Revision (Statistics)

Statistics

Public Examination Questions

HKDSE Paper 1 – Section A(1)
2023/I/9
The stem-and-leaf diagram below shows the distribution of the numbers of working hours of a group of
workers in a week.

Stem (tens) Leaf (units)

2 a 5 5 6 6 8 8
3 3 3 3 4 5 5 9 9
4 0 1 4 4 5 6 7 7 9

The range of the distribution is 27.

(a) Find the mean and the mode of the distribution.
(b) If a worker is randomly selected from the group, find the probability that the number of working
hours of the selected worker in the week exceeds the mode of the distribution. (5 marks)

2021/I/9
The bar chart below shows the distribution of the numbers of books read by a group of students in a year.

If a student is randomly selected from the group, then the probability that the selected student reads fewer
7
than 26 books in the year is .
10
(a) Find k.
(b) Write down the range, the inter-quartile range and the standard deviation of the distribution.
(5 marks)

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 Revision (Statistics)

2020/I/9
The table below shows the distribution of the numbers of subjects taken by a class of students.

Number of subjects taken 4 5 6 7

Number of students 8 12 16 4

(a) Write down the mean, the median and the standard deviation of the above distribution.
(b) A new student now joins the class. The number of subjects taken by the new student is 5. Find the
change in the median of the distribution due to the joining of this student. (5 marks)

2014/I/4
The table below shows the distribution of the numbers of calculators owned by some students.

Number of calculators 0 1 2 3
Number of students 7 14 15 4

Find the median, mode and the standard deviation of the above distribution. (3 marks)

2013/I/9
The bar chart below shows the distribution of the numbers of family members of the employees of
company D.
Distribution of the numbers of family members of the employees of company D
24
Number of employees

20
16
12
8
4
0
1 2 3 4 5 6 7

Number of family members

Figure 2
(c) Find the mean, the interquartile range and the standard deviation of the above distribution.
(d) An employee leaves company D. The number of family members of this employee is 7. Find the
change in the standard deviation of the numbers of family members of the employees of company D
due to the leaving of this employee. (5 marks)

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 Revision (Statistics)

2012/I/7
The box-and-whisker diagram below shows the distribution of the times taken by a large group of
students of an athletic club to finish a 100 m race:

Time(s)
a 12.1 13.2 b 18.1

The inter-quartile range and the range of the distribution are 3.2 s and 6.8 s respectively.
(a) Find a and b.
(b) The students join a training program. It is found that the longest time taken by the students to finish
a 100 m race after the training is 2.9 s less than that before the training. The trainer claims that at
least 25% of the students show improvement in the time taken to finish a 100 m race after the
training. Do you agree? Explain your answer. (4 marks)

HKDSE Paper 1 – Section A(2)

2023/I/11
The table below shows the distribution of the numbers of calculators owned by a class of students.
Number of calculators owned 1 2 3 4
Number of students 8 5 n 1

The mean of the distribution is 2.

(a) Find the median, the inter-quartile range and the variance of the distribution. (5 marks)
(b) Two students now withdraw from the class. It is found that the mean of the distribution remains
unchanged. Is there any change in the range of the distribution due to the withdrawal of the two
students? Explain your answer. (2 marks)

2022/I/11
The stem-and-leaf diagram below shows the distribution of the ages of the players of a football team.
Stem (tens) Leaf (units)
1 7 8 9
2 0 a a 8 8 9
3 b b 5 5 6 6 6 6 7 8
4 3
The inter-quartile range and the median of the distribution are 14 and 31 respectively.
(a) Find a and b. (3 marks)
(b) A player now leaves the football team.
(i) Is there any change in the mode of the distribution due to the leaving of the player? E
(ii) If the range of the distribution is decreased, find the greatest possible standard deviation of the
distribution. (4 marks)
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 Revision (Statistics)

2020/I/11(a)
The stem-and-leaf diagram below shows the distribution of the weights (in grams) of the letters in a bag.
Stem (tens) Leaf (units)
1 1 2 3 3
2 3 3 4 5 6 9 9
3 1 6 7 8 8 8
4 2
5 0 w
It is given that the range of the above distribution is the triple of its inter-quartile range.
(a) Find w. (4 marks)

2019/I/12
The stem-and-leaf diagram below shows the distribution of the results (in seconds) of some boys in a
400 m race.
Stem (tens) Leaf (units)
5 a
6 0 0 3 c c 8 9 9 9
7 0 1 1 1 2 2 5 6 9
8 b
It is given that the inter-quartile range of the distribution is 8 seconds.
(a) Find c. (2 marks)
(b) It is given that the range of the distribution exceeds 34 seconds and the mean of the distribution is 69
seconds. Find
(i) a and b,
(ii) the least possible standard deviation of the distribution. (6 marks)

2018/I/10
The box-and-whisker diagram below shows the distribution of the ages of the clerks in team X of a company.
It is given that the range and the inter-quartile range of this distribution are 43 and 21 respectively.

19 27 38 a b Age

(a) Find a and b. (3 marks)

(b) There are five clerks in team Y of the company and three of them are of age 38. It is given that the
range of the ages of the clerks in team Y is 20. Team X and team Y are now combined to form a section.
The manager of the company claims that the range of the ages of the clerks in the section and the
range of the ages of the clerks in team X must be the same. Do you agree? Explain your answer.
(2 marks)

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 Revision (Statistics)

2018/I/11
The following table shows the distribution of the numbers of children of some families:

Number of children 0 1 2 3 4
Number of families k 2 9 6 7

It is given that k is a positive integer.

(a) If the mode of the distribution is 2, write down
(i) the least possible value of k;
(ii) the greatest possible value of k. (2 marks)
(b) If the median of the distribution is 2, write down
(i) the least possible value of k;
(ii) the greatest possible value of k. (2 marks)
(c) If the mean of the distribution is 2, find the value of k. (2 marks)

2017/I/11
The stem-and-leaf diagram below shows the distribution of the hourly wages (in dollars) of the workers in a
group.
Stem (tens) Leaf (units)
6 1 1 1 3 4 6 8 9 9
7 a 7 7 8
8 1 b

It is given that the mean and the range of the above distribution are $70 and $22 respectively.
(a) Find the median and the standard deviation of the above distribution. (5 marks)
(b) If a worker is randomly selected from the group, find the probability that the hourly wage of the
selected worker exceeds $70. (2 marks)

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2016/I/12
The bar chart below shows the distribution of the ages of the children in a group, where a > 1l and
4 < b < 10 . The median of the ages of the children in the group is 7.5 .

Distribution of the ages of the children in the group

Number of children

Age
(a) Find a and b. (3 marks)
(b) Four more children now join the group. It is found that the ages of these four children are all different
and the range of the ages of the children in the group remains unchanged. Find
(i) the greatest possible median of the ages of the children in the group,
(ii) the least possible mean of the ages of the children in the group. (4 marks)

2015/I/12
The stem-and-leaf diagram below shows the distribution of the weights (in kg) of the students in a
football club.
Stem (tens) Leaf (units)
4 0 2 3 3 3 3 9
5 1 1 2 2 3 7 9
6 3 5 8 9
7 8 9

(a) Find the mean, the median and the range of the above distribution. (3 marks)
(b) Two more students now join the club. It is found that both the mean and the range of the distribution
of the weights are increased by 1 kg. Find the weight of each of these two students.
(4 marks)

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2014/I/11
There are 33 paintings in an art gallery. The box-and-whisker diagram below shows the distribution of the
prices (in thousand dollars) of the paintings in the art gallery. It is given that the mean of this distribution
is 53 thousand dollars.

Price (thousand dollars)

18 42 55 63 91

(a) Find the range and the interquartile range of the above distribution. (3 marks)
(b) Four paintings of respectively prices (in thousand dollars) 32, 34, 58 and 59 are now donated to a
museum. Find the mean and the median of the prices of the remaining paintings in the art gallery.
(3 marks)
2013/I/10
The age of the members of Committee A are shown as follows:
17 18 21 21 22 22 23 23 23 31
31 34 35 36 47 47 58 68 69 69
(a) Write down the median and the mode of the ages of the members of Committee A. (2 marks)
(b) The stem-and-leaf diagram below shows the distribution of the ages of the members of Committee
B. It is given that the range of this distribution is 47.

Stem (tens) Leaf (units)

2 a 5 6 7
3 3 3 8
4 3
5 1 24 9
6 7 b
(i) Find a and b. 6
(ii) From each committee, a member is randomly selected as the representative of that committee.
The two representatives can join a competition when the difference of their ages exceeds 40.
Find the probability that these two representatives can join the competition. (4 marks)

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2012/I/10
Tom conducts a survey on the numbers of hours spent on doing homework in a week by secondary
students. Questionnaires are sent out and twenty of them are returned. The stem-and-leaf diagram below
shows the numbers of hours recorded in the twenty questionnaires:

Stem (tens) Leaf (units)

1 0 0 1 1 2 3 4 5 5 6 6 7 7
2 0 0 0 5 8
3 4 6

(a) Find the mean and the median of the numbers of hours recorded in the twenty questionnaires.
(2 marks)
(b) Tom receives four more questionnaires. He finds that the mean of the numbers of hours recorded in
these four questionnaires is 18. It is found that the numbers of hours recorded in two of these four
questionnaires are 19 and 20.
(i) Write down the mean of the numbers of hours recorded in the twenty-four questionnaires.
(ii) Is it possible that the median of the numbers of hours recorded in the twenty-four
questionnaires is the same as the median found in (a)? Explain your answer. (4 marks)

HKDSE Paper 1 – Section B

2016/I/16
In a test, the mean of the distribution of the scores of a class of students is 61 marks. The standard scores of
Albert and Mary are –2.6 and 1.4 respectively. Albert gets 22 marks. A student claims that the range of the
distribution is at most 59 marks. Is the claim correct? Explain your answer. (3 marks)

2015/I/15
The table below shows the means and the standard deviation of the scores of a large group of students in a
Mathematics examination and a Science examination:
Examination Mean Standard deviation
Mathematics 66 marks 12 marks
Science 52 marks 10 marks

The standard score of David in the Mathematics examination is –0.5

(a) Find the score of David in the Mathematics examination. (2 marks)
(b) Assume that the scores in each of the above examination are normally distributed. David gets 49
marks in the Science examination. He claims that relative to other students, he performs better in
the Science examination than in the Mathematics examination. Is the claim correct? Explain your
answer. (2 marks)

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2013/II/15
The box-and-whisker diagram below shows the distribution of the scores (in marks) of the students of a
class in a test. Susan gets the highest score while Tom gets 65 marks in the test. The standard scores of
Susan and Tom in the test are 3 and 0.5 respectively.

(a) Find the mean of the distribution. (2 marks)

(b) Susan claims that the standard scores of at least half of the students in the test are negative. Do you
agree? Explain your answer. (2 marks)

2012/I/15
The standard deviation of the test scores obtained by a class of students in a Mathematics test is 10 marks.
All the students fail in the test, so the test score of each student is adjusted such that each score is
increased by 20% and then extra 5 marks are added.
(a) Find the standard deviation of the test scores after the score adjustment. (1 mark)
(b) Is there any change in the standard score of each student due to the score adjustment? Explain
your answer. (2 marks)

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 Revision (Statistics)

Answers
Section A(1)
7
2023/I/9 (a) mean = 36, mode = 33 (b)
12
2021/I/9 (a) k = 28 (b) range = 5, I.Q.R. = 2, s.d. = 1.43
2020/I/9 (a) mean = 5.4 median = 5.5 s.d. = 0.917 (b) decrease by 0.5
2014/I/4 median = 1, mode = 2, s.d. = 0.889

7
2013/I/9 (a) mean = , inter-quartile range = 2, s.d. = 1.5 (b) decrease by 0.0485
2
2012/I/7 (a) a = 11.3, b = 15.3 (b) agreed

Section A(2)
2023/I/11 (a) median = 2, inter-quartile range = 2, variance = 0.9 (b) No
2022/I/11 (a) a = 2, b = 1 (b)(i) No (ii) 7.16
2020/I/11 (a) 6

2019/I/12 (a) c=4 (b)(i) 

a=0
b=7
or  a =1
b=6
(ii) 7.34 seconds

2018/I/10 (a) a = 48, b = 62 (b) No

2018/I/11 (a)(i) 1 (ii) 8 (b)(i) 3 (ii) 19 (c) 9
2017/I/11 median = $69, s.d. = $7.33

2016/I/12 (a) a = 12 or a = 13 (b)(i) 8 (ii) 7.6

 
b=8 b=9
2015/I/12 (a) mean = 55, median = 52, range = 39 (b) 52 kg and 80 kg
2014/I/11 (a) range = 73 thousand dollars , IQR = 21 thousand dollars
(b) mean = 54 thousand dollars, median = 55 thousand dollars
a = 0  a = 1 a = 2 8
2013/I/10 (a) median = 31, mode = 23 (b)(i)  ,  or  (ii)
b = 7 b = 8 b = 9 65

2012/I/10 (a) mean = 18, median = 16 (b)(i) mean = 18 (ii) No

Section B
2016/I/16 No 2015/I/15 (a) 60 (b) Yes 2013/I/15 (a) 60 marks (b) Yes
2012/I/15 (a) 12 marks (b) no change

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 Revision (Statistics)

Multiple-choice Questions 2021/II/29

HKDSE – Section A The box-and-whisker diagram below shows the
2023/II/29 distribution of the ages of a group of researchers.
The box-and-whisker diagram below shows the Find the inter-quartile range of the distribution.
distribution of the numbers of training hours of
some engineers in a year. Find the upper quartile
of the distribution.

A. 5 B. 10
C. 20 D. 34
A. 20 B. 40
C. 60 D. 70 2021/II/30
The mean of 70 integers is 32. If the mean of 30 of
2023/II/30 these 70 integers is 24, then the mean of the
There are 14 full-time employees and 56 part-time remaining 40 integers is
employees in a company. The mean salary of the A. 38. B. 40.
full-time employees is $31 530 while the mean C. 43. D. 74.
salary of the part-time employees is $21 525. Find
the mean salary of these employees of the 2020/II/29
company. The bar chart below shows the distribution of the
A. $23 526 B. $25 527 numbers of pens owned by some students. Find
C. $27 528 D. $29 529 the inter-quartile range of the distribution.

2022/II/30
Consider the following positive integers:
2 5 6 6 x x x y
If both the mean and the median of the above
positive integers are 6, which of the following
must be true? A. 1 B. 2
I. The mode of the above positive integers is 6. C. 4 D. 6
II. The least possible range of the above positive
integers is 6.
III. The greatest possible variance of the above
positive integers is 6.
A. I only B. II only
C. I and III only D. II and III only

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2020/II/30 2018/II/29
Consider the following integers: The mean of the numbers of pages of 10
3 3 8 8 8 10 12 m n magazines is 132. If the mean of the numbers of
Let x, y and z be the median, the mean and the pages of 6 of these 10 magazines is 108, then the
mode of the above integers respectively. If the mean of the numbers of pages of the remaining 4
range of the above integers is 9, which of the magazines is
following must be true? A. 148. B. 156.
I. x=8 C. 168. D. 176.
II. y = 8
III. z = 8 2018/II/30
A. I only B. II only The stem-and-leaf diagram below shows the
C. I and III only D. II and III only distribution of the numbers of books read by 20
students in a year.
2019/II/29
Stem (tens) Leaf (units)
Which of the following can be obtained from any
2 1 2 2 8
box-and-whisker diagram?
I. Range 3 a a
II. Standard deviation 4 0 2 4 5 5 7 8
III. Inter-quartile range 5 3
A. I and II only B. I and III only 6 b b 9 9
C. II and III only D. I, II and III 7 0 8

If the inter-quartile range of the above distribution

2019/II/30
is at most 25, which of the following must be true?
The table below shows the distribution of the
I. 5a9
numbers of merits obtained by some students in a
II. 0b4
year.
III. 1  a − b  6
Number of merits 6 7 8 9 10 A. I and II only B. I and III only
obtained C. II and III only D. I, II and III
Number of 32 36 28 18 2
2017/II/29
students
The box-and-whisker diagram below shows the
distribution of the numbers of online hours spent
Which of the following is true?
by a class of students in a certain week. Find the
A. The mode of the distribution is 36.
lower quartile of the distribution.
B. The median of the distribution is 8.
C. The lower quartile of the distribution is 6.
D. The upper quartile of the distribution is 10.

A. 5 B. 15
C. 25 D. 40

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2017/II/30 2015/II/30
Consider the following positive integers: Consider the following integers:
2 3 4 6 7 9 10 m n 2 2 3 3 3 3 3 5 5 6
Let a , b and c be the mode, the median and the 8 8 9 10 m
range of the above positive integers respectively. Let p, q and r be the mean, median and the mode
If the mean of the above positive integers is 5, of the above integers respectively. If 3  m  5,
which of the following must be true? which of the following must be true?
I. a=2 I. p>q
II. b = 4 II. p > r
III. c = 8 III. q > r
A. I only B. II only A. I only B. II only
C. I and III only D. II and III only C. I and III only D. II and III only

2016/II/30 2014/II/29
Consider the following data: The pie chart below shows the expenditure of
32 68 79 86 88 98 98 a b c John in a certain week. John spends $240 on
If the mean and the mode of the above data are 77 clothing that week. Find his expenditure on
and 68 respectively, then the median of the above transportation that week.
data is
A. 76. B. 82.
C. 85. D. 93. Clothing
160o
Transportation
2015/II/29
The box-and-whisker diagram below shows the Meals 50o
distribution of the numbers of books read by some
Others
teachers in a term. Find the inter-quartile-range of
the distribution. A. $40 B. $60
C. $90 D. $135

2014/II/30
The stem-and-leaf diagram below shows the
distribution of the ages of the passengers in a bus.
A. 20 B. 35 If the range of the above distribution is at least 33,
C. 40 D. 45 which of the following must be true?
Stem (tens) Leaf (units) I. 0h3
1 h 4 6 II. 3k9
2 3 3 3 4 6 7 7 III. 3k–h5
3 1 2 2 2 6 8 A. I only B. II only
4 0 k C. I and III only D. II and III only

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 Revision (Statistics)

2013/II/27 2013/II/29
If the mean and the mode of the nine numbers 14, The stem-and-leaf diagram below shows the
6, 4, 5, 7, 5, x, y and z are 8 and 14 respectively, distribution of the hourly wages (in dollars) of
then the median of these nine numbers is some workers.
A. 5 B. 6 Stem (tens) Leaf (units)
C. 7 D. 8 4 0 2 2 2 4 4 4 7
5 0 0 1 2 2 6 8 9
6 3 5 5 7
2013/II/28 7 0
The scatter diagram below shows the relation 8 2 6
between x and y. Which of the following may 9 5
represent the relation between x and y? Which of the following box-and-whisker diagrams
may represent the distribution of their hourly
wages?

A.

B.

A. y increases when x increases.

C.
B. y decreases when x increases.
C. y varies inversely as x2.
D.
D. y varies directly as x–3.

2012/II/29
The bar chart below shows the distribution of the
number of rings owned by the girls in a group.
Find the standard deviation of the distribution
correct to 2 decimal places.

8
Number of girls

6
4
2

0 0 1 2 3 4
Number of rings

A. 1.04 B. 1.16
C. 1.19 D. 2.09

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2012/II/30 2010/II/36
Consider the following data: The box-and-whisker diagram below shows the
19 10 12 12 13 13 14 15 16 m n distribution of the numbers of books read by some
If both the mean and the median of the above data students in a year. Find the inter-quartile range of
are 14, which of the following are true? the numbers of books read.
I. m  14
II. n  16
III. m + n = 30
A. I and II only B. I and III only
C. II and III only D. I, II and III A. 30 B. 40
C. 55 D. 65
HKCEE
2010/II/34 2009/II/35
Let a, b, c and d be the mean, the median, the The mean height of 54 boys and 36 girls is 162cm.
mode and the range of the group of numbers If the mean height of the girls is 153 cm, then the
(x, x, x, x, x, x, x + 1, x + 2, x + 3) respectively. mean height of the boys is
Which of the following must be true? A. 147 cm. B. 157.5 cm.
I. a>b C. 168 cm. D. 175.5 cm.
II. b > c
III. c>d 2009/II/36
A. I only B. II only The box-and-whisker diagram below shows the
C. I and III only D. II and III only distribution of temperatures (in °C) of water in an
experiment under various settings. Which of the
2010/II/35 following are true?
The bar chart below shows the distribution of
scores obtained by a group of students in a test.
Find the standard deviation of the scores correct to
the nearest integer.

I. The range is 40 °C.

II. The median is 24 °C.
III. The inter-quartile range is 12 °C.
A. I and II only B. I and III only
C. II and III only D. I, II and III

A. 12 B. 14
C. 23 D. 33

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2006/II/34 2023/II/45
{x – 6, x – 3, x + 4, x + 5} and {x – 8, x – 1, x + 2, It is given that n is an integer. Let u, v and w be
x + 9} are two groups of numbers. Which of the the standard deviation, the median and the range
following is/are true? of the group of numbers {1 – 9n, 3 – 9n, 4 – 9n,
I. The two groups of numbers have the same 5 – 9n, 7 – 9n} respectively. Which of the
mean. following must be true?
II. The two groups of numbers have the same I. u=2
median. II. v  4
III. The two groups of numbers have the same III. w  6
range. A. I only B. II only
A. I only B. II only C. I and III only D. II and III only
C. I and III only D. II and III only
2022/II/44
2006/II/35 In a test, the median of the test scores of a class of
The box-and-whisker diagram below shows the students is 30 marks. All the students fail in the test,
distribution of the weights (in kg) of some students. so the test score of each student is adjusted such
Find the inter-quartile range of their weights. that each score is increased by 50% and then extra
8 marks are added. Let x marks be the median of
the test scores of the class of students after the
score adjustment. In the test, the standard score of a
50 55 60 65 70 75 80
Weight (kg) student before the score adjustment is –2. Denote
the standard score of this student after the score
A. 5 kg B. 10 kg adjustment by z. Find x and z.
C. 15 kg D. 30 kg A. x = 45 and z = −2
B. x = 45 and z = −1
C. x = 53 and z = −2
HKDSE – Section B D. x = 53 and z = −1
2023/II/44
The table below shows the scores (in marks) and 2022/II/45
the corresponding standard scores of three It is given that d is a real number. Let S1 be a group
students in an examination. of numbers {d – 6, d – 2, d – 1, d + 3, d + 5, d + 7}
Score (marks) 46 x 86 and S2 be another group of numbers {d – 7, d – 5,
Standard score –3 1 2 d – 3, d + 1, d + 2, d + 6}. Which of the following
Find x. is/are true?
A. 64 B. 66 I. The means of S1 and S2 are equal.
C. 70 D. 78 II. The standard deviations of S1 and S2 are equal.
III. The inter-quartile ranges of S1 and S2 are
equal.
A. I only B. II only
C. I and III only D. II and III only

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 Revision (Statistics)

2021/II/44 2019/II/45
In an examination, the mean of the examination The mean, the range and the variance of a set of
scores is 45 marks. A boy gets 25 marks in the numbers are m, r and v respectively. Each number
examination and his standard score is –5. If the of the set is multiplied by 6 and then 5 is added to
standard score of a girl in the examination is 7, each resulting number to form a new set of
then her examination score is numbers. Which of the following is/are true?
A. 4 marks. B. 53 marks. I. The mean of the new set of numbers is
C. 73 marks. D. 80 marks. 6m + 5.
II. The range of the new set of numbers is
2020/II/44 6r + 5.
In a test, the difference of the test scores and the III. The variance of the new set of numbers is
difference of the standard scores of two students 6v + 5.
are 30 marks and 6 respectively. In the test, the A. I only B. II only
standard deviation of the test scores is C. I and III only D. II and Ill only
A. 5 marks. B. 24 marks.
C. 25 marks. D. 36 marks. 2018/II/44
In a test, the mean of the test scores is 68 marks.
2020/II/45 Peter gets 46 marks in the test and his standard
The variance of the six numbers 20a + 3, 20a + 5, score is –2.2. If Susan gets 52 marks in the test,
20a + 9, 20a + 11, 20a + 15 and 20a + 17 is then her standard score is
A. 5. B. 10. A. –2.5. B. –1.6.
C. 25. D. 20a + 25. C. –0.6. D. 1.6.

2019/II/44 2017/II/44
In an examination, the standard deviation of the The standard score of Tom in a Mathematics
examination scores is 8 marks. The examination examination is –2. If the score of Tom in the
score of Mary is 69 marks and her standard score Mathematics examination is 33 marks and the
is 0.5. If the standard score of John in the mean of the scores of the Mathematics
examination is –1.5, then his examination score is examination is 45 marks, then the standard
A. 45 marks. B. 53 marks. deviation of the scores of the Mathematics
C. 65 marks. D. 77 marks. examination is
A. 3 marks. B. 6 marks.
C. 12 marks. D. 36 marks.

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 Revision (Statistics)

2017/II/45 2016/II/45
Let m1 , r1 and v1 be the mode, the inter-quartile The variance of a set of numbers is 49. Each
range and the variance of a group of numbers { x1 , number of the set is multiplied by 4 and then 9 is
x2 , x3 , x4 , x5 , x6 , x7 } respectively while m2 , r2 added to each resulting number to form a new set
and v2 be the mode, the inter-quartile range and of numbers. Find the variance of the new set of
the variance of the group of numbers { 8x1 , 8x2 , numbers.
8x3 , 8x4 , 8x5 , 8x6 , 8x7 } respectively. Which of A. 196 B. 205
the following must be true? C. 784 D. 793
I. m2 = 8m1
II. r2 = 8r1 2015/II/45
III. v2 = 8v1 Let x1, y1 and z1 be the mean, the median and the
A. I and II only B. I and III only variance of a group of numbers {a1, a2, a3, …, a50}
C. II and III only D. I, II and III respectively while x2, y2 and z2 be the mean, the
median and the variance of the group of numbers
2016/II/44 {a1, a2, a3, … a49} respectively. If x1 = a50, which
The stem-and-leaf diagram below shows the of the following must be true?
distribution of the scores (in marks) of a group of I. x1 = x2
students in a test. Ada gets the highest score in II. y1  y2
the test. III. z1  z2
Stem (tens) Leaf (units) A. I and II only B. I and III only
4 5 6 7 8 C. II and III only D. I, II and III
5 5 5 6 8
6 3 5 5 6 9 9 2014/II/44
7 0 0 1 In an examination, Peter gets 55 marks and his
8 0 2 5 standard score is – 3 while Mary gets 95 marks
and her standard score is 2. Find the mean of the
Which of the following is/are true?
examination scores.
I. The upper quartile of the distribution is 55
A. 8 marks B. 64 marks
marks.
C. 75 marks D. 79 marks
II. The standard score of Ada in the test is lower
than 2.
2014/II/45
III. The standard deviation of the distribution is
If the variance of the four numbers a, b, c and d is
greater than 12 marks.
9, then the variance of the four numbers 14 – a,
A. I only B. II only
14 – b, 14 – c and 14 – d is
C. I and III only D. II and III only
A. 5. B. 9.
C. 23. D. 121.

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 Revision (Statistics)

2013/II/45 Practice Paper/II/45

If the variance of the five numbers x1, x2, x3, x4 Let A be a group of numbers { ,  ,  ,  } and B
and x5 is 13, then the variance of the five numbers be another group of numbers { + 2,  + 2, μ + 2,
3x1 + 4, 3x2 + 4, 3x3 + 4, 3x4 + 4 and 3x5 + 4 is  + 2,  + 2} , where     μ    . Which of
A. 39. B. 43. the following must be true?
C. 117. D. 121. I. The median of A is smaller than that of B.
II. The range of A and the range of B are the
2012/II/45 same.
Let m1, r1 and v1 be the mean, the range and the III. The standard deviation of A is greater than
variance of a group of numbers {x1, x2, x3, …, x100} that of B.
respectively. If m2, r2 and v2 are the mean, the A. I and II only B. I and III only
range and the variance of the group of numbers C. II and III only D. I, II and III
{ x1, x2, x3, …, x100, m1} respectively, which of the
following must be true? HKCEE
I. m1 = m2 2006/II/54
II. r1 = r2 The standard deviation of the five numbers
III. v1 = v2 10a + 1, 10a + 3, 10a + 5, 10a + 7 and 10a + 9 is
A. I and II only B. I and III only 12
A. 8. B. .
C. II and III only D. I, II and III 5
C. 10 . D. 2 2.
Practice Paper/II/44
The mean, the variance and the inter-quartile 2004/II/36
range of a set of numbers are 40, 9 and 18 David got 70 marks in a test and his standard
respectively. If 5 is added to each number of the score was –0.625. If the standard deviation of the
set and each resulting number is then tripled to test marks was 8 marks, then the mean mark of the
form a new set of numbers, find the mean, the test was
variance and the inter-quartile range of the new set A. 62 marks. B. 65 marks.
of numbers. C. 75 marks. D. 78 marks.
Mean Variance Inter-quartile range
A. 120 27 69 2003/II/32
B. 120 81 69 The mean mark of a mathematics test was 63
C. 135 27 54 marks. Peter got 75 marks in the test and his
D. 135 81 54 standard score was 0.75. If Mary got 83 marks in
the same test, then her standard score would be
A. 0.83. B. 1.25.
C. 2.22. D. 5.

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 Revision (Statistics)

2003/II/36 Answers
x is the mean of the group of numbers Section A
{a, b, c, d, e}. Which of the following statements HKDSE
about the two groups of numbers {a, b, c, d, e} 2023/II/29 C 2023/II/30 A 2022/II/29 C(79)
and {a, b, c, d, e, x} must be true? 2023/II/30 A(25) 2022/II/30 B(25) 2021/II/29 B(88)
I. The two groups of numbers have the same 2021/II/30 A(77) 2020/II/29 A(69) 2020/II/30 A(49)
mean. 2019/II/29 B(84) 2019/II/30 C(78) 2018/II/29 C(78)
II. The two groups of numbers have the same 2018/II/30 A(43) 2017/II/29 B(76) 2017/II/30 B(51)
range. 2016/II/30 B(76) 2015/II/29 A(91) 2015/II/30 B(45)
III. The two groups of numbers have the same 2014/II/29 C(82) 2014/II/30 B(31) 2013/II/27 B(68)
standard deviation. 2013/II/28 A(80) 2013/II/29 D(57) 2012/II/29 B(71)
A. I only B. III only 2012/II/30 D(47)
C. I and II only D. II and III only HKCEE
2010/II/34 A(60) 2010/II/35 A(67) 2010/II/36 A(84)
2009/II/35 C(81) 2009/II/36 C(86) 2006/II/34 B(70)
2006/II/35 C(84)

Section B
HKDSE
2023/II/44 D 2023/II/45 A 2022/II/44 C(43)
2022/II/45 D(51) 2021/II/44 C(76) 2021/II/45 B(41)
2020/II/44 A(45) 2020/II/45 C(50) 2019/II/44 B(72)
2019/II/45 A(56) 2018/II/44 B(78) 2017/II/44 B(78)
2017/II/45 A(48) 2016/II/44 B(52) 2016/II/45 C(50)
2015/II/45 B(26) 2014/II/44 D(57) 2014/II/45 B(57)
2013/II/45 C(37) 2012/II/45 A(36) PP/II/44 D
PP/II/45 C
HKCEE
2006/II/54 D(63) 2004/II/36 C(61) 2003/II/32 B(62)
2003/II/36 C(35)

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