun Tong is 20% higher than
the mean. Find the standard score of the index in Kwun Tong.
Solution
5. According to the height record of a class, two students Kathy and Mimi have standard scores of
0.5 and 0.2 respectively.
(a) Is Kathy’s height above or below the mean height of her class? Explain your answer.
(b) Determine whether Kathy or Mimi is taller. Explain your answer.
Solution
(a)
(b)
6. In a running competition, the mean finishing time of all athletes is 38 s.
(a) The finishing time of Ryan is 46 s. If his standard score is 1.6, find the standard deviation
of the finishing time in the competition.
(b) If Jeff has a standard score of 0.5, what is his finishing time?
(c) Peter claims that athlete who has higher standard score performs better. Do you agree?
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Measures of Dispersion
Explain your answer.
Solution
(a)
(b)
(c)
7. The monthly income of the families in a certain district is normally distributed with a mean of
$16 000 and a standard deviation of $500. Find the percentage of families with monthly income
(a) between $15 000 and $17 000,
(b) between $16 000 and $16 500,
(c) less than $15 500.
Solution
(a) ∵ $15 000 = $(16 000 – 2 × 500) = x − (
$17 000 = $(16 000 +
)=
∴ The required percentage =
(b)
(c)
66
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5B Section Worksheets
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Measures of Dispersion
NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
9
Measures of Dispersion
Enhanced
Worksheet 9.4
Applications of Standard Deviation
NF
(Refer to Book 5B Ch9 p. 9.45 – 9.59)
Name: _________________________
Class: ___________
Key Points
1. Standard score
For a set of data with mean x and standard deviation , the standard score z of a given datum x is
defined as z =
x−x

.
It measures how far away a datum lies from the mean in units of the standard deviation.
2. Normal distribution
The frequency curve of a normal distribution is represented
by a normal curve.
The characteristics of a normal curve include the following.
I. It is bell-shaped.
II. It has reflectional symmetry about x = x .
III. The mean, the median and the mode are all equal to x .
(In this worksheet, unless otherwise specified, assume that 68%, 95% and 99.7% of the data of a
normal distribution lie within one, two and three standard deviations from the mean respectively.
Moreover, numerical answers should be either exact or correct to 3 significant figures.)
1. In an aptitude test, David obtained a standard score of 0.75. If the mean mark and the standard
deviation of all the candidates are 54 and 12 respectively, how many marks did David obtained?
Solution
2. Mr Mak and Miss Chan are the participants of a 110 m hurdle final. The following table shows
the means and the standard deviation of the finishing times (in s) obtained by the men and the
women.
Men
Women
Mean
17.93
19.45
Standard deviation
1.10
1.75
Mr Mak’s finishing time is 19.91 s while Miss Chan’s finishing time is 22.25 s. Who performs
relatively better among the participants in their own group?
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Measures of Dispersion
Solution
3. The manager of a company plans to employ a trainee. The following table shows the marks of
Adrian in three areas, as well as the means and the standard deviations of all the applicants.
Adrian’s mark
Mean of all
applicants
Standard deviation of
all applicants
Interview
85
80
10
Typing Test
86
82
6
Written Test
83
78
5
(a) Find the standard scores of Adrian in each of the three areas.
(b) Compared to the other applicants, in which area does Adrian perform
(i) the best?
(ii) the worst?
Solution
(a)
(b) (i)
(ii)
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Measures of Dispersion
4. The mean time and the standard deviation for 50 students to finish a fitness test are 20 min and
3.5 min respectively.
(a) Michael’s standard score is 1.4. If the passing time is 23.5 min, can Michael pass the test?
(b) Assume that the finishing times of the students are normally distributed. Find the number
of students passing the fitness test.
Solution
(a)
(b)
5. The box-and-whisker diagram below shows the distribution of score for a group of children in a
game. Peter gets the median marks while Jerry gets 55 marks in the game. The standard scores
of Peter and Jerry are − 1 and 0.5 respectively.
(a) Find the mean and the standard deviation of the distribution.
(b) Peter claims that at least half of the group whose standard score are negative. Do you agree?
Explain your answer.
Solution
(a)
(b)
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Measures of Dispersion
6. The circumferences of a batch of pinewoods are normally distributed with a mean of
153 cm and a standard deviation of 21.4 cm. Only pinewoods with circumferences between
131.6 cm and 174.4 cm are acceptable. If there are 15 pinewoods with circumferences more
than 195.8 cm, find the number of pinewoods which are acceptable.
Solution
7. The weights of n salmons are normally distributed. It is known that 0.15% of them weigh lighter
than 5.4 kg and 16% of them weigh heavier than 23.4 kg.
(a) Find the mean and the standard deviation of their weights.
(b) If 975 salmons are lighter than 27.9 kg, find
(i) the value of n,
(ii) the number of salmons weighing between 9.9 kg and 23.4 kg.
Solution
(a)
(b) (i)
(ii)
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5B Section Worksheets
Basic
Worksheet 9.5
9
Measures of Dispersion
Effects of Data Change on Measures
of Dispersion
NF
(Refer to Book 5B Ch9 p. 9.60 – 9.74)
Name: _________________________
Class: ___________
Key Points
Range
Effects on
the original
measures of
dispersion
after adding a
common constant to
each datum of the
data set
after multiplying
each datum of the
data set by a positive
common constant k
after removing a
datum from the data
set
after adding a datum
to the data set
Inter-quartile range
Standard deviation
remain unchanged
multiplied by k
decrease / remain
unchanged
increase / decrease / remain unchanged
increase / remain
unchanged
increase / decrease / remain unchanged
(In this worksheet, unless otherwise specified, numerical answers should be either exact or correct
to 3 significant figures.)
1. Consider the data set: 1, 4, 6, 9, 10, 12.
(a) Find the mean and the standard deviation of the data set.
(b) Find the new mean and the new standard deviation in each of the following cases.
(i) 3 is added to each datum.
(ii) Each datum is multiplied by 3.
Solution
(a)
(b) (i)
(ii)
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Measures of Dispersion
2. Complete the following table.
Data
Range
(a)
4, 7, 9, 12, 18
(b)
4 + x, 7 + x, 9 + x, 12 + x, 18 + x
(c)
4y, 7y, 9y, 12y, 18y (y > 0)
IQR
Standard deviation
3. The range of the monthly expenditure of 100 families last year is $12 000. If the expenditure of
each family is increased by 5% this year, find the new range of the monthly expenditure this
year.
Solution
4. The following shows the completion time (in s) for 10 students to run 200 m.
28.2
30.4
31.2
35.1
33.5
34.8
29.8
35.5
33.7
32.6
(a) Find the range and the inter-quartile range of the completion time.
(b) Due to the reaction time of the timekeeper, 0.2 s should be added to each datum. Without
actual calculation, determine whether the range and the inter-quartile range of the
completion time will increase, decrease or remain unchanged.
Solution
(a)
(b)
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NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
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Measures of Dispersion
5. The mean mark of S6A students in a Liberal Studies test is 38 and the standard deviation is 6.
The full mark of the test is 50.
(a) Due to a mistake in marking, the teacher deducts 3 marks from each student. Find the new
mean and the new standard deviation.
(b) The teacher adjusts the full mark of the test to 100 by multiplying the mark of each student
by 2. How will the mean and the standard deviation change?
Solution
(a)
(b)
6. Consider the data set: 1, 1, 3, 4, 5, 5, 9, 12.
Will the range and the standard deviation increase, decrease or remain unchanged in the
following situations?
(a) The datum ‘5’ is removed.
Range: _________________
Standard deviation: _________________
(b) The datum ‘5’ is added.
Range: _________________
Standard deviation: _________________
(c) The datum ‘12’ is removed.
Range: _________________
Standard deviation: _________________
(d) The datum ‘13’ is added.
Range: _________________
Standard deviation: _________________
7. Consider the data set: –4, –2, –1, 0, 1, 2, 4.
(a) Find the mean of the data set.
(b) If the datum ‘0’ is removed from the set, will the standard deviation increase, decrease or
remain unchanged? Explain your answer.
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Measures of Dispersion
Solution
(a)
(b) Since the datum removed is (equal to the mean / far away from the mean), removing it will
make the distribution of the data (less concentrate / more concentrate) about the mean.
Therefore, the standard deviation will (increase / decrease / remain unchanged).
8.
The following table shows the number of watches owned by a group of students.
Number of watches
0
1
2
3
4
Frequency
12
20
5
2
1
(a)
Find the inter-quartile range and the standard deviation of the numbers of watches.
(b)
It is given that a student who owns 1 watch joins the group. Will the inter-quartile range
and the standard deviation of the numbers of watches increase, decrease or remain
unchanged?
Solution
(a)
(b)
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NSS Mathematics in Action (2nd Edition)
5B Section Worksheets
9
Measures of Dispersion
Enhanced
Worksheet 9.5
Effects of Data Change on Measures
of Dispersion
NF
(Refer to Book 5B Ch9 p. 9.60 – 9.74)
Name: _________________________
Class: ___________
Key Points
Range
Effects on
the original
measures of
dispersion
after adding a
common constant to
each datum of the
data set
after multiplying
each datum of the
data set by a positive
common constant k
after removing a
datum from the data
set
after adding a datum
to the data set
Inter-quartile range
Standard deviation
remain unchanged
multiplied by k
decrease / remain
unchanged
increase / decrease / remain unchanged
increase / remain
unchanged
increase / decrease / remain unchanged
(In this worksheet, unless otherwise specified, numerical answers should be either exact or correct
to 3 significant figures.)
1. Consider the data set: 1, 3, 9, 12, 14, 18.
(a) Find the inter-quartile range and the standard deviation of the data set.
(b) Using the results of (a), find the inter-quartile range and the standard deviation of the
following data set.
(i) 0, 2, 8, 11, 13, 17
(ii)
1
14
, 1, 3, 4, , 6
3
3
(iii) –1, –3, –9, –12, –14, –18
Solution
(a)
(b) (i)
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Measures of Dispersion
(ii)
(iii)
2. The figure shows the distribution of the salaries of the
workers in factories X and Y. Suppose the number of workers
in two factories are the same.
(a) By inspection, which factory has a less dispersion of
salaries?
(b) If the salary of each worker in these two factories is increased by 30%, how will the ranges,
the inter-quartile ranges and the standard deviations of the salaries of the workers in each
factory change?
Solution
(a)
(b)
3. An art gallery decided to make some picture frames for the paintings recently received. The
following box-and-whisker diagram shows the distribution of the perimeters of those paintings.
(a) Find the range and the inter-quartile range of the perimeters of the paintings.
(b) The cost ($C) of a picture frame can be calculated by using the formula C = 2P + 50, where
P is the perimeter of the painting.
(i) What are the range and the inter-quartile range of the costs of the picture frames?
(ii) Draw a box-and-whisker diagram to present the costs of the picture frames.
Solution
(a)
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Measures of Dispersion
(b) (i)
(ii)
4. The range and the standard deviation of data set X = {a, b, c, d , e} are R and  respectively.
(a) (i) Find the range and the standard deviation of the data set
{a – m, b – m, c – m, d – m , e – m }, where m is a constant.
(ii) Hence, find the range and the standard deviation of the data set
a − m b − m c − m d − m e − m
,
,
,
,

.
3
3
3
3 
 3
(b) It is given that the range and standard deviation of the data set {2, 4, 4, 7, 10} are 8 and 2.8
respectively. Using the results of (a), find the range and the standard deviation of the data
1

set  , 1, 1, 2, 3 .
3

Solution
(a) (i)
(ii)
(b)
5. The following table shows the age distribution of the employees of a company.
Age
Frequency
21 – 30
31 – 40
41 – 50
51 – 60
22
26
15
7
(a) What are the mean and the standard deviation of the ages of the employees?
(b) If a new employee of age 21 joins the company, will the mean and the standard deviation of
the ages of employees increase, decrease or remain unchanged? Explain your answer.
(c) By the end of this year, 3 employees will be 60 years old and they will retire. After they
retire, will the mean and the standard deviation of the ages of employees increase, decrease
or remain unchanged? Explain your answer.
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Measures of Dispersion
Solution
(a)
(b)
(c)
6. The following table shows the marks of S6A students in a Mathematics test.
Marks
Frequency
0 – 20
21 – 40
41 – 60
61 – 80
81 – 100
0
5
13
16
5
It is known that one student was absent on the day which the test was taken.
(a) Find the range and the standard deviation of the marks of students.
(b) State whether the range and the standard deviation of the marks increase, decrease or
remain unchanged in each of the following cases. Explain your answer.
(i) The teacher puts 0 marks as the test result of the absent student.
(ii) The teacher allows the absent student to retake the test and the result of the test, which
is 60 marks, is included in the calculation of the mean and the standard deviation of
the marks of the students.
Solution
(a)
(b) (i)
(ii)
79