Practice › Data Handling › Measures of Central Tendency and Dispersion

Measures of Central Tendency and Dispersion — Paper 1 Long Questions

Eight long questions on the averages and the spread of a data set: reading the mean, median and mode off frequency tables and stem-and-leaf diagrams, weighted and combined means, standard deviation, standard scores and the normal distribution. The set opens gently and gets harder towards the end, as a real Paper 1 does. Final answers are on the page. Detailed solutions can be requested by email.

8 questions · about 55 minutes, 41 marks · answers below each question

All questions are original, written for Math and AI Academy and modelled on past-paper style. They are not reproductions of HKEAA examination questions.

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Question 1 — mean, median and standard deviation from a frequency table; working back from a new mean; the median shifting
Question 2 — two unknown leaves from the median and the mean; adding data without moving the mean
Question 3 — weighted mean; finding a missing component; the effect of changing the weights
Question 4 — standard score to raw score; comparing relative performance across two examinations
Question 5 — standard scores and the normal distribution; estimating counts; testing a claim
Question 6 — an unknown frequency pinned down by the mode and the median together; an unknown frequency from the mean
Question 7 — combined mean of two groups; correcting a wrongly recorded datum
Question 8 — an unknown datum from the standard deviation; choosing between two roots; the effect of multiplying and subtracting on mean and variance

Question 1 · Modelled on 2020 P1 Q9 · 6 marks

The table below shows the distribution of the numbers of books read by a class of students last month.

Number of books read Number of students 0516283645

(a) Write down the mean, the median and the standard deviation of the distribution. (2 marks)

(b) Some new students now join the class. Each of them read $4$ books last month. The mean of the new distribution is $2.5$ . Find the number of new students. (2 marks)

(c) Find the change in the median of the distribution due to the joining of the new students. (2 marks)

Show answer

(a) Mean $= 2$ , median $= 2$ , standard deviation $= \frac{26}{15} \approx 1.32$

(b) $10$

(c) The median increases by $1$ (from $2$ to $3$ ).

Final answers only. Detailed solutions can be requested by email.

Question 2 · Modelled on 2023 P1 Q9 · 6 marks

The stem-and-leaf diagram below shows the distribution of the numbers of hours worked in a week by

15

workers, where

a

and

b

are digits.

Stem (tens) 1234 Leaf (units) 3689 146 a 022 b 8 15

It is given that the median and the mean of the distribution are both

28

(a) Find $a$ and $b$ . (2 marks)

(b) Two more workers now join the group. One of them worked $25$ hours and the other worked $x$ hours in that week. The mean of the distribution is unchanged. Find $x$ . (2 marks)

(c) Is the median of the distribution changed after the two workers join? Explain your answer. (2 marks)

Show answer

(a) $a = 8$ , $b = 7$

(b) $x = 31$

(c) No. The median of the $17$ numbers is the 9th, which is still $28$ .

Final answers only. Detailed solutions can be requested by email.

Question 3 · New sub-type: weighted mean · 4 marks

In a school, the overall score of a student in Mathematics is the weighted mean of the scores in a test, an examination and a project, with weights

30%

50%

and

20%

respectively. Ken scores

70

marks in the test and

90

marks in the project. His overall score is

76

marks.

(a) Find Ken's score in the examination. (2 marks)

(b) The school changes the weights of the test, the examination and the project to $20%$ , $50%$ and $30%$ respectively. Find the change in Ken's overall score. (2 marks)

Show answer

(a) $74$ marks

(b) It increases by $2$ marks (to $78$ marks).

Final answers only. Detailed solutions can be requested by email.

Question 4 · Modelled on 2015 P1 Q15 · 4 marks

The table below shows the means and the standard deviations of the scores of a large group of students in a Mathematics examination and an English examination.

Examination Mathematics English Mean 60 marks 55 marks Standard deviation 8 marks 12 marks

The standard score of Mandy in the Mathematics examination is

1.5

(a) Find Mandy's score in the Mathematics examination. (2 marks)

(b) Mandy's score in the English examination is an integer. Relative to the other students, she performs better in the English examination than in the Mathematics examination. Find the least possible score of Mandy in the English examination. (2 marks)

Show answer

(a) $72$ marks

(b) $74$ marks

Final answers only. Detailed solutions can be requested by email.

Question 5 · Modelled on 2015 P1 Q15 · 6 marks

The heights of the

800

students in a school are normally distributed with mean

165 cm

and standard deviation

6 cm

You may use: for a normal distribution, about 68%, 95% and 99.7% of the data lie within 1, 2 and 3 standard deviations of the mean respectively.

(a) The standard score of Peter's height is $- 1.5$ . Find Peter's height. (2 marks)

(b) Estimate the number of students in the school who are taller than $177 cm$ . (2 marks)

(c) Someone claims that more than $600$ students in the school have heights between $159 cm$ and $171 cm$ . Do you agree? Explain your answer. (2 marks)

Show answer

(a) $156 cm$

(b) About $20$

(c) Disagree. About $68% \times 800 = 544$ students, which is fewer than $600$ .

Final answers only. Detailed solutions can be requested by email.

Question 6 · Modelled on 2018 P1 Q11 · 5 marks

The table below shows the distribution of the numbers of pets kept by some families, where

k

is a non-negative integer.

Number of pets Number of families 06110 2 k 3845

(a) If the mode of the distribution is $1$ and the median of the distribution is $2$ , find all the possible values of $k$ . (3 marks)

(b) If the mean of the distribution is $1.9$ , find $k$ . (2 marks)

Show answer

(a) $k = 4, 5, 6, 7, 8$ or $9$

(b) $k = 11$

Final answers only. Detailed solutions can be requested by email.

Question 7 · Modelled on 2022 P2 Q29 · 4 marks

In a test, the mean score of the

20

students in class A is

64

marks and the mean score of the

30

students in class B is

59

marks.

(a) Find the mean score of all the $50$ students. (2 marks)

(b) It is later found that the score of a student in class B was wrongly recorded as $45$ marks. The correct score is $75$ marks. Find the corrected mean score of class B and the corrected mean score of all the $50$ students. (2 marks)

Show answer

(a) $61$ marks

(b) Class B: $60$ marks; all $50$ students: $61.6$ marks

Final answers only. Detailed solutions can be requested by email.

Question 8 · Modelled on 2019 P2 Q45 · 6 marks

Consider the five numbers

4, 6, 8, 10

and

x

. The standard deviation of these five numbers is

22

(a) Find the two possible values of $x$ . (3 marks)

(b) It is further given that the median of the five numbers is $8$ . Each of the five numbers is multiplied by $3$ and then $1$ is subtracted from each result. Find the mean and the variance of the five new numbers. (3 marks)

Show answer

(a) $x = 2$ or $x = 12$

(b) Mean $= 23$ , variance $= 72$

Final answers only. Detailed solutions can be requested by email.

Continue practising this topic: Try the Paper 1 Challenge for harder variants, or move to Paper 2 multiple-choice — also available as Paper 2 Challenge MC.

Where this sits in the syllabus

Measures of central tendency and dispersion appear in almost every HKDSE paper: usually one short Section A question on Paper 1, and two or three Paper 2 questions near the end of Sections A and B. Quartiles, the inter-quartile range, box-and-whisker diagrams and cumulative frequency belong to the separate topic More about Statistics and are not tested here.

Full coverage of this topic, and the rest of the course, is on the Mathematics page. More sets are listed on the practice index.

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