Practice › Data Handling › Quartiles, Percentiles and Cumulative Frequency

Quartiles, Percentiles and Cumulative Frequency: Paper 1 Challenge

The Challenge tier. These questions state a condition rather than a method: unknown frequencies that must place a quartile, ties that break the idea of an exact middle half, a negative scaling that swaps the quartiles, and claims to prove or disprove. Work the standard Paper 1 set first. Detailed solutions can be requested by email.

10 questions · about 90 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 — what a box-and-whisker diagram guarantees about counts; claims to test
Question 2 — an unknown datum whose position in the ordered list is not fixed; inter-quartile range as a function of x
Question 3 — a negative linear change of scale swaps the quartiles; median and inter-quartile range of the new data
Question 4 — conditions on where the median and quartiles fall, turned into inequalities on cumulative frequencies
Question 5 — unknown frequencies fixed by the inter-quartile range; mean equal to median
Question 6 — removing the greatest and least data: a proof about the inter-quartile range
Question 7 — two cumulative frequency polygons on one grid, read in both directions
Question 8 — tied data at the quartiles: how many data lie in the ‘middle half’
Question 9 — percentile from a cumulative frequency polygon, then a linear score adjustment read backwards
Question 10 — two unknown data fixed by the range and the inter-quartile range: are they determined?

Jump to Answer Key

Question 1 · Modelled on 2019 P2 Q29

The box-and-whisker diagram below shows the distribution of the scores (all integers) of 30 students in a quiz.

Box-and-whisker diagram of 30 quiz scores: 12, 20, 27, 35, 50

(a) Someone claims that at least 8 students scored 35 or above. Is the claim correct? Explain your answer. (2 marks)

(b) Someone claims that exactly 7 students scored less than 20. Is the claim correct? Explain your answer. (2 marks)

(c) Find the least possible number of students who scored at least 27. (1 mark)

Show answer

(a) Correct: the upper quartile is the 23rd score, so the 23rd to 30th scores (8 students) are at least 35. (b) Not correct: the lower quartile is the 8th score, which only shows that at most 7 students scored less than 20. For example, the 7th score may also be 20. (c) 15.

Question 2 · Modelled on 2022 P1 Q11

Consider the following eight numbers:

$3$ $7$ $8$ $12$ $15$ $18$ $20$ $x$

where $x$ is a real number.

(a) If $x > 20$ , find the inter-quartile range of the eight numbers. (1 mark)

(b) Find all the possible values of $x$ such that the inter-quartile range of the eight numbers is 10. (3 marks)

(c) Someone claims that the inter-quartile range of the eight numbers cannot be less than 9. Do you agree? Explain your answer. (2 marks)

Show answer

(a) 11.5. (b) $x = 6$ or $x = 17$ . (c) Agree. The least possible inter-quartile range is 9, reached when $8 \leq x \leq 15$ .

Question 3 · Modelled on 2017 P2 Q45

The box-and-whisker diagram below shows the distribution of the times taken (in seconds) by 40 runners in a race.

Box-and-whisker diagram of 40 race times: 50, 54, 57, 61, 70 seconds

Each runner is awarded $p$ points, where $p = 200 - 2 t$ and $t$ seconds is the time taken by the runner.

(a) Find the median and the inter-quartile range of the points awarded to the 40 runners. (3 marks)

(b) Someone claims that the upper quartile of the points is $200 - 2 (61)$ . Do you agree? Explain your answer. (2 marks)

Show answer

(a) Median 86 points, inter-quartile range 14 points. (b) No. Since $p$ decreases as $t$ increases, the order is reversed: the upper quartile of the points comes from the lower quartile of the times, $200 - 2 (54) = 92$ .

Question 4 · Modelled on 2022 P1 Q9

The table below shows the distribution of the distances (correct to the nearest km) travelled by 40 cyclists in a day, where $p$ and $q$ are positive integers.

Distance (km)	Frequency
1 – 10	$5$
11 – 20	$2 p$
21 – 30	$12$
31 – 40	$q$
41 – 50	$4$

It is given that the median of the distribution lies in the class 21 – 30 and the upper quartile lies in the class 31 – 40.

(a) Find the range of possible values of $p$ . (4 marks)

(b) If the lower quartile also lies in the class 11 – 20, find the number of possible values of $p$ . (2 marks)

Show answer

(a) $2 \leq p \leq 6$ . (b) 4.

Question 5 · Modelled on 2023 P1 Q11

The table below shows the distribution of the scores of 30 students in a quiz, where $m$ and $n$ are non-negative integers.

Score	$1$	$2$	$3$	$4$	$5$
Number of students	$4$	$m$	$7$	$n$	$3$

(a) It is given that the inter-quartile range of the distribution is 2. Find the range of possible values of $m$ . (4 marks)

(b) Find the value of $m$ such that the mean of the distribution is equal to its median. (2 marks)

Show answer

(a) $4 \leq m \leq 11$ . (b) $m = 7$ .

Question 6 · Modelled on 2022 P2 Q45

The weights (in kg) of 10 parcels are

$2$ $3$ $5$ $6$ $8$ $9$ $11$ $12$ $15$ $20$

(a) The heaviest parcel and the lightest parcel are removed. Find the inter-quartile range of the weights before and after the removal. (2 marks)

(b) Let $x_{1} \leq x_{2} \leq \dots \leq x_{10}$ be any 10 numbers. Prove that removing $x_{1}$ and $x_{10}$ cannot increase the inter-quartile range. (3 marks)

(c) Someone claims that removing the greatest and the least of 10 numbers always decreases the inter-quartile range. Do you agree? Explain your answer. (1 mark)

Show answer

(a) Before: 7 kg; after: 6 kg. (b) Before: $x_{8} - x_{3}$ . After: $\frac{x _{7} + x _{8}}{2} - \frac{x _{3} + x _{4}}{2}$ . The difference is $\frac{( x _{8} - x _{7} ) + ( x _{4} - x _{3} )}{2} \geq 0$ . (c) No. If $x_{3} = x_{4}$ and $x_{7} = x_{8}$ , the inter-quartile range is unchanged; for example 1, 2, 3, 3, 4, 5, 6, 6, 7, 8.

Question 7 · Modelled on 2016 P1 Q9

The cumulative frequency polygons below show the distributions of the scores of the 40 students of class A (solid line) and the 40 students of class B (dashed line) in a test.

Cumulative frequency polygons of the scores of class A (solid) and class B (dashed), 40 students each

(a) Which class has the more dispersed scores? Explain your answer. (2 marks)

(b) The teacher sets the passing score so that exactly 75% of the students of class A pass. Estimate the percentage of the students of class B who pass. (3 marks)

(c) Mary of class A scored 70. Someone claims that Mary’s score would be above the upper quartile of class B. Do you agree? Explain your answer. (1 mark)

Show answer

(a) Neither: both inter-quartile ranges are 20. (b) Passing score 50; about 85% of class B pass. (c) No. The upper quartile of class B is 80; a score of 70 is only the median of class B.

Question 8 · Modelled on 2021 P2 Q28

The stem-and-leaf diagram below shows the distribution of the ages of 16 members of a choir.

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

(a) Find the lower quartile and the upper quartile of the distribution. (2 marks)

(b) Find the number of members whose ages are not less than the lower quartile and not greater than the upper quartile. (1 mark)

(c) Someone claims that for any 16 numbers, exactly 8 of them are not less than the lower quartile and not greater than the upper quartile. Do you agree? Explain your answer. (2 marks)

Show answer

(a) Lower quartile 21, upper quartile 35. (b) 11. (c) No. The choir data are a counterexample: 11 of the 16 ages lie from 21 to 35 inclusive.

Question 9 · Modelled on 2022 P2 Q44

The cumulative frequency polygon below shows the distribution of the scores of 200 candidates in an examination.

Cumulative frequency polygon of the examination scores of 200 candidates

(a) Find the median and the inter-quartile range of the scores. (2 marks)

(b) An award is given to the candidates with the top 15% of the scores. Estimate the least score needed for the award. (2 marks)

(c) Each score $x$ is adjusted to $0.8 x + 20$ .

(i) Find the inter-quartile range of the adjusted scores.

(ii) Someone claims that more than 30% of the candidates have adjusted scores higher than 64. Do you agree? Explain your answer. (3 marks)

Show answer

(a) Median 50, inter-quartile range 20. (b) 70. (c) (i) 16 (ii) Agree. An adjusted score of 64 corresponds to an original score of 55; about 125 candidates scored below 55, so about 37.5% are above 64.

Question 10 · Modelled on 2024 P1 Q11

Consider the following eight integers:

$2$ $5$ $9$ $11$ $13$ $16$ $x$ $y$

where $2 \leq x < y$ . It is given that the range and the inter-quartile range of the eight integers are 20 and 9 respectively.

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

(b) Someone claims that $x$ can be uniquely determined. Do you agree? Explain your answer. (4 marks)

Show answer

(a) $y = 22$ . (b) No. Both $x = 6$ and $x = 16$ give range 20 and inter-quartile range 9.

Answer Key

(a) Correct: the upper quartile is the 23rd score, so the 23rd to 30th scores (8 students) are at least 35. (b) Not correct: the lower quartile is the 8th score, which only shows that at most 7 students scored less than 20. For example, the 7th score may also be 20. (c) 15.
(a) 11.5. (b) $x = 6$ or $x = 17$ . (c) Agree. The least possible inter-quartile range is 9, reached when $8 \leq x \leq 15$ .
(a) Median 86 points, inter-quartile range 14 points. (b) No. Since $p$ decreases as $t$ increases, the order is reversed: the upper quartile of the points comes from the lower quartile of the times, $200 - 2 (54) = 92$ .
(a) $2 \leq p \leq 6$ . (b) 4.
(a) $4 \leq m \leq 11$ . (b) $m = 7$ .
(a) Before: 7 kg; after: 6 kg. (b) Before: $x_{8} - x_{3}$ . After: $\frac{x _{7} + x _{8}}{2} - \frac{x _{3} + x _{4}}{2}$ . The difference is $\frac{( x _{8} - x _{7} ) + ( x _{4} - x _{3} )}{2} \geq 0$ . (c) No. If $x_{3} = x_{4}$ and $x_{7} = x_{8}$ , the inter-quartile range is unchanged; for example 1, 2, 3, 3, 4, 5, 6, 6, 7, 8.
(a) Neither: both inter-quartile ranges are 20. (b) Passing score 50; about 85% of class B pass. (c) No. The upper quartile of class B is 80; a score of 70 is only the median of class B.
(a) Lower quartile 21, upper quartile 35. (b) 11. (c) No. The choir data are a counterexample: 11 of the 16 ages lie from 21 to 35 inclusive.
(a) Median 50, inter-quartile range 20. (b) 70. (c) (i) 16 (ii) Agree. An adjusted score of 64 corresponds to an original score of 55; about 125 candidates scored below 55, so about 37.5% are above 64.
(a) $y = 22$ . (b) No. Both $x = 6$ and $x = 16$ give range 20 and inter-quartile range 9.

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

Where this sits in the syllabus

Everything here stays inside More about Statistics in the HKDSE Compulsory Part. Harder means less scaffolded, not out of syllabus: every question can be answered from the definition of the median and the quartiles, carefully applied.

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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