Measures of Central Tendencies: Mathematics Tingkatan 2 questions

Section A: objective questions

12 marks

Choose the correct answer. Each question carries 1 mark.

Which statement describes the mode of a set of data?

The value that appears most often
The value in the middle after the data is arranged in order
The sum of the values divided by the number of values
The largest value minus the smallest value

Show answerHide answer

The mode is the value with the highest frequency. The second option is the median, the third is the mean and the last is the range.

For the teacher: Remembering, low difficulty, standard 12.1.1

Textbook p. 247

A canteen records the drinks bought by 8 students: Teh tarik, Milo, Teh tarik, Kopi, Milo, Teh tarik, Kopi, Milo. State the mode.

Teh tarik
Teh tarik and Milo
Milo
Kopi

Show answerHide answer

Teh tarik appears 3 times, Milo 3 times and Kopi 2 times. The highest frequency is shared, so there are two modes.

For the teacher: Understanding, low difficulty, standard 12.1.1

Textbook p. 247

Find the median of 12, 7, 9, 15, 8, 20, 11.

Show answerHide answer

Arrange in order: 7, 8, 9, 11, 12, 15, 20. There are 7 values, so the median is the 4th value, 11. 15 is the middle of the list as written, before ordering.

For the teacher: Applying, low difficulty, standard 12.1.1

Textbook p. 249

Find the median of 14, 9, 11, 16, 9, 12, 18, 10.

11
12
12.5
11.5

Show answerHide answer

Ordered: 9, 9, 10, 11, 12, 14, 16, 18. There are 8 values, so the median is the average of the 4th and 5th: (11 + 12) ÷ 2 = 11.5.

For the teacher: Applying, low difficulty, standard 12.1.1

Textbook p. 249

Diagram 1 shows the number of goals scored in 20 football matches. Find the mean number of goals per match.

Bar chart of goals scored in 20 matches. 0 goals: 4 matches, 1 goal: 6 matches, 2 goals: 7 matches, 3 goals: 3 matches — Diagram 1

1.45
1.5
1.75
2.0

Show answerHide answer

Total goals = 0 × 4 + 1 × 6 + 2 × 7 + 3 × 3 = 29. Mean = 29 ÷ 20 = 1.45. 1.5 comes from (0 + 1 + 2 + 3) ÷ 4, which ignores the frequencies.

For the teacher: Applying, medium difficulty, standard 12.1.1

Textbook p. 253

Which value is an extreme value in this set of data: 21, 23, 24, 25, 26, 4?

Show answerHide answer

4 is much smaller than all the other values, which lie between 21 and 26.

For the teacher: Understanding, low difficulty, standard 12.1.1

Textbook p. 253

The data 4, 5, 5, 6, 8 has its largest value changed from 8 to 28. Which statement is correct?

The mean, median and mode all change
The median changes but the mean and mode stay the same
The mode changes but the mean and median stay the same
The mean changes but the median and mode stay the same

Show answerHide answer

Mean: 28 ÷ 5 = 5.6 changes to 48 ÷ 5 = 9.6. Median stays 5 and mode stays 5. An extreme value affects the mean but not the median or mode.

For the teacher: Analysing, medium difficulty, standard 12.1.1

Textbook p. 254

A set of data has mean 8, median 7 and mode 6. Every value is increased by 5. What are the new mean, median and mode?

40, 35, 30
13, 7, 6
13, 12, 6
13, 12, 11

Show answerHide answer

A uniform change of +5 changes all three measures by +5: 8 + 5 = 13, 7 + 5 = 12, 6 + 5 = 11. The first option multiplies by 5.

For the teacher: Applying, medium difficulty, standard 12.1.2

Textbook p. 255

A class in a grouped frequency table is 40–44. What is its midpoint?

Show answerHide answer

Midpoint = (40 + 44) ÷ 2 = 42. 84 is the sum, which has not been divided by 2.

For the teacher: Understanding, low difficulty, standard 12.1.4

Textbook p. 260

Marks	10–19	20–29	30–39	40–49	50–59
Frequency	3	7	12	6	2

The table shows the marks of 30 students. What is the modal class?

20–29
30–39
40–49
12

Show answerHide answer

The modal class is the class with the highest frequency, 12, so the modal class is 30–39. 12 is the frequency, not the class.

For the teacher: Applying, medium difficulty, standard 12.1.4

Textbook p. 259

In which situation is the mode the most appropriate measure of central tendency?

The marks of 30 students in a test
The monthly pay of 6 workers where one earns far more than the rest
The favourite sport chosen by 40 students
The heights of 20 saplings

Show answerHide answer

Favourite sport is category data, so only the mode makes sense. The pay data has an extreme value (use the median). Marks and heights have no extreme value, so the mean is suitable.

For the teacher: Evaluating, medium difficulty, standard 12.1.5

Textbook p. 262

Score	0	1	2	3
Frequency	4	x	5	3

The mean score is 1.5. Find the value of x.

Show answerHide answer

(0 + x + 10 + 9) ÷ (12 + x) = 1.5, so x + 19 = 18 + 1.5x. Then 1 = 0.5x and x = 2. Check: 21 ÷ 14 = 1.5.

For the teacher: Applying, high difficulty, standard 12.1.6

Textbook p. 265

Section B: structured questions

28 marks

Answer all questions. Show your working.

Books borrowed	1	2	3	4
Number of students	4	8	6	6

The table shows the number of books borrowed by 24 students in a week.

[2 marks] (a)Calculate the median.

Show answerHide answer

n = 24, so the median is the average of the 12th and 13th values [1]1
12th = 2 (positions 5–12), 13th = 3; (2 + 3) ÷ 2 = 2.5 [1]1

Tick each point you got.

[2 marks] (b)Calculate the mean, correct to 2 decimal places.

Show answerHide answer

Sum = 1 × 4 + 2 × 8 + 3 × 6 + 4 × 6 = 62 [1]1
62 ÷ 24 = 2.58 [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, standard 12.1.1

Textbook p. 250

Diagram 2 shows the number of books read in a month by 20 students.

Dot plot of books read by 20 students. 0 books: 2 dots, 1: 5 dots, 2: 6 dots, 3: 4 dots, 4: 2 dots, 5: 1 dot — Diagram 2

[1 mark] (a)State the mode.

Show answerHide answer

2 books (highest number of dots, 6) [1]1

Tick each point you got.

[2 marks] (b)Calculate the mean number of books read.

Show answerHide answer

0 × 2 + 1 × 5 + 2 × 6 + 3 × 4 + 4 × 2 + 5 × 1 = 42 [1]1
42 ÷ 20 = 2.1 books [1]1

Tick each point you got.

[2 marks] (c)Find the median.

Show answerHide answer

n = 20: average of the 10th and 11th values [1]1
Both are 2 (positions 8–13), so the median = 2 [1]1

Cumulative frequencies: 2, 7, 13, so positions 8 to 13 are 2.

Tick each point you got.

For the teacher: Applying, low difficulty, standard 12.1.6

Textbook p. 264

The weekly pocket money, in RM, of 5 students is 6, 7, 7, 8, 27.

[2 marks] (a)Calculate the mean and the median.

Show answerHide answer

Mean = 55 ÷ 5 = RM11 [1]1
Median = RM7 [1]1

Tick each point you got.

[1 mark] (b)Which measure, the mean or the median, better represents the pocket money of these students? Give a reason.

Show answerHide answer

Median (RM7), because the mean RM11 is higher than four of the five students receive, as it is pulled up by the extreme value RM27 [1]1

Tick each point you got.

For the teacher: Analysing, low difficulty, standard 12.1.5

Textbook p. 262

The scores of 5 students in a quiz are 2, 4, 4, 6, 9.

[3 marks] (a)Calculate the mean, median and mode.

Show answerHide answer

Mean = 25 ÷ 5 = 5 [1]1
Median = 4 [1]1
Mode = 4 [1]1

Tick each point you got.

[2 marks] (b)Each score is multiplied by 3. State the new mean, median and mode.

Show answerHide answer

Mean = 15 [1]1
Median = 12 and mode = 12 [1]1

A uniform change of × 3 multiplies all three measures by 3.

Tick each point you got.

For the teacher: Understanding, low difficulty, standard 12.1.2

Textbook p. 255

Time (minutes)	1–5	6–10	11–15	16–20	21–25
Frequency	3	6	8	5	2

The table shows the time taken by 24 students to walk to school.

[1 mark] (a)State the modal class.

Show answerHide answer

11–15 (highest frequency 8) [1]1

Tick each point you got.

[3 marks] (b)Calculate the mean time, correct to 1 decimal place.

Show answerHide answer

Midpoints 3, 8, 13, 18, 23 [1]1
Σfx = 9 + 48 + 104 + 90 + 46 = 297 [1]1
297 ÷ 24 = 12.4 minutes [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 12.1.4

Textbook p. 260

A teacher chooses one archer for a competition. Their scores in five rounds are:
Aini: 70, 72, 74, 76, 78
Balqis: 55, 90, 62, 91, 72

[2 marks] (a)Calculate the mean score of each archer.

Show answerHide answer

Aini: 370 ÷ 5 = 74 [1]1
Balqis: 370 ÷ 5 = 74 [1]1

Tick each point you got.

[1 mark] (b)Calculate the range of each archer's scores.

Show answerHide answer

Aini: 78 − 70 = 8; Balqis: 91 − 55 = 36 [1]1

Tick each point you got.

[1 mark] (c)Which archer should be chosen? Give a reason.

Show answerHide answer

Aini: the means are equal but her scores have a much smaller range (8 against 36), so she is more consistent [1]1

Tick each point you got.

For the teacher: Evaluating, medium difficulty, standard 12.1.7

Textbook p. 266

Write a set of five whole numbers that has a mode of 4, a median of 5 and a mean of 6.

[3 marks] (a)Write your five numbers and show that they satisfy all three conditions.

Show answerHide answer

4, 4, 5, 8, 9 (other correct sets are accepted)

Sum of the five numbers = 5 × 6 = 30 [1]1
Middle (3rd) value in order is 5 and 4 appears more often than any other value [1]1
A correct set, e.g. 4, 4, 5, 8, 9: mean = 30 ÷ 5 = 6, median = 5, mode = 4 [1]1

Tick each point you got.

For the teacher: Creating, high difficulty, standard 12.1.1

Textbook p. 249

Section A: objective questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Section B: structured questions

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19