Practice › Data Handling › Measures of Central Tendency and Dispersion

Measures of Central Tendency and Dispersion — Paper 1 Challenge Questions

Eight harder long questions on the same topic. Most of them state a condition, such as “the mean equals the median”, rather than telling you which statistic to calculate, and several have a second answer that is easy to miss. Final 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 65 minutes, 44 marks · 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.

Jump to

Question 1 — a frequency pinned down by the median and the mode together; a mean that can never equal the median
Question 2 — two unknown leaves from “the mean equals the median” and a range; which averages survive a correction
Question 3 — recovering the mean and standard deviation from two standard scores; a boundary-case claim; a shift that leaves standard scores alone
Question 4 — the range and extreme standard scores give the standard deviation; how removing one datum moves the mean
Question 5 — a linear transformation recovered from its mean and standard deviation; the second solution with a negative multiplier
Question 6 — mean and standard deviation from two percentages of a normal distribution; an asymmetric interval
Question 7 — working with the sum of squared deviations; keeping two statistics fixed at once; proving an impossibility
Question 8 — mean equals median with the unknown in three different positions; comparing standard deviations

Question 1 · Modelled on 2025 P1 Q9 — two averages fixed at once · 5 marks

The table below shows the distribution of the numbers of pens owned by some students, where

s

is a non-negative integer.

Number of pens Number of students 3447 5 s 6675

The median and the mode of the distribution are both

5

(a) Find the least possible value of $s$ . (2 marks)

(b) Someone claims that for a suitable value of $s$ , the mean of the distribution is equal to its median. Do you agree? Explain your answer. (3 marks)

Show answer

(a) $8$

(b) Disagree. The mean is $5 + \frac{1}{22 + s}$ , which is greater than $5$ for every value of $s$ .

Final answers only. Detailed solutions can be requested by email.

Question 2 · Modelled on 2023 P1 Q9 — conditions instead of given values · 5 marks

The stem-and-leaf diagram below shows the distribution of the scores of

11

players in a game, where

a

and

b

are digits.

Stem (tens) 1234 Leaf (units) 57 a 689 2249 b

The range of the distribution is

30

, and the mean of the distribution is equal to its median.

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

(b) It is later found that the score $34$ was wrongly recorded. The correct score is $43$ . Determine which of the mean, the median, the mode and the range of the distribution are changed, and find the new mean. (2 marks)

Show answer

(a) $a = 2$ , $b = 5$

(b) Only the mean changes. The new mean is $\frac{328}{11} \approx 29.8$ .

Final answers only. Detailed solutions can be requested by email.

Question 3 · Modelled on 2019 P2 Q44 — mean and standard deviation recovered from two students · 6 marks

In a test, Ada scores

62

marks and her standard score is

- 0.5

. Ben scores

86

marks and his standard score is

1.5

(a) Find the mean and the standard deviation of the scores in the test. (2 marks)

(b) The standard score of Carl in the test is $2.25$ . Carl claims that he scores more than $95$ marks. Do you agree? Explain your answer. (2 marks)

(c) The teacher adds $5$ marks to the score of every student in the test. Does Ada's standard score change? Explain your answer. (2 marks)

Show answer

(a) Mean $= 68$ marks, standard deviation $= 12$ marks

(b) Disagree. Carl scores exactly $68 + 2.25 \times 12 = 95$ marks.

(c) No. The mean rises by $5$ marks and the standard deviation is unchanged, so her standard score stays $- 0.5$ .

Final answers only. Detailed solutions can be requested by email.

Question 4 · Modelled on 2016 P1 Q16 — the range read through standard scores · 5 marks

In a test, the range of the scores of a class of students is

60

marks. The lowest score is

30

marks and its standard score is

- 2

. The standard score of the highest score is

3

(a) Find the mean and the standard deviation of the scores. (3 marks)

(b) A student who scores $60$ marks leaves the class. Someone claims that the mean score of the remaining students is higher than the mean found in (a). Do you agree? Explain your answer. (2 marks)

Show answer

(a) Mean $= 54$ marks, standard deviation $= 12$ marks

(b) Disagree. The student removed scored above the mean ( $60 > 54$ ), so the mean of the rest falls below $54$ marks.

Final answers only. Detailed solutions can be requested by email.

Question 5 · Modelled on 2022 P2 Q44 — the negative multiplier the standard set avoids · 5 marks

The mean and the standard deviation of the ten numbers

x_{1}, x_{2}, \dots, x_{10}

are

12

and

3

respectively. Let

y_{i} = a x_{i} + b

for

i = 1, 2, \dots, 10

, where

a

and

b

are constants and

a \neq = 0

. The mean and the standard deviation of

y_{1}, y_{2}, \dots, y_{10}

are

30

and

6

respectively.

(a) Find all the possible pairs of values of $a$ and $b$ . (3 marks)

(b) It is further given that the median of $x_{1}, x_{2}, \dots, x_{10}$ is $14$ and the median of $y_{1}, y_{2}, \dots, y_{10}$ is $34$ . Find $a$ and $b$ . (2 marks)

Show answer

(a) $a = 2, b = 6$ or $a = - 2, b = 54$

(b) $a = 2, b = 6$

Final answers only. Detailed solutions can be requested by email.

Question 6 · New technique — the normal distribution read backwards · 6 marks

The weights of

1000

bags of rice are normally distributed. About

16%

of the bags are heavier than

520 g

and about

2.5%

of the bags are lighter than

490 g

You may use: for a normal distribution, about 68%, 95% and 99.7% of the data lie within 1, 2 and 3 standard deviations of the mean respectively.

(a) Find the mean and the standard deviation of the weights of the bags. (4 marks)

(b) Estimate the number of bags whose weights are between $500 g$ and $530 g$ . (2 marks)

Show answer

(a) Mean $= 510 g$ , standard deviation $= 10 g$

(b) About $815$

Final answers only. Detailed solutions can be requested by email.

Question 7 · New technique — adding data while keeping the mean and standard deviation fixed · 6 marks

The mean and the standard deviation of nine numbers are

20

and

4

respectively.

(a) Two numbers $p$ and $q$ , where $p < q$ , are added to the nine numbers. The mean and the standard deviation of the eleven numbers are still $20$ and $4$ respectively. Find $p$ and $q$ . (4 marks)

(b) Someone claims that it is impossible to add just one number to the original nine numbers so that both the mean and the standard deviation are unchanged. Do you agree? Explain your answer. (2 marks)

Show answer

(a) $p = 16$ , $q = 24$

(b) Agree. To keep the mean $20$ , the added number must be $20$ ; the standard deviation then becomes $14.4 \approx 3.79$ , not $4$ .

Final answers only. Detailed solutions can be requested by email.

Question 8 · Modelled on 2023 P2 Q45 — a group of numbers with an unknown member · 6 marks

Consider the five numbers

2, 5, 7, 9

and

k

, where

k

is a real number. The mean of the five numbers is equal to their median.

(a) Find all the possible values of $k$ . (4 marks)

(b) For which of the values of $k$ in (a) is the standard deviation of the five numbers the least? (2 marks)

Show answer

(a) $k = 2$ , $k = \frac{23}{4}$ or $k = 12$

(b) $k = \frac{23}{4}$

Final answers only. Detailed solutions can be requested by email.

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

Where this sits in the syllabus

This set sits alongside the standard Paper 1 long questions on the same topic. Where the standard set names the statistic to find, these questions give a condition and leave the method to you, which is closer to how recent HKDSE papers set their harder statistics questions. Quartiles and cumulative frequency are left to the separate topic More about Statistics.

Full coverage of this topic, and the rest of the course, is on the Mathematics page. More sets are listed on the practice index.

Start with a free discovery session

No cost, no obligation. We look at where your child actually is, and show you what a term of real progress would look like.

Book a free discovery session See the programmes