Download NTM5_supp_E16_OEQ

16.54
New Trend Mathematics S5A — Supplement
OPEN-ENDED QUESTION ZONE
Write down two different sets of data with the same standard deviation.
3 marks: 2 marks for correct answer; 1 mark for clear explanation.
S UGGESTED SOLUTION
[ Analysis: We have learned that the standard deviations of x 1 , x 2 , x3 ,  , x n and x 1  k, x 2  k, x 3  k,  , x n  k
are the same. ]
 The standard deviations of 1, 2, 3, 4 and 1  1, 2  1, 3  1, 4  1 are the same.
 1, 2, 3, 4 and 2, 3, 4, 5 are two different sets of data with the same standard deviation.
S OLUTION
OF A STUDENT
Data 1 : 1, 3, 5, 7, 9
Data 2 : 2, 4, 6, 8

both standard deviation  5

They have the same dispersion and thus their standard deviations are the same.
(Translated from a Student's Solution in Chinese)
Comments
This student may mistakenly regard the mean as standard deviation. In fact,
1. Mean of data 1 = 5; standard deviation of data 1 
(5  1) 2  (5  3) 2  (5  5) 2  (5  7) 2  (5  9) 2
5
 8
(5  2) 2  (5  4) 2  (5  6) 2  (5  8) 2
4
 5
2. Mean of data 2 = 5; standard deviation of data 2 
Explain how to write down two different sets of data A and B such that the inter-quartile range
of A is greater than that of B, but the standard deviation of A is smaller than that of B.
6 marks: 2 marks for correct answer; 4 marks for clear explanation.
S UGGESTED SOLUTION
[ Analysis: Consider the differences in calculation between inter-quartile range and standard deviation, e.g.
inter-quartile range drops out the extreme cases but standard deviation takes all data into account. Try to
make use of these differences to achieve the purpose. ]
Let set A be 1, 1, 3, 3, 3, 4, 4.
Inter-quartile range  4  1  3
Standard deviation  1.16 (corr. to 2 d.p.)
Chapter 16 Measures of Dispersion
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Let set B be a, 2, 3, 3, 3, 4, b.
Inter-quartile range  4  2  2
Take a  100 and b  100,
standard deviation  53.47 (corr. to 2 d.p.)
 1, 1, 3, 3, 3, 4, 4 and 100, 2, 3, 3, 3, 4, 100 are two required sets of data.
S OLUTION
OF A STUDENT
Q1
Q2
Q3
Q4
Q1
Q2
Q3
Q4
A | Q 3  Q 1 |B | Q 3  Q 1 |
 The inter-quartile range of A is greater.
B Q 4 A Q 4

 The standard deviation of B is greater than that of A.

(Translated from a Student's Solution in Chinese)
Comments
This student cannot give two specific sets of data of which their inter-quartile ranges and standard deviation s can be
calculated. The following are some suggestion s to this student.
1. Q 1 , Q 2 , Q 3 are quartiles based on a set of data but they may not be the data in the set. Therefore a set of data
referred by each diagram should be clearly stated.
2. ‘Q 4 ’ may better be written as ‘maximum value’.
3. The value of standard deviation depends on the mean, and the square of the distance between any datum and the
mean. Even though the maximum value of set B is greater than that of set A, the standard deviation of set B may
not be greater than that of set A.
Exercise of Open-ended Questions 16
Level 1
1. Write down two different sets of data with 0 as the upper quartile.
3 marks: 2 marks for correct answer; 1 mark for clear explanation.
2. Write down two different sets of data with the same standard deviation
2.
4 marks: 2 marks for correct answer; 2 marks for clear explanation.
16.56
New Trend Mathematics S5A — Supplement
3. Consider the following set of data arranged in ascending order.
1, 2, a, b, c, d, 8, 9
Suppose a, b, c and d are integers, find two possible sets of a, b, c and d such that the
inter-quartile range is less than 4.
4 marks: 2 marks for correct answer; 2 marks for clear explanation.
4. The following shows the distribution of the ages of a group of people.
Age
13
14
15
16
17
18
Frequency
6
3
4
x
1
2
Write down two possible values of x if the inter-quartile range is 2 years old.
4 marks: 2 marks for correct answer; 2 marks for clear explanation.
5. Give two sets of data which can be represented by the following box-and-whisker diagram
and explain briefly.
0
2
4
6
8
10
12
14
16
Waiting time (minute)
18
20
22
24
4 marks: 2 marks for correct answer; 2 marks for clear explanation.
6. The figure shows the histogram of a frequency distribution.
8
Frequency
7
6
5
4
3
2
1
0
1
2
3
4
5
6
x
With the total frequency and the values of x remain unchanged,
(a) draw a histogram with a standard deviation greater than that of the given histogram.
(b) draw a histogram with a standard deviation smaller than that of the given histogram.
4 marks: 2 marks for correct answer; 2 marks for clear explanation.
7. Given a set of numbers 36, 45, 48, 51, 60, 63, 90, 99, 159, 177 , insert a number in this set
such that the new mean and standard deviation are less than the original mean and standard
deviation respectively.
4 marks: 2 marks for correct answer; 2 marks for clear explanation.
Chapter 16 Measures of Dispersion
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Level 2
8. Compare the differences between central tendency and dispersion.
5 marks: 5 marks for clear explanation.
9. Give one set of values of a, b, c and d such that the standard deviations of a, b, c, d and
3a  1, 3a  2, 3a  3, 3a  4 are the same.
6 marks: 2 marks for correct answer; 4 marks for clear explanation.
10. Compare the differences among range, inter-quartile range and standard deviation.
6 marks: 6 marks for clear explanation.