Practice › Data Handling › Measures of Central Tendency and Dispersion

Measures of Central Tendency and Dispersion — Paper 2 Challenge MC Questions

Eight harder multiple-choice questions. Each one hides a distractor that is the right number to the wrong question: a variance offered as a standard deviation, one case taken for every case, a negative multiplier treated as positive. Answers are on the page. Detailed solutions can be requested by email.

Challenge tier. These questions state a condition rather than naming a method, and several use ideas the standard set does not touch at all. Try the standard set first if you have not already.

8 questions · about 16 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 — mean and median both fixed; listing every admissible case before judging the mode, range and variance
Question 2 — the full range of an unknown frequency that keeps the median fixed, odd and even totals both checked
Question 3 — statistics of a group that slides with an integer parameter
Question 4 — standard deviation from a difference of scores and a difference of standard scores
Question 5 — mean and standard deviation hidden in two percentages; then an interval
Question 6 — removing data equal to the mean: the sum of squared deviations is unchanged but shared by fewer numbers
Question 7 — mean, range and variance under a transformation with a negative multiplier
Question 8 — combined means with the group sizes unknown; a ratio first, then a count

Question 1 · Modelled on 2022 P2 Q30 — every case, not one case

Consider the following positive integers:

24557 x x y

If both the mean and the median of the above positive integers are

5

, which of the following are true?

I. The mode of the above positive integers is $5$ .
II. The least possible range of the above positive integers is $5$ .
III. The greatest possible variance of the above positive integers is $6$ .

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

Show answer

D. II and III only

Detailed solutions can be requested by email.

Question 2 · Modelled on 2018 P1 Q11 — both ends of the range

The table below shows the distribution of a set of data, where

k

is a non-negative integer.

Datum Frequency 16 2 k 354854

If the median of the distribution is

3

, then

$2 \leq k \leq 10$ .
$3 \leq k \leq 10$ .
$2 \leq k \leq 11$ .
$3 \leq k \leq 11$ .

Show answer

A. $2 \leq k \leq 10$

Detailed solutions can be requested by email.

Question 3 · Modelled on 2023 P2 Q45

It is given that

n

is an integer. Consider the group of numbers

{n - 4, n - 1, n, n + 2, n + 8}

. Which of the following must be true?

I. The mean of the group exceeds its median by $1$ .
II. The standard deviation of the group does not depend on $n$ .
III. The range of the group is twice its standard deviation.

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

Show answer

A. I and II only

Detailed solutions can be requested by email.

Question 4 · Modelled on 2025 P2 Q44 — no mean given at all

In a test, the standard scores of Alan and Betty are

1.2

and

- 0.8

respectively, and Alan scores

10

marks more than Betty. Carmen scores

20

marks more than Alan. The standard score of Carmen is

$3.2$ .
$4.2$ .
$5.2$ .
$6.2$ .

Show answer

C. $5.2$

Detailed solutions can be requested by email.

Question 5 · New technique — the normal distribution read backwards

The scores of a large group of candidates in an examination are normally distributed. Half of the candidates score more than

60

marks, and about

16%

of the candidates score more than

68

marks. About what percentage of the candidates score between

44

marks and

60

marks?

$34%$
$47.5%$
$68%$
$95%$

Show answer

B. $47.5%$

Detailed solutions can be requested by email.

Question 6 · New technique — removing data at the mean

The mean and the standard deviation of ten numbers are

8

and

2

respectively. Two of the numbers, both equal to

8

, are removed. The variance of the remaining eight numbers is

$5$ .
$3.2$ .
$4$ .
$5$ .

Show answer

D. $5$

Detailed solutions can be requested by email.

Question 7 · Modelled on 2019 P2 Q45 — with a negative multiplier

The mean and the variance of a group of numbers

x_{1}, x_{2}, \dots, x_{n}

are

3

and

4

respectively. Let

y_{i} = 5 - 2 x_{i}

for

i = 1, 2, \dots, n

. Which of the following are true?

I. The mean of $y_{1}, y_{2}, \dots, y_{n}$ is $- 1$ .
II. The range of $y_{1}, y_{2}, \dots, y_{n}$ is $- 2$ times the range of $x_{1}, x_{2}, \dots, x_{n}$ .
III. The variance of $y_{1}, y_{2}, \dots, y_{n}$ is $16$ .

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

Show answer

B. I and III only

Detailed solutions can be requested by email.

Question 8 · Modelled on 2022 P2 Q29 — two conditions, two unknown group sizes

In a test, the mean score of the boys in a class is

60

marks, the mean score of the girls is

70

marks, and the mean score of the whole class is

66

marks. Later,

5

more boys join the class; the mean score of these

5

boys is

60

marks. The mean score of the whole class then becomes

65

marks. How many girls are there in the class?

$10$
$12$
$15$
$18$

Show answer

C. $15$

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 1 long questions.

Where this sits in the syllabus

This set sits alongside the standard Paper 2 multiple-choice questions on the same topic. The mathematics is the same; what changes is that each question rewards the second check, the one that catches a missing case or a statistic confused with its square.

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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