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Quartiles, Percentiles and Cumulative Frequency: Paper 1 long questions

Ten long questions on the lower and upper quartiles, the inter-quartile range, box-and-whisker diagrams, percentiles and cumulative frequency tables and polygons. Each question opens gently and ends with a step that asks you to explain, not just compute. Detailed solutions can be requested by email.

10 questions · about 80 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.

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Question 1 — median and inter-quartile range from a frequency table; effect of two new data
Question 2 — box-and-whisker diagram with unknown lower quartile and maximum; range after new data
Question 3 — frequency and cumulative frequency tables with unknowns; median class; probability
Question 4 — reading the median, quartiles and a percentile from a cumulative frequency polygon
Question 5 — stem-and-leaf diagram with unknown leaf; ratio of range to inter-quartile range
Question 6 — stem-and-leaf diagram with two unknown leaves; condition that keeps the median unchanged
Question 7 — comparing two box-and-whisker diagrams; what a quartile guarantees about counts
Question 8 — frequency table with two unknown frequencies; median condition; greatest inter-quartile range
Question 9 — stem-and-leaf diagram with unknown leaves from the range and the mean; comparing with a box-and-whisker diagram
Question 10 — cumulative frequency polygon against a box-and-whisker diagram; percentile; comparing two schools

Jump to Answer Key

Question 1 · Modelled on 2023 P1 Q11

The table below shows the distribution of the numbers of books read in a month by a group of 24 students.

Number of books read	$0$	$1$	$2$	$3$	$4$	$5$
Number of students	$3$	$5$	$7$	$4$	$3$	$2$

(a) Find the median and the inter-quartile range of the distribution. (3 marks)

(b) Two more students now join the group. One of them read 0 books and the other read 5 books in the month. Is there any change in the inter-quartile range of the distribution? Explain your answer. (2 marks)

Show answer

(a) Median $= 2$ , inter-quartile range $= 2$ . (b) No. The lower quartile is still $1$ and the upper quartile is still $3$ , so the inter-quartile range remains $2$ .

Question 2 · Modelled on 2018 P1 Q10

The box-and-whisker diagram below shows the distribution of the ages of the members of a hiking club. It is given that the range and the inter-quartile range of the distribution are $40$ and $17$ respectively.

Box-and-whisker diagram of ages: minimum 18, lower quartile a, median 35, upper quartile 44, maximum b

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

(b) Two new members, aged 16 and 57, now join the club. Someone claims that the range of the distribution becomes 41. Do you agree? Explain your answer. (2 marks)

Show answer

(a) $a = 27$ , $b = 58$ . (b) No. The greatest age is still 58, so the new range is $58 - 16 = 42$ , not 41.

Question 3 · Modelled on 2022 P1 Q9

The frequency distribution table and the cumulative frequency distribution table below show the distribution of the times taken (correct to the nearest minute) by 40 students to travel to school.

Time taken (minutes)	Frequency
1 – 10	$4$
11 – 20	$a$
21 – 30	$12$
31 – 40	$b$
41 – 50	$6$

Time taken less than (minutes)	Cumulative frequency
$10.5$	$4$
$20.5$	$13$
$30.5$	$x$
$40.5$	$34$
$50.5$	$40$

(a) Find $a$ , $b$ and $x$ . (3 marks)

(b) Write down the class containing the median of the distribution. (1 mark)

(c) If a student is randomly selected from the 40 students, find the probability that the time taken by the selected student is less than 30.5 minutes but not less than 10.5 minutes. (1 mark)

Show answer

(a) $a = 9$ , $b = 9$ , $x = 25$ . (b) 21 – 30 minutes. (c) $\frac{21}{40}$ .

Question 4 · Modelled on 2016 P1 Q9

The cumulative frequency polygon below shows the distribution of the marks obtained by 80 candidates in a test.

Cumulative frequency polygon of marks of 80 candidates

(a) Find the median and the inter-quartile range of the distribution. (3 marks)

(b) The passing mark of the test is 45. Estimate the number of candidates who failed the test. (2 marks)

(c) The candidates with the top 10% of the marks are awarded grade A. Estimate the least mark needed for grade A. (1 mark)

Show answer

(a) Median $= 60$ marks, inter-quartile range $= 20$ marks. (b) 14. (c) 80 marks.

Question 5 · Modelled on 2026 P1 Q10

The stem-and-leaf diagram below shows the distribution of the numbers of minutes spent on homework in a day by 16 students.

Stem (tens)	Leaf (units)
$1$	$1$ $5$ $8$
$2$	$0$ $3$ $3$ $7$
$3$	$1$ $4$ $6$ $8$
$4$	$0$ $w$ $5$
$5$	$1$ $9$

The ratio of the range to the inter-quartile range of the distribution is $12 : 5$ .

(a) Find $w$ . (3 marks)

(b) (i) Find the median of the distribution.

(ii) If a student is randomly selected from the 16 students, find the probability that the number of minutes spent by the selected student exceeds the upper quartile of the distribution. (2 marks)

Show answer

(a) $w = 3$ . (b) (i) 32.5 minutes (ii) $\frac{1}{4}$ .

Question 6 · Modelled on 2022 P1 Q11

The stem-and-leaf diagram below shows the distribution of the waiting times (in minutes) of 20 customers in a restaurant.

Stem (tens)	Leaf (units)
$0$	$5$ $7$ $8$
$1$	$2$ $3$ $a$ $6$ $9$
$2$	$0$ $1$ $4$ $4$ $7$ $8$
$3$	$b$ $3$ $5$ $9$
$4$	$1$ $6$

It is given that the upper quartile and the inter-quartile range of the distribution are 33 minutes and 19 minutes respectively.

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

(b) The waiting times of two more customers are now recorded. One of them waited 10 minutes and the other waited $x$ minutes. It is found that the median of the distribution remains unchanged. Find the least possible value of $x$ . (2 marks)

Show answer

(a) $a = 5$ , $b = 3$ . (b) 24.

Question 7 · Modelled on 2025 P1 Q12

The box-and-whisker diagrams below show the distributions of the scores of the 40 students of class A and the 40 students of class B in a test.

Box-and-whisker diagrams: class A 32, 48, 56, 66, 90; class B 20, 52, 60, 64, 82

(a) Find the inter-quartile range of the scores of each class. (2 marks)

(b) By comparing the inter-quartile ranges, determine which class has the less dispersed scores. (2 marks)

(c) The passing score of the test is 52. The teacher claims that at least 30 students of class B passed the test. Is the claim correct? Explain your answer. (2 marks)

Show answer

(a) Class A: 18; class B: 12. (b) Class B. (c) Yes. The lower quartile of class B is the average of the 10th and 11th scores and equals 52, so the 11th to 40th scores (30 students) are all at least 52.

Question 8 · Modelled on 2025 P1 Q9

The table below shows the distribution of the numbers of pets kept by 24 households, where $k$ and $h$ are positive integers.

Number of pets	$0$	$1$	$2$	$3$	$4$
Number of households	$3$	$k$	$8$	$h$	$2$

It is given that the median of the distribution is 2.

(a) Find the least possible value and the greatest possible value of $k$ . (3 marks)

(b) Find the greatest possible inter-quartile range of the distribution. (3 marks)

Show answer

(a) Least $k = 2$ , greatest $k = 8$ . (b) 2.

Question 9 · Modelled on 2025 P1 Q12

A farmer grows 12 plants with a new fertiliser. The stem-and-leaf diagram below shows the distribution of the heights (in cm) of the plants after 4 weeks.

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

It is given that the range and the mean of the distribution are 30 cm and 37 cm respectively.

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

The box-and-whisker diagram below shows the distribution of the heights (in cm) of the same 12 plants at the start.

Box-and-whisker diagram of heights at the start: 12, 18, 23, 27, 35

(b) (i) Find the increase in the median height of the plants over the 4 weeks.

(ii) Are the heights of the plants after 4 weeks less dispersed than those at the start? Explain your answer. (3 marks)

Show answer

(a) $a = 6$ , $b = 4$ . (b) (i) 13.5 cm (ii) No. The inter-quartile range after 4 weeks is 15 cm, greater than 9 cm at the start.

Question 10 · Modelled on 2016 P1 Q9 and 2025 P1 Q12

The cumulative frequency polygon below shows the distribution of the heights of 200 students in school X.

Cumulative frequency polygon of heights of 200 students in school X

(a) Find the median and the inter-quartile range of the heights of the students in school X. (3 marks)

(b) Estimate the 90th percentile of the heights of the students in school X. (2 marks)

The box-and-whisker diagram below shows the distribution of the heights of 160 students in school Y.

Box-and-whisker diagram of heights in school Y: 145, 156, 164, 168, 185

(c) (i) Which school has the more dispersed heights? Explain your answer.

(ii) Someone claims that the percentage of students taller than 168 cm in school Y is greater than that in school X. Do you agree? Explain your answer. (3 marks)

Show answer

(a) Median $= 165$ cm, inter-quartile range $= 10$ cm. (b) 177.5 cm. (c) (i) School Y: its inter-quartile range is 12 cm, greater than 10 cm. (ii) No. In school X about 35% of students are taller than 168 cm; in school Y at most 25% are.

Answer Key

(a) Median $= 2$ , inter-quartile range $= 2$ . (b) No. The lower quartile is still $1$ and the upper quartile is still $3$ , so the inter-quartile range remains $2$ .
(a) $a = 27$ , $b = 58$ . (b) No. The greatest age is still 58, so the new range is $58 - 16 = 42$ , not 41.
(a) $a = 9$ , $b = 9$ , $x = 25$ . (b) 21 – 30 minutes. (c) $\frac{21}{40}$ .
(a) Median $= 60$ marks, inter-quartile range $= 20$ marks. (b) 14. (c) 80 marks.
(a) $w = 3$ . (b) (i) 32.5 minutes (ii) $\frac{1}{4}$ .
(a) $a = 5$ , $b = 3$ . (b) 24.
(a) Class A: 18; class B: 12. (b) Class B. (c) Yes. The lower quartile of class B is the average of the 10th and 11th scores and equals 52, so the 11th to 40th scores (30 students) are all at least 52.
(a) Least $k = 2$ , greatest $k = 8$ . (b) 2.
(a) $a = 6$ , $b = 4$ . (b) (i) 13.5 cm (ii) No. The inter-quartile range after 4 weeks is 15 cm, greater than 9 cm at the start.
(a) Median $= 165$ cm, inter-quartile range $= 10$ cm. (b) 177.5 cm. (c) (i) School Y: its inter-quartile range is 12 cm, greater than 10 cm. (ii) No. In school X about 35% of students are taller than 168 cm; in school Y at most 25% are.

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

Quartiles and cumulative frequency belong to More about Statistics in the HKDSE Compulsory Part. Paper 1 usually asks for them in Section A: a stem-and-leaf diagram, a frequency table or a box-and-whisker diagram with one or two unknowns, then a short claim to test. Most marks are lost on the position of a quartile, not on the arithmetic.

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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