Practice › Data Handling › Quartiles, Percentiles and Cumulative Frequency

Quartiles, Percentiles and Cumulative Frequency: Paper 2 Challenge MC

The Challenge tier for Paper 2. Each question hides a trap: a scaling that reverses the order of the data, a median that need not be a datum, an inter-quartile range that stays constant over a whole interval, and statements that are true only for some data sets. Try the standard Paper 2 set first. Detailed solutions can be requested by email.

10 questions · about 25 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 — a negative change of scale reverses the order: which quartile becomes the upper quartile
Question 2 — what the quartiles of 20 integers guarantee about counts
Question 3 — adding one datum below the least and one above the greatest: range, median and inter-quartile range
Question 4 — percentage of data between two values, read from a cumulative frequency polygon
Question 5 — decreasing every datum below the median: which measures must stay the same
Question 6 — comparing two box-and-whisker diagrams: statements that must be true
Question 7 — how many new top scores are needed to move the upper quartile
Question 8 — statements about quartiles that hold for every data set
Question 9 — an inter-quartile range that is constant over an interval: how many solutions
Question 10 — a median that is not one of the data; statements from a frequency table

Jump to Answer Key

Question 1 · Modelled on 2017 P2 Q45

The box-and-whisker diagram below shows the distribution of a group of 24 numbers $x_{1}, x_{2}, \dots, x_{24}$ . Let $y_{k} = 60 - 2 x_{k}$ for $k = 1, 2, \dots, 24$ . Find the inter-quartile range and the upper quartile of $y_{1}, y_{2}, \dots, y_{24}$ .

Box-and-whisker diagram: 20, 26, 31, 38, 45

inter-quartile range $= 12$ , upper quartile $= 8$
inter-quartile range $= 24$ , upper quartile $= - 16$
inter-quartile range $= 24$ , upper quartile $= 8$
inter-quartile range $= - 24$ , upper quartile $= 8$

Show answer

C. The order reverses, so the upper quartile of $y$ comes from the lower quartile of $x$ : $60 - 2 (26) = 8$ . The inter-quartile range is $2 \times (38 - 26) = 24$ . The option $- 16 = 60 - 2 (38)$ is the lower quartile of $y$ .

Question 2 · Modelled on 2019 P2 Q29

The box-and-whisker diagram below shows the distribution of 20 integers. Which of the following must be true?

Box-and-whisker diagram of 20 integers: 5, 9, 14, 18, 30

I. At least 5 of the integers are not less than 18.
II. Exactly 10 of the integers are less than 14.
III. At most 5 of the integers are less than 9.

I and II only
I and III only
II and III only
I, II and III

Show answer

B. The upper quartile 18 is the average of the 15th and 16th integers, so the 16th to 20th are at least 18 (I). The lower quartile 9 is the average of the 5th and 6th, so the 6th is at least 9 and at most 5 integers are below 9 (III). If the 10th and 11th are both 14, fewer than 10 are below 14, so II can fail.

Question 3 · Modelled on 2023 P1 Q11

Consider the following 12 numbers:

$4$ $6$ $7$ $9$ $10$ $12$ $15$ $16$ $18$ $21$ $23$ $27$

The numbers 1 and 30 are now added to the group. Which of the following are true?

I. The range is increased.
II. The median is unchanged.
III. The inter-quartile range is unchanged.

I and II only
I and III only
II and III only
I, II and III

Show answer

A. The range goes from 23 to 29. The median stays $\frac{12 + 15}{2} = 13.5$ . But the quartiles move from 8 and 19.5 to 7 and 21, so the inter-quartile range goes from 11.5 to 14.

Question 4 · Modelled on 2016 P1 Q9

The cumulative frequency polygon below shows the distribution of the marks of 400 students in a test. Estimate the percentage of the students whose marks are between 30 and 70.

Cumulative frequency polygon of the marks of 400 students

$40%$
$50%$
$62.5%$
$70%$

Show answer

C. At 30 marks the cumulative frequency is 80, and at 70 marks it is 330. So about $\frac{330 - 80}{400} = 62.5%$ of the students scored between 30 and 70.

Question 5 · Modelled on 2022 P2 Q45

For a group of 40 numbers, each number that is less than the median is decreased by 5. Which of the following must be true?

I. The upper quartile is unchanged.
II. The median is unchanged.
III. The inter-quartile range is increased by 5.

I only
II only
I and III only
II and III only

Show answer

A. The 21st to 40th numbers are not less than the median, so they are unchanged and the upper quartile stays. If the 20th number is less than the median, it drops by 5 and the median changes. If the 10th and 11th numbers equal the median (ties), the lower quartile does not change.

Question 6 · 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. Which of the following must be true?

Box-and-whisker diagrams: class A 30, 45, 55, 68, 90; class B 20, 35, 42, 50, 66

I. At least half of the students of class A scored more than 50.
II. At least 25% of the students of class A scored more than every student of class B.
III. The scores of class B are more dispersed than those of class A.

I only
II only
I and II only
I, II and III

Show answer

C. Class A’s median 55 means its 21st to 40th scores are at least 55 (I). Its upper quartile 68 means its 31st to 40th scores are at least 68, above class B’s highest score 66 (II). Class B’s inter-quartile range is 15, less than class A’s 23 (III is false).

Question 7 · Modelled on 2023 P1 Q11

The table below shows the distribution of the scores of 28 students in a quiz.

Score	$1$	$2$	$3$	$4$	$5$
Number of students	$3$	$6$	$8$	$7$	$4$

Now $k$ more students, each scoring 5, join the group. Find the least value of $k$ such that the upper quartile of the distribution becomes 5.

Show answer

B. With $k = 4$ (32 scores) the upper quartile is the average of the 24th and 25th scores, 4 and 5, giving 4.5. With $k = 5$ (33 scores) it is the 25th score, which is 5.

Question 8 · Modelled on 2019 P2 Q29

Which of the following must be true for any group of numbers?

I. The lower quartile is not greater than the median, and the median is not greater than the upper quartile.
II. The inter-quartile range is not greater than the range.
III. If every number is multiplied by $- 1$ , the inter-quartile range is unchanged.

I only
I and II only
II and III only
I, II and III

Show answer

D. The lower half lies below the upper half, so $Q_{1} \leq$ median $\leq Q_{3}$ . Both quartiles lie between the least and greatest numbers, so $Q_{3} - Q_{1} \leq$ range. Multiplying by $- 1$ turns the quartiles into $- Q_{3}$ and $- Q_{1}$ , and $(- Q_{1}) - (- Q_{3}) = Q_{3} - Q_{1}$ .

Question 9 · Modelled on 2022 P1 Q11

Consider the following eight numbers:

$7$ $9$ $12$ $15$ $18$ $21$ $25$ $x$

where $x$ is a real number. If the inter-quartile range of the eight numbers is 9, how many possible values of $x$ are there?

$1$
$2$
$3$
infinitely many

Show answer

D. For $12 \leq x \leq 18$ , the lower quartile is $\frac{9 + 12}{2} = 10.5$ and the upper quartile is $\frac{18 + 21}{2} = 19.5$ , so the inter-quartile range is 9 for every such $x$ .

Question 10 · Modelled on 2019 P2 Q30

The table below shows the distribution of the numbers of cars owned by 24 families.

Number of cars	$0$	$1$	$3$	$4$
Number of families	$5$	$7$	$8$	$4$

Which of the following are true?

I. The median of the distribution is 2.
II. The inter-quartile range of the distribution is 3.
III. Exactly half of the families own fewer than 2 cars.

I only
I and III only
II and III only
I, II and III

Show answer

B. The 12th and 13th data are 1 and 3, so the median is 2 even though no family owns 2 cars. The quartiles are 1 and 3, so the inter-quartile range is 2. Families with fewer than 2 cars: $5 + 7 = 12$ , exactly half.

Answer Key

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

Where this sits in the syllabus

These questions test the definitions behind More about Statistics rather than new techniques. A statement that must be true has to hold for every data set that fits the information; one counterexample is enough to rule it out.

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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