Book 6 Chapter 21 Measures of Dispersion

1
21
Measures of Dispersion
Case Study
21.1 Range and Inter-quartile Range
21.2 Box-and-whisker Diagrams
21.3 Standard Deviation
21.4 Applications of Standard Deviation
21.5 Effects on the Dispersion with a Change in
Data
Chapter Summary
2
Case Study
Although both drivers got the same total mark, it
does not mean that both of them had a consistent
performance for all ten dives.
If we plot a broken line graph for both divers,
we find that the marks of diver A fluctuate more
than those of diver B.
In the above example, we consider the spread of
the data.
In this chapter, we will learn how to represent
this by statistical methods.
3
21.1 Range and Inter-quartile Range
In junior forms, we learnt three measures of
central tendency of a set of data, namely mean,
median and mode. However, these measures tell us
only limited information about the data.
Consider two boxes of apples A and B.
The weight (in g) of each apple in each box is
given below Box A 100, 100, 100, 104, 110,
110 Box B 85, 95, 100, 104, 116, 124
The mean and the median of the weights for both
boxes of apples is 104 g and 102 g respectively,
and the weights of the apples in Box B are more
widely spread than those in Box A.
The spread or variability of data is called the
dispersion of the data.
In this chapter, we are going to learn the
following measures of dispersion 1. Range
2. Inter-quartile range 3. Standard deviation
4
21.1 Range and Inter-quartile Range
A. Range
The range is a simple measure of the dispersion
of a set of data.
1. For ungrouped data, the range is the
difference between the largest value and the
smallest value in the set of data.
Range ? Largest value Smallest value
2. For grouped data, the range is the difference
between the highest class boundary and the lowest
class boundary
Range ? Highest class boundary Lowest class
boundary
5
21.1 Range and Inter-quartile Range
A. Range
Example 21.1T
The weights (in g) of eight pieces of meat are
given below 210, 230, 245, 180, 220, 240, 175,
195 (a) Find the range of the weights. (b) If
the meat is sold at 3 per 100 g, find the range
of the prices of the meat.
Solution
(a) Range
? (245 ? 175) g
(b) Range of the prices
6
21.1 Range and Inter-quartile Range
A. Range
Example 21.2T
The following table shows the weights of the boys
in S6A. (a) Write down the upper class
boundary of the class 70 kg 74 kg. (b) Write
down the lower class boundary of the class 50 kg
54 kg. (c) Hence find the range of the
weights.
Weight (kg) 50 54 55 59 60 64 65 69 70 74
Frequency 3 6 8 5 2
Solution
(a) 74.5 kg
(b) 49.5 kg
(b) Range
? (74.5 ? 49.5) kg
7
21.1 Range and Inter-quartile Range
B. Inter-quartile Range
When a set of data is arranged in ascending order
of magnitude, the quartiles divide the data into
four equal parts.
Full set of data arranged in order of magnitude
25 of data 25 of data 25 of data 25 of data
Q1 Q2 Q3
Q1 lower quartile ? 25 of data less than it Q2
median ? 50 of data less than it Q3 upper
quartile ? 75 of data less than it
The inter-quartile range is defined as the
difference between the upper quartile and the
lower quartile of the set of data.
Inter-quartile range ? Q3 Q1
8
21.1 Range and Inter-quartile Range
B. Inter-quartile Range
Example 21.3T
The marks of 13 boys in a Chinese test are
recorded below 72 78 80 65 62 62
78 81 84 70 68 69 60 (a) Arrange
the marks in ascending order. (b) Find the
median mark. (c) Find the range and the
inter-quartile range.
Solution
(a) Arrange the marks in ascending order
60, 62, 62, 65, 68, 69, 70, 72, 78, 78, 80, 81,
84
(b) Median
(c) Range
? Inter-quartile range
9
21.1 Range and Inter-quartile Range
B. Inter-quartile Range
Example 21.4T
The cumulative frequency polygon shows the
lifetimes (in hours) of 80 bulbs. (a) Find the
range of the lifetimes. (b) Find the median
lifetime. (c) Find the inter-quartile range.
Solution
(a) Range
? (780 ? 140) hours
(b) From the graph,
Median
(c) Q1 ? 240 hours,
Q3 ? 600 hours
? Inter-quartile range
? (600 ? 240) hours
10
21.1 Range and Inter-quartile Range
B. Inter-quartile Range
Example 21.5T
Consider the ages of passengers in two
mini-buses. Mini-bus A 18, 24, 25, 19, 12, 10,
34, 39, 45, 23, 34, 40, 24, 28 Mini-bus B 23,
26, 28, 32, 38, 34, 19, 26, 29, 32, 35, 30, 29,
22 By comparing the ranges and the
inter-quartile ranges of the ages, determine
which group of passengers has a larger dispersion
of ages.
Solution
For Mini-bus A,
For Mini-bus B,
the range ? 45 ? 10
? 35
the range ? 38 ? 19
? 19
For Mini-bus A,
For Mini-bus B,
the inter-quartile range ? 34 ? 19
the inter-quartile range ? 32 ? 26
? 15
? 6
Since the range and the inter-quartile range of
the ages of passengers of mini-bus A are larger,
the passengers on mini-bus A have a larger
dispersion.
11
21.2 Box-and-whisker Diagrams
A box-and-whisker diagram is a statistical
diagram that provides a graphical summary of the
set of data by showing the quartiles and the
extreme values of the data.
A box-and-whisker diagram shows the greatest
value, the least value, the median, the lower
quartile and the upper quartile of a set of data.
12
21.2 Box-and-whisker Diagrams
Example 21.6T
The following box-and-whisker diagram shows the
number of family members of a class of students.
(a) Find the median and the range of the number
of family members. (b) Find the inter-quartile
range.
Solution
(a) Median
Maximum value ? 6 and minimum value ? 1
Range
(b) Q1 ? 2 and Q3 ? 4
Inter-quartile range
13
21.2 Box-and-whisker Diagrams
Example 21.7T
The following shows the measurements of the
waists (in inches) of the students in a class.
Girls 27 26 25 23 26 28 32 24 25 29
30 28 22 25 26 Boys 32 32 30 29 28 25
28 26 29 31 32 34 28 27 28 (a) Find the
median, the lower quartile and the upper quartile
of the waists for both boys and girls.
(b) Draw box-and-whisker diagrams of their waist
measurements on the same graph paper.
Solution
(a) Arrange the measurements in ascending order
Girls 22, 23, 24, 25, 25, 25, 26, 26, 26, 27,
28, 28, 29, 30, 32 Boys 25, 26, 27, 28, 28, 28,
28, 29, 29, 30, 31, 32, 32, 32, 34
For girls,
For boys,
14
21.2 Box-and-whisker Diagrams
Example 21.7T
The following shows the measurements of the
waists (in inches) of the students in a class.
Girls 27 26 25 23 26 28 32 24 25 29
30 28 22 25 26 Boys 32 32 30 29 28 25
28 26 29 31 32 34 28 27 28 (a) Find the
median, the lower quartile and the upper quartile
of the waists for both boys and girls.
(b) Draw box-and-whisker diagrams of their waist
measurements on the same graph paper.
Solution
(b) For boys, minimum ? 25 inches maximum ? 34
inches
For girls, minimum ? 22 inches maximum ? 32
inches
Refer to the figure on the right.
15
21.3 Standard Deviation
A. Standard Deviation for Ungrouped Data
Standard deviation describes how the spread out
of the data are around the mean. It is usually
denoted by ?.
Consider a set of ungrouped data x1, x2, , xn.
Notes The quantity ? 2 is called the variance of
the data.
16
21.3 Standard Deviation
A. Standard Deviation for Ungrouped Data
Example 21.8T
Six students joined the inter-school
cross-country race. The times taken (in min) to
complete the race are recorded below 45,
46, 49, 50, 52, y If the mean time is 49.5 min,
find (a) the value of y and (b) the standard
deviation of the times taken. (Give the answer
correct to 3 significant figures.)
Solution
(a)
(b)
Standard deviation
(cor. to 3 sig. fig.)
17
21.3 Standard Deviation
A. Standard Deviation for Ungrouped Data
Example 21.9T

• The following table shows the marks of Eric in
  five tests of two subjects.
• Find the standard deviations of the marks of each
  subject.
• (Give the answers correct to 3 significant
  figures.)
• (b) In which subject is his performance more
  consistent?

Test 1 Test 2 Test 3 Test 4 Test 5
Chinese 65 70 76 68 78
English 72 81 85 90 80
Solution

1. For Chinese, mean

Standard deviation
(cor. to 3 sig. fig.)
18
21.3 Standard Deviation
A. Standard Deviation for Ungrouped Data
Example 21.9T

• The following table shows the marks of Eric in
  five tests of two subjects.
• Find the standard deviations of the marks of each
  subject.
• (Give the answers correct to 3 significant
  figures.)
• (b) In which subject is his performance more
  consistent?

Test 1 Test 2 Test 3 Test 4 Test 5
Chinese 65 70 76 68 78
English 72 81 85 90 80
Solution
For English, mean
Standard deviation
(cor. to 3 sig. fig.)
19
21.3 Standard Deviation
A. Standard Deviation for Ungrouped Data
Example 21.9T

• The following table shows the marks of Eric in
  five tests of two subjects.
• Find the standard deviations of the marks of each
  subject.
• (Give the answers correct to 3 significant
  figures.)
• (b) In which subject is his performance more
  consistent?

Test 1 Test 2 Test 3 Test 4 Test 5
Chinese 65 70 76 68 78
English 72 81 85 90 80
Solution
(b) For Chinese, standard deviation ? 4.88
For English, standard deviation ? 5.95
The standard deviation of Chinese is smaller
than that of English, so Erics performance in
Chinese is more consistent.
20
21.3 Standard Deviation
A. Standard Deviation for Ungrouped Data
Example 21.10T
The numbers of students in five classes are given
as y 15, y 6, y 9, y 20, y 15 (a)
Find the mean and the standard deviation. (Give
the answers correct to 3 significant figures if
necessary.) (b) Find the range and the median if
the mean is 34.
Solution
(a) Mean
Standard deviation
(b) ? y ? 1 ? 34
? y ? 35
The five numbers are 20, 41, 44, 15 and 50.
? Range
and Median
(cor. to 3 sig. fig.)
21
21.3 Standard Deviation
B. Standard Deviation for Grouped Data
For a set of grouped data, we have to consider
the frequency of each group.
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21.3 Standard Deviation
B. Standard Deviation for Grouped Data
Example 21.11T
The following table shows the ages of 50 workers
in a company. (a) Find the mean age of the
workers. (b) Find the standard deviation of the
ages. (Give the answer correct to 3
significant figures.)
Age 11 20 21 30 31 40 41 50 51 60
Number of workers 4 24 12 8 2
Solution
(a)
Class mark 15.5 25.5 35.5 45.5 55.5
Frequency 4 24 12 8 2
Mean age
(b)
Standard deviation
(cor. to 3 sig. fig.)
23
21.3 Standard Deviation
C. Finding Standard Deviation by a Calculator
In actual practice, it is quite difficult to
calculate the standard deviation if the amount of
data is very large.
In such circumstances, a calculator can help us
to find the standard deviation.
In order to use a calculator, we have to set the
function mode of the calculator to standard
deviation SD. We also have to clear all the
previous data in the SD mode.
For both ungrouped and grouped data, we can use a
calculator to find the mean and the standard
deviation.
24
21.3 Standard Deviation
The following table summarizes the advantages and
disadvantages of the three different measures of
dispersion.
Measure of dispersion Advantage Disadvantage
1. Range Only two data are involved, so it is the easiest one to calculate. Only extreme values are considered which may give a misleading impression.
2. Inter-quartile range It only focuses on the middle 50 of data, thus avoiding the influence of extreme values. Cannot show the dispersion of the whole group of data.
3. Standard deviation It takes all the data into account that can show the dispersion of the whole group of data. Difficult to compute without using a calculator.
25
21.4 Applications of Standard Deviation
A. Standard Scores
Standard score is used to compare data in
relation with the mean and the standard deviation
?.
Notes The standard score may be positive,
negative or zero.
A positive standard score means the given value
is z times the standard deviation above the mean
while a negative standard score means the given
value is z times the standard deviation below the
mean.
26
21.4 Applications of Standard Deviation
A. Standard Scores
Example 21.12T
Ryan sat for a mathematics examination which
consisted of two papers. The following table
shows his marks as well as the means and the
standard deviations of the marks for the whole
class in these papers. (a) Find his standard
scores in the two papers. (Give the answers
correct to 3 significant figures.) (b) In which
paper did he perform better?
Paper I Paper II
Marks 66 71
Mean 60.8 62.4
Standard deviation 4.2 6.4
Solution
(a) Paper I
(cor. to 3 sig. fig.)
Paper II
(cor. to 3 sig. fig.)
(b) Since 1.34 ? 1.24, Ryan performed better in
Paper II than in Paper I.
27
21.4 Applications of Standard Deviation
A. Standard Scores
Example 21.13T
Given that the standard scores of Doriss marks
in Art and Music are 2.3 and 1.4 respectively,
find (a) her mark in Art if the mean and the
standard deviation of the marks are 30 and 2
respectively (b) the mean mark of Music if
Doris got 41.5 marks and the standard deviation
of the marks is 3.5.
Solution
(a) Doriss mark in Art
? (?2.3) ? 2 30
(b) The mean mark of Music
? 41.5 ? 1.4 ? 3.5
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21.4 Applications of Standard Deviation
B. Normal Distribution
For a large number of the frequency distributions
we meet in our daily life, their frequency curves
have the shape of a bell
The bell can have different shapes. This
bell-shaped frequency curve is called the normal
curve and the corresponding frequency
distribution is called the normal distribution.
For a normal distribution, the mean, median and
the mode of the data lie at the centre of the
distribution.
Therefore, the normal curve is symmetrical about
the mean, i.e., the axis of symmetry for the
normal curve is x ? .
29
21.4 Applications of Standard Deviation
B. Normal Distribution
In addition, we can tell the percentage of the
data lie within a number of standard deviations
from the mean
1. About 68 of the data lie within one standard
deviation from the mean, i.e., ? and
?.
2. About 95 of the data lie within two standard
deviations from the mean, i.e., 2? and
2?.
3. About 99.7 of the data lie within three
standard deviations from the mean, i.e., 3?
and 3?.
30
21.4 Applications of Standard Deviation
B. Normal Distribution
Example 21.14T
The heights of some soccer players are normally
distributed with a mean of 180 cm and a standard
deviation of 8 cm. Find the percentage of players
(a) whose heights are between 172 cm and 188 cm,
(b) whose heights are greater than 188 cm.
Solution
Given ? 180 and s ? 8.
(a) 172 ? 180 ? 8
? 34 of the players heights lie between (
) cm and cm.
188 ? 180 8
? 34 of the players heights lie between cm
and ( ) cm.
Percentage of players whose heights are between
172 cm and 188 cm
? 34 34
(b) 188 ? 180 8
? Percentage of players whose heights are
greater than 188 cm
? 50 ? 34
31
21.4 Applications of Standard Deviation
B. Normal Distribution
Example 21.15T
The weights of 2000 children are normally
distributed with a mean of 56.4 kg and a
standard deviation of 6.2 kg. (a) How many
children have weights between 50.2 kg and 68.8
kg? (b) How many children are heavier than 50.2
kg?
Solution
Given ? 56.4 and s ? 6.2.
(a) 50.2 ? 56.4 ? 6.2
68.8 ? 56.4 2 ? 6.2
81.5 of children have weights between (
) kg and ( ) kg.
? Number of children
? 2000 ? 81.5
(b) 50.2 ? 56.4 ? 6.2
Percentage of children who are heavier than 50.2
kg
? 50 34
? 84
? Number of children
? 2000 ? 84
32
21.5 Effects on the Dispersion with a
Change in Data
A. Removal of the Largest or Smallest item
from the Data
In junior forms, we learnt that if we remove a
datum greater than the mean of the data set, then
the mean will decrease.
Similarly, if we remove a datum less than the
mean of the data set, then the mean will increase.
We can deduce that
If the greatest or the least value (assuming the
removed datum is unique) in a data set is
removed, then
1. the range will decrease
2. the inter-quartile range may increase,
decrease or remain unchanged
3. the standard deviation may increase or
decrease.
33
21.5 Effects on the Dispersion with a
Change in Data
B. Adding a Common Constant to the Whole
Set of Data
We have the following conclusion
If a constant k is added to each datum in a set
of data, then the following measures of
dispersion will not change
1. the range,
2. the inter-quartile range and
3. the standard deviation
34
21.5 Effects on the Dispersion with a
Change in Data
C. Multiplying the Whole Set of Data by a
Common Constant
We have the following conclusion
The range, the inter-quartile range and the
standard deviation will be k times the original
values if each datum in a set of data is
multiplied by a constant k.
Notes If the quartiles are not members of the
data set, the conclusion will be the same.
35
21.5 Effects on the Dispersion with a
Change in Data
C. Multiplying the Whole Set of Data by a
Common Constant
Example 21.16T
Consider the nine different numbers 20, 45,
25, 30, 32, 28, 35, 51, 40 (a) Find the range,
the inter-quartile range and the standard
deviation. (b) Find the new range, the new
inter-quartile range and the new standard
deviation of the new set of data if (i) 10
is subtracted from each data (ii) each
number is halved (iii) the datum 45 is
removed. (Give the answers correct to 3
significant figures if necessary.)
Solution
(a) Range
Inter-quartile range
Standard deviation
(cor. to 3 sig. fig.)
36
21.5 Effects on the Dispersion with a
Change in Data
C. Multiplying the Whole Set of Data by a
Common Constant
Example 21.16T
Consider the nine different numbers 20, 45,
25, 30, 32, 28, 35, 51, 40 (a) Find the range,
the inter-quartile range and the standard
deviation. (b) Find the new range, the new
inter-quartile range and the new standard
deviation of the new set of data if (i) 10
is subtracted from each data (ii) each
number is halved (iii) the datum 45 is
removed. (Give the answers correct to 3
significant figures if necessary.)
Solution
(b) (i) If 10 is subtracted from each datum, the
range, the inter-quartile range and the standard
deviation of the new data remain unchanged.
Range
Inter-quartile range
Standard deviation
(cor. to 3 sig. fig.)
37
21.5 Effects on the Dispersion with a
Change in Data
C. Multiplying the Whole Set of Data by a
Common Constant
Example 21.16T
Consider the nine different numbers 20, 45,
25, 30, 32, 28, 35, 51, 40 (a) Find the range,
the inter-quartile range and the standard
deviation. (b) Find the new range, the new
inter-quartile range and the new standard
deviation of the new set of data if (i) 10
is subtracted from each data (ii) each
number is halved (iii) the datum 45 is
removed. (Give the answers correct to 3
significant figures if necessary.)
Solution
(b) (ii) If each datum is halved, the range, the
inter-quartile range and the standard deviation
of the new data are multiplied by 0.5.
Range
Inter-quartile range
Standard deviation
(cor. to 3 sig. fig.)
38
21.5 Effects on the Dispersion with a
Change in Data
C. Multiplying the Whole Set of Data by a
Common Constant
Example 21.16T
Consider the nine different numbers 20, 45,
25, 30, 32, 28, 35, 51, 40 (a) Find the range,
the inter-quartile range and the standard
deviation. (b) Find the new range, the new
inter-quartile range and the new standard
deviation of the new set of data if (i) 10
is subtracted from each data (ii) each
number is halved (iii) the datum 45 is
removed. (Give the answers correct to 3
significant figures if necessary.)
Solution
(b) (iii) The remaining data are 20, 25, 28, 30,
32, 35, 40, 51.
Range
Inter-quartile range
Standard deviation
(cor. to 3 sig. fig.)
39
21.5 Effects on the Dispersion with a
Change in Data
D. Insertion of Zero in the Data Set
We have the following conclusion
If a zero value is inserted in a non-negative
data set, then
1. the range may increase or remain unchanged
2. the inter-quartile range may increase,
decrease or remain unchanged
3. the standard deviation may increase or
decrease.
40
21.5 Effects on the Dispersion with a
Change in Data
D. Insertion of Zero in the Data Set
Example 21.17T
The daily income () of a hawker during the last
two weeks was 200, 220, 230, 240, 250, 320,
340, 360, 380, 400, 450, 580, 650 (a) Find the
inter-quartile range and the standard deviation.
(b) There was a thunderstorm last Monday and the
income on that day was zero. If the income
from last Monday is also considered, find the
new inter-quartile range and the new standard
deviation. (Give the answers correct to 1
decimal place if necessary.)
Solution
(a)
(b)
Inter-quartile range
Inter-quartile range
The standard deviation
The standard deviation
(cor. to 1 d. p.)
(cor. to 1 d. p.)
41
Chapter Summary
21.1 Range and Inter-quartile Range
1. The range is the difference between the
largest value (highest class boundary) and the
smallest value (lowest class boundary) in a set
of ungrouped (grouped) data.
2. The inter-quartile range is the difference
between the upper quartile Q3 and the lower
quartile Q1 of a set of data.
42
Chapter Summary
21.2 Box-and-whisker Diagrams
A box-and-whisker diagram illustrates the spread
of a set of data. It shows the greatest value,
the least value, the median, the lower quartile
and the upper quartile of the data.
43
Chapter Summary
21.3 Standard Deviation
Standard deviation ? is the measure of dispersion
that describes how spread out a set of data is
around the mean value.
1. For ungrouped data
2. For grouped data
Larger values for the range, the inter-quartile
range and the standard deviation of the data
indicate a larger dispersion and vice versa.
44
Chapter Summary
21.4 Applications of Standard Deviation
1. The standard score z is the number of standard
deviations that a given value is above or below
the mean, and is given by
2. The curve of a normal distribution is
bell-shaped and is called the normal curve.
In the normal distribution, different percentages
of data lie within different standard deviations
from the mean.
45
Chapter Summary
21.5 Effects on the Dispersion with a Change
in Data
1. If the greatest or the least value (assuming
both are unique) in a data set is removed, then
the range will decrease. However, the
inter- quartile range may increase, decrease or
remain unchanged and the standard deviation may
increase or decrease.
2. If a constant k is added to each datum in a
set of data, then the range, the inter-quartile
range and the standard deviation will not change.
3. If each item in the data is multiplied by a
positive constant k, then the range, the
inter-quartile range and the standard deviation
will be k times their original values.
4. If a zero value is inserted in a non-negative
data set, then the range may increase or remain
unchanged, the inter-quartile range may
increase, decrease or remain unchanged and the
standard deviation may increase or decrease.
46
Follow-up 21.1
21.1 Range and Inter-quartile Range
A. Range
The results (in m) of the best eight boys in the
long jump are 5.2, 5.6, 4.8, 4.2, 5.3, 4.5,
5.0, 4.8 Find the range of the results.
Solution
? (5.6 ? 4.2) m
Range
47
Follow-up 21.2
21.1 Range and Inter-quartile Range
A. Range
The following table shows the heights of the
students in S6B. (a) What is the upper class
boundary of the class 171 cm 175 cm? (b) What
is the lower class boundary of the class 151 cm
155 cm? (c) Hence find the range of the heights.
Height (cm) 151 155 156 160 161 165 166 170 171 175
Frequency 1 3 6 2 4
Solution
(a) 175.5 cm
(b) 150.5 cm
(c) Range
? (175.5 ? 150.5) cm
48
Follow-up 21.3
21.1 Range and Inter-quartile Range
B. Inter-quartile Range
The air pollution indices of a city are recorded
at noon every day. The following are the indices
in the last 15 days. 56 58 62 58 60
64 63 63 59 56 70 69 67 69 65
(a) Arrange the data in ascending order. (b)
Find the median. (c) Find the inter-quartile
range.
Solution
(a) Arrange the marks in ascending order 56,
56, 58, 58, 59, 60, 62, 63, 63, 64, 65, 67, 69,
69, 70
(b) Median
(c) Q3 ? 67 and Q1 ? 58
Inter-quartile range
49
Follow-up 21.4
21.1 Range and Inter-quartile Range
B. Inter-quartile Range
The figure shows the cumulative frequency polygon
of the heights (in cm) of 100 trees. (a) Find
the lower quartile and the upper quartile of the
heights. (b) Find the inter-quartile range.
Solution
(a) From the graph,
(b) Inter-quartile range
50
Follow-up 21.5
21.1 Range and Inter-quartile Range
B. Inter-quartile Range
The following table shows the distribution of
members ages in two sports clubs. (a) How
many members are there in each of the clubs?
Age 12 13 14 15 16 17
Badminton Club 14 10 8 10 6 2
Basketball Club 0 8 12 17 13 0
Solution
(a) For Badminton Club,
For Basketball Club,
number of members
number of members
? 14 10 8 10 6
? 0 8 12 17 13 0
51
Follow-up 21.5
21.1 Range and Inter-quartile Range
B. Inter-quartile Range
The following table shows the distribution of
members ages in two sports clubs. (b) Find
the range and the inter-quartile range of the
distribution of ages of the members in each
club. (c) Which club has a larger dispersion of
ages?
Age 12 13 14 15 16 17
Badminton Club 14 10 8 10 6 2
Basketball Club 0 8 12 17 13 0
Solution
(b) For Badminton Club,
range
? 17 12
inter-quartile range
? 15 12
For Basketball Club,
range
? 16 13
inter-quartile range
? 16 14
(c) Since the range and the inter-quartile range
of the ages of the members in the Badminton Club
are larger, Badminton Club has a larger
dispersion of ages.
52
Follow-up 21.6
21.2 Box-and-whisker Diagrams
The following box-and-whisker diagram shows the
lengths (in cm) of the hands of a class of
students. (a) Find the median length of the
hands. (b) Find the range of the lengths of the
hands. (c) Find the inter-quartile range of the
lengths of the hands.
Solution
(a) Median
(b) Maximum length ? 19.4 cm and
minimum length ? 17.3 cm
Range ? (19.4 ? 17.3) cm
(c)
Inter-quartile range
? (19.0 ? 18.2) cm
53
Follow-up 21.7
21.2 Box-and-whisker Diagrams
Kelvin and Johnny are comparing their results on
10 mathematics tests. The following are their
marks. Kelvins marks 54, 70, 67, 92, 75, 80,
84, 78, 66, 82 Johnnys marks 70, 74, 76, 78,
78, 79, 80, 78, 72, 75 (a) Find the median, the
lower quartile and the upper quartile of the
marks for each student.
Solution
(a) Arrange the marks in ascending order
Kelvins marks 54, 66, 67, 70, 75, 78, 80, 82,
84, 92
Median
Johnnys marks 70, 72, 74, 75, 76, 78, 78, 78,
79, 80
Median
54
Follow-up 21.7
21.2 Box-and-whisker Diagrams
Kelvin and Johnny are comparing their results on
10 mathematics tests. The following are their
marks. Kelvins marks 54, 70, 67, 92, 75, 80,
84, 78, 66, 82 Johnnys marks 70, 74, 76, 78,
78, 79, 80, 78, 72, 75 (b) Draw box-and-whisker
diagrams on the same graph to compare the
results. (c) Which student performs more
consistently in the tests?
Solution
(b) For Kelvin, minimum ? 54 marks maximum ? 92
marks
For Johnny, minimum ? 70 marks maximum ? 80
marks
Refer to the figure on the right.
(c) Johnny performs more consistently.
55
Follow-up 21.8
21.3 Standard Deviation
A. Standard Deviation for Ungrouped Data
Emily got the following marks in five English
tests. 59, 65, 76, 68, 82 (a) Find the mean
mark. (b) Find the standard deviation of the
marks. (Give the answer correct to 3
significant figures.)
Solution
(a) Mean
(b)
? 121 25 36 4 144
? 330
Standard deviation
(cor. to 3 sig. fig.)
56
Follow-up 21.9
21.3 Standard Deviation
A. Standard Deviation for Ungrouped Data
The following table shows the marks of two
students in five tests. (a) Find the
standard deviations of the marks of the two
students. (Give the answers correct to 3
significant figures.) (b) Which student has a
more consistent performance?
Test 1 Test 2 Test 3 Test 4 Test 5
Winnie 80 65 82 66 78
Cherry 76 74 80 72 72
Solution
(a) For Winnie,
For Cherry,
standard deviation
standard deviation
(cor. to 3 sig. fig.)
(cor. to 3 sig. fig.)
(b) Since the standard deviation of Cherrys
marks is smaller than Winnies, Cherry has a more
consistent performance.
57
Follow-up 21.10
21.3 Standard Deviation
A. Standard Deviation for Ungrouped Data
The following are the ages of five people. 66
x , 65 3x, 71 x , 68 x , 70 x (a) Find
the mean age in terms of x. (b) If x ? 4, find
the standard deviation. (Give the answer
correct to 3 significant figures.)
Solution
(a) Mean
(b) ? x ? 4
? Mean ? 68 4 ? 72 and the five numbers are
70, 77, 67, 72 and 74.
? Standard deviation
(cor. to 3 sig. fig.)
58
Follow-up 21.11
21.3 Standard Deviation
B. Standard Deviation for Grouped Data
The following table shows the air pollution index
recorded daily in a particular district at 500
p.m. in June. (a) Find the value of y. (b)
Find the mean. (c) Find the standard deviation.
Index 51 55 56 60 61 65 66 70
Number of days 8 10 7 y
Solution
(a) 8 10 7 y ? 30
(b) Mean
(c) Standard deviation
(cor. to 3 sig. fig.)
59
Follow-up 21.12
21.4 Applications of Standard Deviation
A. Standard Scores
The following table shows the means and the
standard deviations of the marks for the whole
class in two subjects, as well as Helens marks.
(a) Find the standard scores of Helens
marks in these subjects. (Give the answers
correct to 3 significant figures.) (b) In which
subject did she perform better?
Subject Chinese History
Helens mark 85 70
Mean 79 62
Standard deviation 8.2 4.3
Solution
(a) Chinese
History
(cor. to 3 sig. fig.)
(cor. to 3 sig. fig.)
(b) Since 1.86 ? 0.732, Helen performed better in
History than in Chinese.
60
Follow-up 21.13
21.4 Applications of Standard Deviation
A. Standard Scores
Given that the standard scores of Kelvins marks
in the Chinese reading and written tests are 1
and 1 respectively, find (a) his mark in the
reading test if the mean and the standard
deviation of the marks for his whole class are
6.9 and 1.1 respectively (b) the mean mark of
the written test if his mark in the test and
the standard deviation of the marks are 5.5 and
2 respectively.
Solution
(a) Kelvins mark in the reading test
(b) The mean mark of the written test
61
Follow-up 21.14
21.4 Applications of Standard Deviation
B. Normal Distribution
The foot sizes of a group of children in a child
care centre are normally distributed with a mean
of 20 cm and a standard deviation of 2.5 cm.
(a) Find the percentage of children having foot
sizes between 15 cm and 22.5 cm. (b) Find the
percentage of children having foot sizes less
than 12.5 cm.
Solution
(a) Given ? 20 and s ? 2.5.
15 ? 20 ? 2 ? 2.5
? 47.5 of the childrens foot sizes lie between
( ) cm and cm.
22.5 ? 20 2.5
? 34 of the childrens foot sizes lie between
cm and ( ) cm.
Percentage of children having foot sizes between
15 cm and 22.5 cm
? 47.5 34
(b)
12.5 ? 20 ? 3 ? 2.5
Percentage of children having foot sizes less
than 12.5 cm
? 50 49.85
62
Follow-up 21.15
21.4 Applications of Standard Deviation
B. Normal Distribution
The lifetimes of a pack of 1500 bulbs is normally
distributed with a mean of 1200 hours and a
standard deviation of 50 hours. (a) How many
bulbs have a lifetime between 1100 hours and 1300
hours? (b) How many bulbs have a lifetime less
than 1150 hours?
Solution
Since ? 1200 and s ? 50,
(a) 1100 ? 1200 ? 2 ? 50
1300 ? 1200 2 ? 50
95 of bulbs have a lifetime between (
) hours and ( ) hours.
? Number of bulbs
? 1500 ? 95
(b) 1150 ? 1200 ? 50
Percentage of bulbs having a lifetime less than
1150 hours
? 50 34
? 16
? Number of bulbs
? 1500 ? 16
63
Follow-up 21.16
21.5 Effects on the Dispersion with a
Change in Data
C. Multiplying the Whole Set of Data by a
Common Constant
Consider the following ten numbers 0 2 3
5 6 8 8 9 10 12 (a) Find the range,
the inter-quartile range and the standard
deviation. (b) Find the change in the range, the
inter-quartile range and the standard
deviation after each of the following changes
(i) 1 is added to each datum (ii) each datum
is multiplied by 5 (c) Find the range, the
inter-quartile range and the standard deviation
of the positive numbers. (Give the answers
correct to 3 significant figures if necessary.)
Solution
(a) Range
Inter-quartile range
Standard deviation
(cor. to 3 sig. fig.)
64
Follow-up 21.16
21.5 Effects on the Dispersion with a
Change in Data
C. Multiplying the Whole Set of Data by a
Common Constant
Consider the following ten numbers 0 2 3
5 6 8 8 9 10 12 (a) Find the range,
the inter-quartile range and the standard
deviation. (b) Find the change in the range, the
inter-quartile range and the standard
deviation after each of the following changes
(i) 1 is added to each datum (ii) each datum
is multiplied by 5 (c) Find the range, the
inter-quartile range and the standard deviation
of the positive numbers. (Give the answers
correct to 3 significant figures if necessary.)
Solution
(b) (i) If 1 is added to each datum, the range,
the inter-quartile range and the standard
deviation of the new data remain unchanged.
(ii) If each datum is multiplied by 5, the
range, the inter-quartile range and the standard
deviation of the new data are 5 times the
original values.
65
Follow-up 21.16
21.5 Effects on the Dispersion with a
Change in Data
C. Multiplying the Whole Set of Data by a
Common Constant
Consider the following ten numbers 0 2 3
5 6 8 8 9 10 12 (a) Find the range,
the inter-quartile range and the standard
deviation. (b) Find the change in the range, the
inter-quartile range and the standard
deviation after each of the following changes
(i) 1 is added to each datum (ii) each datum
is multiplied by 5 (c) Find the range, the
inter-quartile range and the standard deviation
of the positive numbers. (Give the answers
correct to 3 significant figures if necessary.)
Solution
(c) Range
Inter-quartile range
Standard deviation
(cor. to 3 sig. fig.)
66
Follow-up 21.17
21.5 Effects on the Dispersion with a
Change in Data
D. Insertion of Zero in the Data Set
The figure shows the stem-and-leaf diagram of the
marks of the girls who took the same English
test. (a) Find the inter-quartile range and the
standard deviation of the marks. (b) Fiona was
also absent, so she got a zero mark. If Fionas
mark is also considered, find the new
inter-quartile range and the new
standard deviation of the marks. (Give the
answers correct to 3 significant figures if
necessary.)
Stem (Tens digit) Leaf (Units digit)
0 0 0 0 0 2 6 7 8
1 1
2 2 4
3 1 1
Solution
(a) The inter-quartile range
The standard deviation
(cor. to 3 sig. fig.)
(b) The total number of girls is 14.
The new inter-quartile range
The new standard deviation
(cor. to 3 sig. fig.)