Chapter 8 II Statistics III ENRICH

8/3/2019 Chapter 8 II Statistics III ENRICH

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Enrichment

CHAPTER 8: STATISTICS III

8.1 Understand and use the concept of class interval.

1. Data obtained from the measurement of certain quantities can be grouped and arranged

into several classes. The range of each class is called the class interval.

2. The lower limit is the lowest value, whereas the upper limit is the highest value of the

class interval.

3. Lower boundary of a class interval

=2

1 × (lower limit of the class + upper limit of the previous class interval)

4. Upper boundary of a class interval

= 2

1 ×(upper limit of the class + lower limit of the next class interval)

5. Size of class interval = (Upper boundary – Lower boundary )

8.2 Understand and use the concept of mode and mean of grouped data

1. Modal class is the class with the highest frequency.

2. Midpoint of class =2

1 × ( Lower class limit + Upper class limit )

Mean =Sum of the values of (midpoint frequency) of all the classes

Sum of frequencies of all the classes

×

or x =∑∑

f

fx

8.3 Histogram

Histogram with class intervals of the same size – The frequency for every class interval is

represented by a rectangle of the same width and its height proportional to the frequency.

Steps to draw a histogram:

1. Determine the upper and the lower boundaries for every class interval.

2. By using the given scale for the x-axis, mark the lower and upper

boundaries. Note that the upper boundary of the first class interval is thelower boundary of the second class interval and so on.

3 By using the given scale for the y-axis, mark the frequencies.

4. Draw rectangles by plotting the height which corresponds to the frequency

for every class interval.

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8. 4 Frequency Polygon

A frequency polygon is a graph obtained by joining all the class marks (or mid points of

all the class intervals) at the tips of all the rectangles in a histogram.

Steps to draw a frequency polygon:

1. Add an extra class interval with frequency 0 before the first class and after the last class.

2. Calculate the midpoint or the class mark for every class interval.

3. By using the given scale for the x-axis, mark the midpoint of the class on the x-axis.

4. By using the given scale for the y-axis, mark the frequency on the y-axis.

5. Plot the corresponding frequency against the midpoint (class mark).

6. Join all the points by using a ruler.

7. Both ends of the frequency polygon must be on the x-axis.

8. 5 Ogive

The cumulative frequency for a class interval is the sum of its frequency and the

frequencies of all the class intervals before it.

An ogive is also known as the cumulative frequency curve.

Steps to draw an ogive:

1. Add one class interval with frequency 0 before the first class.

2. Calculate the upper boundaries for all the class intervals.3. Calculate the cumulative frequency for every class interval.

3. By using the given scale for the x-axis, mark the upper boundary.

4. By using the given scale for the y-axis, mark the cumulative frequency.

5. Plot the corresponding cumulative frequency against the upper boundary.6. Join all the points by drawing a smooth curve passing through them.

8.6 Understand and use the concept of measures of dispersion to solve problems.

a) Range

(i) Range of the ungrouped data refers to the difference between the highest value and the

lowest value of the data.

(ii) Range for the grouped data refers to the difference between the midpoint of the last

class and the midpoint of the first class.

b) Median is the value in the middle of the set when the data are arranged in

ascending numerical order.

c) First Quartile (Q1) is the value of data such that one-quarter of the set of data have

values less than or equal to it.

d) Third Quartile (Q3) is the value of data such that three-quarters of the set of data have

values less than or equal to it.

e) Inter Quartile Range (IQR) is the difference between the Third Quartile and the First

Quartile.

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8.7 Questions based on the examination format

1. (a) The table shows the speed, in km h-1, of 56 cars on a road.

Speed ( km h-1 ) Frequency Midpoint

51 – 55 2

56 – 60 6

61 – 65 18

66 – 70 15

71 – 75 9

76 – 80 5

81 – 85 1

(i) Find the midpoints of the class intervals.

(ii) Calculate the mean speed of the cars.

(b) Complete the table given in the answer space below.

(c) For this part of question, use the graph paper. By using a scale of 2 cm to 5 km h-1 on the x axis and 2 cm to 5 cars on the y-axis,

draw an ogive for the data.

Hence, find

(i) the median(ii) the interquartile range.

Solution:

(a) (i) and (b)

Speed (km h-1 ) Frequency Midpoint Upper

Boundary

Cumulative

Frequency46 – 50 0 48 50.5 0

51 – 55 2 53 55.5 2

56 – 60 6 58 60.5 8

61 – 65 18 63 65.5 26

66 – 70 15 68 70.5 41

71 – 75 9 73 75.5 50

76 – 80 5 78 80.5 55

81 – 85 1 83 85.5 56

(ii) Mean =

56

)183()578()973()1568()1863()658()253( ×

=56

3738

= 66.75 km h-1.

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(c)

(a) Median = 66.5 km h-1

(b) Interquartile range = 71.0 – 62.5

= 8.5 km h-1

2. The data above shows the pocket money, in RM, per week of a group of 40 students.

10 13 18 7 19 5 21 21

18 16 18 10 15 19 20 84 14 12 6 9 20 20 15

8 10 16 11 13 17 9 11

19 15 17 12 14 11 16 17

(a) Based on the data, complete the table below.

Pocket money (RM) Frequency Midpoint

1 – 3 0

4 – 6

(b) Based on the table,(i) state the modal class,

(ii) calculate the mean,

of the data.

(c) For this part of the question, use a graph paper.By using a scale of 2 cm to RM 3 on the x-axis and 2 cm to 1 student on the y-axis, drawa histogram and a frequency polygon on the same graph paper.

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0

5

10

15

20

25

3035

40

45

50

55

60

50.5 55.5 60.5 65.5 70.5 75.5 80.5 85.5 90.5

Speed (km/h)

C u m u l a t i v e f r e q u e n c y From the ogive,

First quartile = 62.5Median = 66.5

Third Quartile = 71.0



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3. (a) The table above shows the scores of a group of players in the game.

Score 10 11 12 13 14 15

Frequency 4 7 3 x 5 8 (i) Given the median is 12.5, find the value of x.

(ii) Given the mode is 15, find the maximum value of x.

(b) The data shows the mass, in kg, of a group of students.

49 45 47 66 66 62 68 43 46

53 50 61 65 67 63 68 65 44

48 63 62 45 61 67 56 43 44

57 64 60 52 58 46 59 41 43

64 65 47 46 52 50 54 42 60

(i) Based on the data, complete the frequency table below.

Mass (kg) Frequency

41 – 44

65 – 68

(ii) For this part of question, use the graph paper.(a) By using a scale of 2 cm to 4 kg on the x-axis and 2cm to 1 student on

the y-axis, draw a histogram for the data.(b) State one information obtained from the histogram.

4. The histogram shows the donations from a group of 40 students to a school fund.

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0

2

4

6

8

10

12

14

0.5 4.5 8.5 12.5 16.5 20.5 24.5 28.5

Donation (RM)

F r e q u

e n c y

0.5 4.5 8.5 12.5 16.5 20.5 24.5 28.5



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(a) Based on the histogram, state the modal class of the data.

(b) Complete the frequency table below.

Donation (RM) Frequency Cumulative frequency Upper boundary

0 0

1 – 4 2

(c) For this part of question, use a graph paper.

(i) By using a scale of 2 cm to RM4 on the x-axis and 2 cm to 5 students on the

y-axis, draw an ogive for the data.

(ii) From the ogive, find the inter quartile range.

5. The ogive shows the scores of a group of 48 students in a Mathematics test.

(a) Based on the ogive, calculate the percentage of students who scored more than 80.

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0

10

20

30

40

50

60

64.5 69.5 74.5 79.5 84.5 89.5 94.5 99.5

Score

C u m u l a t i v e f r e q u e

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(b) Complete the table below.

Score Cumulative frequency Frequency Midpoint

60 – 64 0

65 – 69

(c) Calculate the mean of the data.

(d) For this part of question, use the graph paper.

By using a scale of 2 cm to 5 scores on the x-axis and 2 cm to 2 students on the y-

axis, draw a frequency polygon for the data.

6. The frequency table above shows the masses, in kg, of watermelons in a lorry.

Mass (kg) Frequency

1.5 – 1.9 4

2.0 – 2.4 10

2.5 – 2.9 26

3.0 – 3.4 84

3.5 – 3.9 50

4.0 – 4.4 15

4.5 – 4.9 8

5.0 – 5.4 3

(a) From the table, state

(i) the size of the class interval,

(ii) the midpoint of the modal class.

(b) (i) Based on the information, complete the table below.

Upper

boundary1.45 5.45

Cumulative

Frequency0 200

(ii) For this part of question, use a graph paper.By using a scale of 2 cm to 0.5 kg on the x-axis and 2cm to 20 watermelons

on the y-axis, draw an ogive for the data.(iii) From the ogive,

a) state the median of the data,

b) find the number of watermelons with the weight less than 3.8 kg.

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7. The cumulative frequency table shows the marks of a group of 40 students in a test.

Mark 60 – 64 65 – 69 70 – 74 75 – 79 80 – 84 85 – 89 90 – 94

Cumulative

frequency7 11 17 26 34 37 40

(a) Based on the information, complete the table in the answer corner.

(b) Find

(i) the size of the class interval,

(ii) the modal class of the data,

(iii) the midpoint of the modal class.

(c) Calculate the mean of the data.

(d) For this part of question, use the graph paper.By using a scale of 2 cm to 5 marks on the x-axis and 2 cm to 1 student on the

y-axis, draw a histogram for the data.

8. The data shows the marks of a group of 40 students in a Mathematics test.

71 20 44 48 47 37 70 56

33 61 52 21 36 31 55 56

53 66 72 38 45 49 50 50

83 29 55 46 59 63 60 48

42 69 43 44 58 78 40 38

(a) Using the data and a class interval of 10 marks, complete the table below.

Mark Frequency Cumulative frequency Upper boundary

1 – 10

81 - 90

(b) For this part of question, use a graph paper.

By using a scale of 2 cm to 10 marks on the x-axis and 2 cm to 5 students on the y-axis,

draw an ogive for the data.

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(c) From your ogive,

(i) find the interquartile range,

(ii) find the median, hence, explain briefly the meaning of the median.

9. The data shows the ages of a group of 40 workers in a factory.

38 41 32 39 32 50 31 38

36 51 31 27 44 30 49 28

26 31 22 36 32 33 34 43

31 27 46 41 35 35 34 31

40 35 25 27 50 37 33 45

(a) Find the range of the data.

(b) Based on the data and by using a class interval of 5 years, complete the table below.

Ages (years) Frequency Midpoint

21 – 25

26 – 30

(c) From the table,

(i) state the modal class,(ii) calculate the mean,

of the data.

(d) For this part of question, use a graph paper.By using a scale of 2 cm to 5 years on the x-axis and 2 cm to 2 workers on the y-axis,

draw a frequency polygon for the data.

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10. The frequency polygon shows the daily incomes, in RM, of 28 workers.

(a) Based on the frequency polygon, complete the table below.

Daily income

(RM)Midpoint

Upper

boundaryFrequency

Cumulative

frequency

16 – 20 18 20.5 0 0

(b) Based on the table,

(i) state the modal class,

(ii) calculate the mean, of the data.

(c) For this part of question, use a graph paper.

(i) By using a scale 2 cm to RM5 on the x-axis and 2 cm to 4 workers on the y-axis,draw an ogive for the data.

(ii) From the ogive, find the number of workers whose daily income is more than

RM38.

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0

2

4

6

8

10

12

18 23 28 33 38 43 48

Daily income

F r e q u e n c y



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8.8 Past Year SPM Questions

Paper 2

1. November 2003

The data below shows the donations, in RM, of 40 families to their children’s school welfare

fund.

40 24 17 30 22 26 35 19

23 28 33 33 39 34 39 28

27 35 45 21 38 22 27 35

30 34 31 37 40 32 14 28

20 32 29 26 32 22 38 44

(a) Using the data above and a class interval of RM5, complete the following table.

[4 marks]

Donation (RM) Frequency Cumulative Frequency

11 – 15

16 – 20

(b) For this part of question, use a graph paper.

By using a scale of 2 cm to RM5 on the x-axis and 2 cm to 5 families on the y-axis, drawan ogive based on the data.

[6 marks]

(c) From your ogive in (b),

(i) find the third quartile,

(ii) hence, explain briefly the meaning of the third quartile.

[2 marks]

2. July 2004

The data in Diagram 7 shows the number of durian trees planted by 44 farmers.

Diagram 7

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52 33 48 22 34 42 57 51 51 65 4166 54 66 53 53 34 46 52 65 75 52

25 68 48 63 62 43 52 56 59 49 58

43 58 36 72 68 54 62 40 73 38 63

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(a) (i) Based on the data in Diagram 7 and by using a class interval of 10, complete Table 3

provided in the answer space.

(ii) Hence, state the modal class. [6marks]

Answer :Class Interval Upper Boundary Frequency Cumulative Frequency

11 – 20 20.5 0 0

21 – 30

(b) For this part of the question, use the graph paper on page 41 .

By using a scale of 2 cm to 10 trees on the x-axis and 2 cm to 5 farmers on the y-axis,

draw an ogive for the data. [4marks]

(c) Based on the ogive in (b), Ahmad concludes that 50% of the farmers planted less than 52

durian trees.

Determine whether the conclusion is correct or not and give a reason.

[2marks]

3. November 2004

The data below shows the masses, in kg, of suitcases for a group of tourists. Each tourist

has one suitcase.

27 10 22 28 21 14 29 25

29 18 22 13 20 21 24 2727 25 16 19 16 24 26 27

29 19 33 25 23 24 26 31

(a) Based on the data above and by using a class interval of 3, complete the table below.

[4 marks]

Class Interval Frequency Midpoint

10 – 12

13 – 15

(b) Based on the table in (a), calculate the estimated mean mass of the suitcases.

[3 marks]

(c) For this part of the question, use a graph paper.

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0

5

10

15

20

25

30

115 120 125 130 135 140 145 150

Height (in cm)

N u

m b e r o f P u p i l s

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By using a scale of 2 cm to 3 kg on the x-axis and 2 cm to 1 suitcase on the y-axis, draw

the histogram for the data. [3 marks]

(d) State one information obtained based on the histogram in (c).

[2 marks]

4. July 2005

iagram 9 is a frequency polygon which represents the heights, in cm, for a group of 80

pupils.

(a) (i) Based on the information from the frequency polygon, complete Table 2 in the

answer space.

(iii)Hence, calculate the mean height, in cm, of the pupils. Give your answer correct to 2decimal places.

[6 marks]

Answer:

Height

(in cm)

Upper

Boundary

Midpoint Frequency Cumulative

Frequency

118 – 122 122.5 120 2 2

123 – 127

128 – 132

133 – 137138 – 142

143 – 147

(b) For this part of the question, use the graph paper on page 45. You may use a flexible

curve rule.

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Diagram 9



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By using a scale of 2 cm to a height of 5 cm on the horizontal axis and 2 cm to 10 pupils

on the vertical axis, draw an ogive for the data.

[5 marks]

(c) Give one information that can be obtained from the ogive in (b).

[1 mark]

5. SPM November 2005

The data in Diagram shows the marks for an English Language monthly test for 42 pupils.

(a) Using data in

diagram and a class

interval of 5 marks,complete Table in the

answer space.

[4marks]

Marks Midpoint Frequency

20 – 24 22

25 – 29

(b) Based on your table in (a),

(i) state the modal class,

(ii) calculate the mean mark for the English Language monthly test and give your answer

correct to decimal places.

[4marks]

(c) For this part of the question, use the graph paper provided .

By using a scale of 2cm to 5 marks on the horizontal axis and 2 cm to 1 pupil on the

vertical axis, draw a histogram for the data. [4marks]

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51 20 45 31 26 40 30

25 32 37 41 21 36 38

46 38 28 37 39 23 39

33 35 42 29 38 31 23

42 34 26 35 43 28 22

25 47 31 48 44 34 54

15



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6. SPM July 2006

\ Table 3 shows the distribution of age for a group of 40 tourists who visited a museum in a

particular day.

Age(Year) 20 – 24 25 – 29 30 – 34 35 – 39 40 – 44 45 – 49 50 – 54

Frequency 2 6 9 10 7 4 2Table 3

(a) Based on the distribution of age in Table 3, complete Table 4 in the answer space.

[2marks]

Answer:

Age (Year) Midpoint Upper Boundary Cumulative Frequency

20 – 24 22

25 – 29 27

30 – 34 32

35 – 39 37

40 – 44 42

45 – 49 47

50 - 54 52

Table 4

(b) Calculate the estimated mean age of the tourists. [3marks]

(c) For this part of the question, use the graph paper .

By using the scale of 2 cm to 5 years on the horizontal axis and 2 cm to 5 tourists on the

vertical axis, draw an ogive for the data in table 3. [5marks]

(d) From your ogive constructed in (c),

(i) find the first quartile,

(ii) state one piece of information about the first quartile [2marks]

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8. SPM Jun 2007, No 15

Diagram 8 shows the mass, in kg, of newspapers collected over the period of 40 days.

79 70 75 79 83 74 66 58 65 8584 76 56 80 64 57 78 67 52 59

72 58 57 75 69 76 55 75 60 7383 63 74 65 78 58 73 64 75 59

Diagram 8

Based on the data in Diagram 8, complete Table 2 in the answer space.

[4 marks]

(b) Hence, calculate the estimated mean of the masses of the newspapers collected.

[3 marks]

(c) For this part of the question, use the graph paper provided.

By using the scale of 2 cm to 5 kg on the horizontal axis and 2 cm to 1 day on the vertical

axis, draw a histogram for the data.

[4 marks]

(d) Based on your histogram in 15(c), give one information about the modal class of the data.

[1 mark]

Answer :

Mass (kg) Frequency Midpoint

52 – 56

57 – 61

9. SPM Nov 2007, No 16

Table 2 shows the frequency distribution of the mass, in kg, of a group of 80 students.

Mass (kg) Frequency

30 – 34 5

35 – 39 8

40 – 44 11

45 – 49 21

50 – 54 22

55 – 59 10

60 – 64 3

Table 2

(a) (i) State the modal class

(ii) Calculate the estimated mean of the mass of the group of students.

[4 marks]

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(b) Based on Table 2, complete Table 3 in the answer space to show the cumulative frequency

distribution of the masses.

[3 marks]

(c) For this part of the question, use the graph paper provided .By using the scale of 2 cm to 5 kg on the horizontal axis and 2 cm to 10 students on the

vertical axis, draw an ogive for the data.[4 marks]

(d) 25% of all the students in the group have a mass of less than p kg. These students will be

supplied with nutritional food.

Using the ogive you had drawn in 16(c), find the value of p.

[1 mark]

Answer:

(a) (i)

(ii)

(b)

Upper Boundary (kg) Cumulative frequency

29.5 0

34.5

Table 3

10. SPM Jun 2008, No 14

Data in Diagram 14 shows the number of books read by 40 students in a reading programme in a

particular class.

18 14 11 16 13 17 12 23

12 19 17 15 11 21 7 14

15 11 5 18 9 22 24 19

17 9 14 12 6 15 10 13

19 15 20 10 14 8 12 16Diagram 14

(a) Based on the data in Diagram 14 and by using the class interval of 3, complete Table 14

in the answer space.

[3 marks]

(b) Based on Table 14 in 14(a), calculate the estimated mean of the books read by a student.

[3 marks]Statistics III 19



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(c) By using the scale of 2 cm to 3 books on the horizontal axis and 2 cm to one student on

the vertical axis, draw the frequency polygon for this data.

[5 marks]

(d) Based on the frequency polygon in 14(c), give one information about this reading

programme.[1 mark]

Answer :(a)

Class interval Frequency Midpoint

5 – 7 6

8 – 10

Table 14

(b)

(c)

(d)

11. SPM Nov 2008, No 14.

The data below shows the payment, in RM, by 40 drivers at a toll booth.

38 34 41 25 19 32 28 42

32 25 32 27 42 23 18 46

21 42 36 30 33 37 43 25

24 18 26 35 47 22 38 33

30 23 37 48 39 34 41 27

(a) Based on the data, complete Table 14 in the answer space.

[3 marks]

(b) Based on Table 14 in 14(a), calculate the estimated mean of the toll paid by a driver.

[3 marks]

(c) For this part of the question, use the graph paper provided .

By using the scale of 2 cm to RM5 on the horizontal axis and 2 cm to 1 driver on the

vertical axis, draw a frequency polygon for the data.

[5 marks]

(d) Based on the frequency polygon in 14(c), state the number of drivers who paid more than

RM34 for the toll.

[1 mark]

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Answer:

a)

Class Interval Midpoint Frequency

15 – 19 17

b)

d)

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ANSWERS

8.7 Examination Format Questions

2. (a)

Pocket money(RM) Frequency Midpoint

1 – 3 0 2

4 – 6 3 5

7 – 9 5 8

10 – 12 8 11

13 – 15 7 14

16 – 18 9 17

19 – 21 8 20

(b) (i) Modal class = 16 – 18

(ii) Mean =

8978530

)820()917()714()811()58()35(

+×

=40

554

= RM 13.85

(iii)

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0

1

2

3

4

5

6

7

8

9

10

3.5 6.5 9.5 12.5 15.5 18.5 21.5

Pocket money

F r e q u e n c y

3.5 6.5 9.5 12.5 15.5



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3. (a) (i) x = ( 4+ 7 +3 ) – ( 5 + 8 )

= 14 – 13

= 1

(ii) Maximum value of x = 7

(b) (i)

Mass (kg) Frequency

41 – 44 7

45 – 48 8

49 – 52 5

53 – 56 3

57 – 60 5

61 – 64 8

65 – 68 9

(ii)

(iii) The modal class of the data is 65 – 68 kg.

4. (a) Modal class = 9 – 12

(b)

Donation (RM) Frequency Cumulativefrequency

Upper boundary

0 0 0 0.51 – 4 2 2 4.5

5 – 8 5 7 8.5

9 – 12 12 19 12.5

13 – 16 8 27 16.5

17 – 20 5 32 20.5

21 – 24 3 35 24.5

25 – 28 1 36 28.5

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0

1

2

3

4

5

6

7

8

910

4 4 . 5

5 2 . 5

6 0 . 5

6 8 . 5

Mass(kg)

F r e q u e n c y

40.5 44.5 48.8 52.5 56.5 60.5



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(c) (i)

(ii) Interquartile range = 16.5 – 9.3

= RM 7.20

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0

5

10

15

20

25

30

35

40

0.5 4.5 8.5 12.5 16.5 20.5 24.5 28.5

Donation (RM)

C u m u l a t i v e f r e q u e n c y Third uartile

First

9. 16.5



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5. (a) The percentage of students who scored more than 80

=48

1948 −x 100%

=48

29x 100%

= 60.42 %

(b)

Score Cumulative frequency Frequency Midpoint

60 – 64 0 0 62

65 – 69 2 2 67

70 – 74 7 5 72

75 – 79 17 10 77

80 – 84 34 17 82

85 – 89 43 9 87

90 – 94 47 4 92

95 – 99 48 1 97

(c) Mean =1491710520

)197()492()987()1782()1077()572()267(

+×

=48

3906

= 81.375

(d)

6. (a) (i) Size of class interval = 1.95 – 1.45Statistics III 25

0

2

4

6

8

10

12

14

16

18

62 67 72 77 82 87 92 97 102

Score

F r e q u e n c y



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= 0.5 kg

(ii) Modal class = 3.0 – 3.4

Midpoint of the modal class =2

4.30.3 +

= 3.2

(b) (i)

Upper boundary Cumulative frequency

1.45 0

1.95 4

2.45 14

2.95 40

3.45 124

3.95 174

4.45 189

4.95 1975.45 200

(ii)

(iii) (a) Median = 3.40 kg

(b) The number of watermelons with the weight less than 3.8 kg = 200 – 164

= 36.

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0

20

40

60

80

100

120

140

160

180

200

220

1.45 1.95 2.45 2.95 3.45 3.95 4.45 4.95 5.45

Mass (kg)

C u m m u l a t i v e f r e q u e n c y

Median

164

3 . 4

0



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7. (a)

Mark Cumulative

frequency

Frequency Midpoint

55 – 59 0 0 57

60 – 64 7 7 62

65 – 69 11 4 67

70 – 74 17 6 72

75 – 79 26 9 77

80 – 84 34 8 82

85 – 89 37 3 87

90 - 94 40 3 92

(b) (i) Size of class interval = 59.5 – 54.5

= 5 marks

(ii) Modal class = 75 – 79

(iii) Midpoint of the modal class =2

7975 +

= 77

(c) Mean =33896470

)392()387()882()977()672()467()762(

+×

=40

3020

= 75.5 marks

(d)

8. (a)

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0

1

2

3

4

5

6

7

8

9

10

54.5 59.5 64.5 69.5 74.5 79.5 84.5 89.5 94.5

Mark

F r e q u e n c y

54.5 59.5 64.5 69.5 74.5 79.5 84.5 89.5 94.5



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Mark Frequency Cumulative

frequency

Upper boundary

1 – 10 0 0 10.5

11 – 20 1 1 20.5

21 – 30 2 3 30.5

31 – 40 7 10 40.5

41 – 50 12 22 50.5

51 – 60 9 31 60.5

61 – 70 5 36 70.5

71 – 80 3 39 80.5

81 – 90 1 40 90.5

(b)

(c) (i) Interquartile range = 59.5 – 40.5

= 19.0 marks

(ii) Median = 49.5 marks

The median means 20 of the students scored less than 49.5 marks.

Statistics III 28

0

5

10

15

20

25

30

35

40

45

10.5 20.5 30.5 40.5 50.5 60.5 70.5 80.5 90.5

Mark

C u m u l a t i v e f r e q u e n c y

First quartile

Median

Third quartile

4 9 . 5

5 9 . 5



Enrichment

9. (a) Range = 51 – 22

= 29 years.

(b)

Ages (years) Frequency Midpoint

21 – 25 2 2326 – 30 6 28

31 – 35 15 33

36 – 40 7 38

41 – 45 5 43

46 – 50 4 48

51 – 55 1 53

(c) (i) Modal class = 31 – 35

(ii) Mean =

14571562

)153()448()543()738()1533()628()223(

+×

=40

1435

= 35.875 years

(d)

0

2

4

6

8

10

12

14

16

18 23 28 33 38 43 48 53 58

Age (years)

F r e q u e n c y

Statistics III 29



Enrichment

10. (a)

Daily

income(RM)

Midpoint Upper

boundary

Frequency Cumulative

frequency

16 – 20 18 20.5 0 0

21 – 25 23 25.5 4 4

26 – 30 28 30.5 10 14

31 – 35 33 35.5 8 22

36 – 40 38 40.5 4 26

41 – 45 43 45.5 2 28


(ii) Mean =2481040

)243()438()833()1028()423(

+×

=28

874

= RM 31.21

(c) (i)

0

4

8

12

16

20

24

28

20.5 25.5 30.5 35.5 40.5 45.5

Daily income (RM)

C u m u l a t i v e

f r e q u e n c y

(ii) The number of workers whose daily income is more than RM 38 = 28 – 24

= 4

Statistics III 30



Enrichment

8.8 SPM PAST YEAR QUESTIONS

1. SPM 2003

(a)

Donation (RM) Frequency Cumulative frequency

11 – 15 1 1

16 – 20 3 421 – 25 6 10

26 – 30 10 20

31 – 35 11 31

36 – 40 7 38

41 – 45 2 40

(b)

0

5

10

15

20

25

30

35

40

45

10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5

Donation (RM)

C u m u l a t i v e f r e q u e n c y

(c) (i) 35

(ii) 30 families donated RM 35 or less

3. (SPM 2004)

(a)

Class interval Frequency Midpoint

10 – 12 1 1113 – 15 2 14

16 – 18 3 17

19 – 21 5 20

22 – 24 6 23

25 – 27 9 26

28 – 30 4 29

31 - 33 2 32

Statistics III 31



Enrichment

(b) Mean =

24965321

)232()429()926()623()520()317()214()111(

+×

=32

742

= 23.19

(c)

0

1

2

3

4

5

6

7

8

9

10

12.5 18.5 24.5 30.5

Mass (kg)

F r e q u e n c y

9.5 12.5 15.5 18.5 21.5 24.5 27.5 30.5 33.5

(d) The modal class for the mass of suitcases is 25 – 27

6. (SPM 2006J)

(a)Age ( Year) Midpoint Upper boundary Cumulative frequency

20 – 24 22 24.5 2

25 – 29 27 29.5 8

30 – 34 32 34.5 17

35 -39 37 39.5 27

40 - 44 42 44.5 34

45 – 49 47 49.5 38

50 - 54 52 54.5 40

(b) Mean =

40

)522()474()427()3710()329()276()222( ×

=40

10418829437028816244 +

=40

1450

= 36.25

Statistics III 32



Enrichment

5. SPM 2005N

(a)

Marks Midpoint Frequency

20 – 24

25 – 29

30 – 34

35 – 3940 – 44

45 – 49

50 – 54

22

27

32

3742

47

52

5

7

8

106

4

2


(ii) Mean =24610875

)52()474()426()3710()328()277()225(

+×

=42

1469

= 34.98 ( 2 d.p)

(c)

Statistics III 33

0

1

2

3

4

5

6

7

8

9

10

11

19.5 24.5 29.5 34.5 39.5 44.5 49.5 54.5

Mark

F r e q u e n c y

19.5 24.5 29.5 34.5 39.5 44.5 49.5 54.5



Enrichment

7. November 2006

a)

Class Interval Midpoint Frequency

21 – 25 23 5

26 – 30 28 6

31 – 35 33 8

36 – 40 38 10

41 – 45 43 7

46 – 50 48 4

b) mean =23 5 28 6 33 8 38 10 43 7 48 4

40

× + × + × + × + × + ×

=1420

40= 35.5

c)

d) Any correct information that can be obtained from the frequency polygon, eg. 4 students

donated more than RM48.

Statistics III 34



Enrichment

8. SPM June 2007

a)

Mass (kg) Frequency Midpoint Freq x midpoint

52 – 56 3 54.5 163.5

57 – 61 8 59.5 47662 – 66 6 64.5 387

67 – 71 3 69.5 208.5

72 – 76 11 74.5 819.5

77 – 81 5 79.5 397.5

36 2452

ii. Mean =sum of (frequency midpoint)

sum of data

×

2452

36

68.11

=

=

c)

b) Between 72 – 76 kg of newspapers was collected over the most number of days (11 days)

Statistics III 35



Enrichment

9. SPM Nov 2007

a) i) Modal class is 50 – 54 kg

ii)

Mass (kg) Midpoint Frequency Midpoint x

frequency

30 – 34 32 5 16035 – 39 37 8 296

40 – 44 42 11 462

45 – 49 47 21 987

50 – 54 52 22 1144

55 – 59 57 10 570

60 – 64 62 3 186

80 3805

Mean =

3805

80

47.56

=

=

b)

Boundaries Frequency Cumulative

frequency

29.5 0 0

34.5 5 5

39.5 8 13

44.5 11 24

49.5 21 45

54.5 22 67

59.5 10 77

64.5 3 80

c)

d) at 25%, 20 of the students have a mass of less than p = 43 kg

10. SPM June 2008 Q14Statistics III 36



Enrichment

a)

Class interval Frequency Midpoint Freq x midpoint

5 – 7 3 6 18

8 – 10 5 9 45

11 – 13 9 12 108

14 – 16 10 15 150

17 – 19 8 18 144

20 – 22 3 21 63

23 – 25 2 24 48

40 576

b)

576mean

40

14.4

=

=

c)

d) The modal class for the number of books read by these students is 14 – 16 books.

Statistics III 37



Enrichment

11. SPM Nov 2008 Q14

a)

Class Interval Midpoint Frequency Midpoint x frequency

15 – 19 17 3 51

20 – 24 22 5 110

25 – 29 27 7 189

30 – 34 32 9 288

35 – 39 37 7 259

40 – 44 42 6 252

45 – 49 47 3 141

40 1290

b)

1290Mean =

40

= 32.25

c)

d) The number of drivers who paid more than RM34 for the toll

= 7 + 6 + 3

= 16

Documents

Chapter 8 II Statistics III ENRICH