Practice › Data Handling › Data Presentation and Interpretation

Data Presentation and Interpretation: Paper 1 Challenge

Challenge tier. These questions state a condition, not a method: a pie chart whose unknown is never named, leaves pinned down by counting, a class boundary that moves a value into the next class, and claims about charts to accept or reject. Work the standard Paper 1 set first. Answers sit under each question. Detailed solutions can be requested by email.

10 questions · about 70 minutes · 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.

Question 1 · Challenge — runs HKDSE 2017 Paper 1 Q7 without naming x, then adds a boundary trap: the new angle lands exactly on 180°

The pie chart shows the distribution of the means of transport to school of the students in a school. The number of students who travel by MTR is 60 more than the number of students who walk.

Pie chart of modes of transport: Bus (3x + 6)°, MTR 5x°, Walk 2x°, Others 54°

(a) Find the number of students who travel by bus. (4 marks)

(b) Later, 20 students who walk change to travel by MTR. Someone claims that the angle of the MTR sector will then be greater than $18 0^{\circ}$ . Is the claim correct? Explain your answer. (2 marks)

Show answer

(a) 64 (here $x = 30$ and there are 240 students)
(b) Not correct. The MTR group becomes 120 of the 240 students, so the angle is exactly $18 0^{\circ}$ , not greater.

Question 2 · Challenge — new idea relative to the standard set: comparing two distributions on one pair of axes, with a claim to test

The frequency polygons show the distributions of the test scores of the 40 students in Class A and the 40 students in Class B. The scores are whole numbers, and the classes are 40–49, 50–59, …, 90–99.

Two frequency polygons of test scores for Class A (solid) and Class B (dashed); class marks 44.5 to 94.5

(a) Which class has more students scoring 80 marks or above? By how many? (2 marks)

(b) Someone claims that, for every class interval from 60–69 upwards, Class B has more students than Class A. Is the claim correct? Explain your answer. (2 marks)

(c) The two classes are combined. Find the percentage of the 80 students who scored below 60 marks. (2 marks)

Show answer

(a) Class B, by 8 (20 students against 12)
(b) Not correct. In 60–69, Class A has 9 students and Class B has only 6.
(c) $13.75%$

Question 3 · Challenge — new idea relative to the standard set: unknown leaves pinned down by counting conditions alone, one of them only partly determined

The stem-and-leaf diagram shows the distribution of the weights (in kg) of 15 parcels, where

a

b

and

c

are digits. Exactly 4 parcels weigh from 20 kg to 25 kg inclusive, and exactly 6 parcels weigh 36 kg or more.

Stem (tens)	Leaf (units)
1	2 5 $a$
2	0 3 3 $b$ 8
3	1 $c$ 6 6 9
4	0 4

(a) Find $c$ . (2 marks)

(b) Find all the possible values of $b$ . (2 marks)

(c) Someone claims that one third of the parcels may weigh less than 23 kg. Is the claim correct? Explain your answer. (2 marks)

Show answer

(a) $c = 6$
(b) 3, 4 or 5
(c) Not correct. Whatever the digits, exactly 4 parcels (12, 15, $10 + a$ and 20 kg) weigh less than 23 kg, and one third of 15 is 5.

Question 4 · Challenge — a technique the standard set never touches: a pictogram that scales in three dimensions, linked to similar solids

A poster shows the oil output of a company with two similar cylindrical drums. The output was 2 million barrels in 2025 and 4 million barrels in 2026. The designer doubled both the radius and the height of the drum for 2026.

Two drums: the 2026 drum has twice the radius and twice the height of the 2025 drum

(a) Find the ratio of the volume of the 2026 drum to that of the 2025 drum, as drawn. (2 marks)

(b) Explain why the poster is misleading. (1 mark)

(c) The designer wants the volumes of the two similar drums to be in the true ratio of the outputs. By what factor should the radius of the 2025 drum be multiplied to give the radius of the 2026 drum? Give your answer correct to 3 significant figures. (2 marks)

Show answer

(a) $8 : 1$
(b) The eye compares the sizes of the drums, which suggest output grew 8 times; it only doubled.
(c) $32 \approx 1.26$

Question 5 · Challenge — a genuine trap: class limits versus class boundaries when data are rounded, with a value that looks as if it belongs to the lower class

The weights of 40 students are recorded to the nearest kg and grouped in the table below.

Weight (kg)	40–44	45–49	50–54	55–59
Number of students	5	11	14	10

(a) Peter draws a histogram whose bars run from 40 to 44, 45 to 49, 50 to 54 and 55 to 59 on the horizontal axis, leaving gaps between the bars. Explain what is wrong, and write down the correct class boundaries of the class 45–49. (2 marks)

(b) The actual weight of a student is 49.7 kg. In which class is this student counted? (1 mark)

(c) Find the percentage of students whose actual weights are less than 49.5 kg. (2 marks)

Show answer

(a) The bars should run between class boundaries, so adjacent bars touch with no gaps; the class 45–49 has boundaries 44.5 and 49.5.
(b) 50–54, since 49.7 kg is recorded as 50 kg
(c) $40%$

Question 6 · Challenge — more justify and explain: correlation read from a scatter diagram, and a causation claim to reject

The scatter diagram shows the numbers of air-conditioners and the numbers of bottles of cold drinks sold by a shop in each of 12 months.

Scatter diagram of monthly air-conditioner sales against monthly cold drink sales for 12 months

(a) Describe the relationship between the two quantities. (1 mark)

(b) A reporter claims that buying air-conditioners makes people drink more cold drinks. Comment on the claim. (2 marks)

(c) In how many months were more than 300 air-conditioners sold but fewer than 5000 bottles of cold drinks sold? (2 marks)

Show answer

(a) Positive correlation: months with higher air-conditioner sales also tend to have higher cold drink sales.
(b) Correlation does not show cause. Both are likely driven by a third factor, hot weather.
(c) 2

Question 7 · Challenge — a genuine trap: counts and proportions point in opposite directions when two groups differ in size

The bar chart shows the numbers of the 80 boys and the 60 girls of a school year who joined each of four clubs. Each student joined exactly one club.

Comparative bar chart of club membership. Boys: Chess 12, Music 16, Sports 32, Science 20. Girls: Chess 9, Music 18, Sports 15, Science 18

(a) Someone claims, ‘More boys than girls joined the Science club, so a boy in this year is more likely than a girl to join the Science club.’ Is the claim correct? Explain your answer. (2 marks)

(b) The data for boys and for girls are shown in two separate pie charts. Find the difference between the angles of the two Music sectors. (2 marks)

(c) For which club are the angles of the two sectors equal? (2 marks)

Show answer

(a) Not correct. 20 of 80 boys is $25%$ , but 18 of 60 girls is $30%$ .
(b) $3 6^{\circ}$ ( $7 2^{\circ}$ for boys, $10 8^{\circ}$ for girls)
(c) Chess ( $15%$ of each group)

Question 8 · Challenge — modelled on HKDSE 2016 Paper 1 Q9 (a frequency table with unknowns) — changed: the unknowns are fixed by a percentage, and the answer is read off a frequency polygon the student must picture

The table shows the distribution of the times spent on homework last night by 45 students. The times are recorded to the nearest minute, and

40%

of the students spent more than 90.5 minutes.

Time spent (minutes)	1–30	31–60	61–90	91–120	121–150
Number of students	3	8	$k$	$m$	4

(a) Find the coordinates of the highest point of the frequency polygon of the distribution. (3 marks)

(b) Find the angle of the sector representing the class 91–120 in a pie chart of the distribution. (1 mark)

(c) Someone claims that at least one third of the students spent more than one and a half hours on homework. Do you agree? Explain your answer. (2 marks)

Show answer

(a) $(75.5, 16)$ (here $k = 16$ , $m = 14$ )
(b) $11 2^{\circ}$
(c) Agree. 18 of the 45 students ( $40%$ ) spent more than 90.5 minutes, which is more than one third.

Question 9 · Challenge — more justify and explain: two sources of bias, an estimate from a random sample, and a claim that treats an estimate as exact

A gaming website asks its visitors, ‘How many hours a day do you play video games?’ The results are used to describe the gaming habits of all teenagers in Hong Kong.

(a) Give two reasons why the results may not represent all teenagers in Hong Kong. (2 marks)

(b) A school instead chooses 80 of its 960 students at random. 28 of them play video games for at least 2 hours a day. Estimate the number of students in the school who play video games for at least 2 hours a day. (1 mark)

(c) A teacher says, ‘Because the sample was chosen at random, exactly that number of students in the school play for at least 2 hours a day.’ Comment on the statement. (2 marks)

Show answer

(a) Visitors to a gaming website play more than typical teenagers; responses are voluntary; visitors need not be teenagers or live in Hong Kong. (Any two.)
(b) 336
(c) Not correct. A random sample gives an estimate; a different random sample would usually give a slightly different number.

Question 10 · Challenge — a genuine trap: a strict inequality where the boundary case gives exactly 45°

The pie chart shows the distribution of the 120 members of a club among four interest groups A, B, C and D. New members then join, all of them in group C; no one leaves.

Pie chart of 120 members in four interest groups: A 90°, B 120°, C 60°, D 90°

(a) If 30 new members join, find the new angle of the sector for group C. (2 marks)

(b) Find the number of new members needed for the sectors of groups B and C to be equal. (2 marks)

(c) Find the least number of new members needed for the angle of the sector for group A to be less than $4 5^{\circ}$ . (2 marks)

Show answer

(a) $12 0^{\circ}$
(b) 20
(c) 121 (120 new members make the angle exactly $4 5^{\circ}$ )

Continue practising this topic: Try the Paper 1 long questions to build the basics, or move to Paper 2 multiple-choice — also available as Paper 2 Challenge MC.

Where this sits in the syllabus

Recent HKDSE Paper 1 questions give fewer named steps and more single unguided asks, often ending with a claim to check. With charts, the skill tested is judgement: whether a comparison should use counts or proportions, and whether a picture tells the truth about the numbers behind it.

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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