CHAPTER 6 Statistics III (Notes and Exercises)

Mathematics Form 4 Chapter 6 : Statistics III

CHAPTER 6 : STATISTICS III (a)

Mode of a ungrounded data Mode = the value of data with the highest frequency Example 1 :

Example 2 :

6, 7, 7, 11, 5, 6, 11, 13, 14, 11, 8

Score Frequency

⇒ 5, 6, 6, 7, 7, 8, 11, 11, 11, 13, 14

2 3

mode = 11

8 12

10 9

2 0

3 x

4 2

Example 4 : 0 1

1 3

2 7

3 x

4 5

Score Frequency

If mode = 2, the maximum value of x = ??? x7

Median of a ungrounded data Median = the middle value when a set of data is arranged in ascending order Example 1 :

Example 2 : 5, 3, 3, 5, 7, 7, 1

24, 23, 12, 19, 16, 17

⇒ 1, 3, 3, 5, 5, 7, 7

⇒ 12, 16, 17, 19, 23, 24

median = 5

median =

17 + 19 = 18 2

Example 3 : Number of books Number of pupils

1 3

2 0

3 1

4 5

5 6

⇒ 1, 1, 1, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5 median = 4 Example 4 :

8 8

Saiz of shoes

1

2

3

4

5

Number of students

8

14

12

x

3

median = 3, range of x = ???

14 14

1 11 11 1

x x

3 3

.

8 + 14 = 11 + x + 3 22 = x + 14 8=x

8 + 14 + 11 = x + 3 33 = x + 3 30 = x

∴ 8 ≤ x ≤ 30

© Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013

Mathematics Form 4 Chapter 6 : Statistics III (c)

Mean of a ungrounded data mean =

sum of all the values of data the number of data

mean =

Example 1 :

sum of (value × frequency ) total frequency

Example 2 : 68, 62, 84, 75, 78, 89

mean =

Mark Frequency

68 + 62 + 84 + 75 + 78 + 79 6

mean =

= 76

74 5

78 10

82 2

86 3

74 (5) + 78 (10) + 82 (2) + 86 (3) 5 + 10 + 2 + 3

= 78.6 (d)

Measure of Dispersion ~ range, first / lower quartile (Q1), third / upper quartile (Q3), interquartile range

• range = ( largest − smallest ) value of data • Q1 = the value that divides the values of data that are less than median into 2 equal parts • Q3 = the value that divides the values of data that are greater than median into 2 equal parts • Interquatile range = Q3 − Q1 Example 1 :

Example 2 : 8, 12, 6, 10, 6, 7, 13, 3, 8, 10, 13, 19

5, 30, 45, 29, 25, 6, 21, 8, 28, 4

⇒ 3, 6, 6, 7, 8, 8, 10, 10, 12, 13, 13, 19 ⇒ 4, 5, 6, 8, 21, 25, 28, 29, 30, 45

(e)

∴

range = 19 − 3 = 16

∴

Q1 =

6+7 = 6.5 2 12 + 13 = = 12.5 2

∴

range = 45 − 4 = 41

∴

Q1 = 6

∴

Q3 = 29

∴

Q3

∴

Interquartile range = 29 − 6 = 23

∴

Interquartile range = 12.5 − 6.5 = 6

Solve problem involving ungorounded data

Example 1 : Score ⇒ third quartile = 11, x =1 ??? 3 Frequency

1

1

6

x

2

3

12

14

1

1

.

∴ ⇒ 1,

3,

6,

6,

x,

x,

x,

12,

14

x + 12 2

= 11

x + 12 = 22 x = 22 − 12 x = 10

Example 2 : 3, 3, 6, x, x, 3 © Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013

Mathematics Form 4 Chapter 6 : Statistics III mode = 3, median = 4.

Two new pieces of data, 4 and 7 put into the set, mean = ??? 3+ x 2

3, 3, 3, x, x, 6

(f)

mean

=4

3+3+3+5+5+6+ 4+7 8

3+x=8

=

x = 8 −3 x=5

= 4.5

Class interval, lower / upper limit, lower / upper boundary, size of class interval, midpoint Example : Class interval

11 – 15

16 – 20

21 – 25

26 – 30

31 – 35

36 – 40

41 – 45

.

• lower limit = 16,

• lower boundary = 15.5,

upper limit= 20

• midpoint =

• size of class interval = 5

(g)

upper boundary = 20.5

lower lim it + upper lim it 2 lower boundary + upper boundary 2

= upper. B − lower. B

=

= midpoint2 − midpoint1

= 18

Frequency table, Cumulative Frequency, Modal class, Mean, Range Example : Donation, x

11 – 15

16 – 20

21 – 25

26 – 30

31 – 35

36 – 40

41 – 45

Frequency, f

1

3

6

10

11

7

2

Cumulative F.

1

4

10

20

31

38

40

` .

• modal class = the class interval with the highest frequency = 31 – 35 • mean =

∑ fx ∑f

=

sum of (midpo int × frequency ) sum of frequency

=

13 (1) + 18 (3) + 23 (6) + 28 (10) + 33 (11) + 38 (7) + 43 (2) 40

• range = midpoint of (hightest − lowest ) class = 43 − 13 = 30

(h)

Histogram, frequency polygons, ogive, first quartile, third quartile, interquartile range Histogram

Example :

[ base on frequency table in (g) ]

• lower / upper boundary. • frequency ** the frequency polygon can be constructed based on a histogram

© Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013

= 30

Mathematics Form 4 Chapter 6 : Statistics III Frequency Histogram 10

Frequency polygon

8 6 4 2

Frequency Polygons

Example :

Donation

45.5

40.5

35.5

30.5

25.5

20.5

15.5

10.5

0

[ base on frequency table in (g) ]

Frequency

• midpoint

10

• frequency

8

** the frequency polygon should add a class with zero frequency before the first class and after the last class

6 4 2 0

Ogive • upper boundary

Example :

48

43

38

33

28

23

18

13

8

Donation

[ base on frequency table in (g) ]

Cumulative Frequency

1 N 4

Q1 25.5

20.5

15.5

10.5

0

med Q3 . 34.5 30.5

Donation

© Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013

45.5

10

1 N 2

40.5

20

35.5

30

3 N 4

30.5

** add a class with zero frequency before the first class

40

25.5

• cumulative frequency

Mathematics Form 4 Chapter 6 : Statistics III

EXERCISES of SPM PAPER 2 FORMAT : (STATISTICS III) 1

The data shows the ages, in years, of 30 workers in a carpenter factory.

(a)

21

38

25

39

31

23

29

40

47

28

48

29

34

45

26

20

36

33

31

20

38

34

24

26

28

31

43

27

32

25

State the range of the data.

(Ans : 28)

[1 mark]

Answer : (a)

(b)

Based on the data above and by using a class interval of 5 years, complete the table in the answer space. [4 marks] Answer : (b) Age (years)

Frequency

Mid-point

Upper Boundary

20 – 24 25 – 29

(c)

Based on the table in (b), (i)

state the modal class,

(Ans : 25 – 29)

(ii)

calculate the estimated mean ages of the workers and given your answer correct to 3 decimal places. (Ans : 31.667) [3 marks]

Answer : (c)

(i) © Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013

Mathematics Form 4 Chapter 6 : Statistics III (ii) (d)

For this part of the question, use the graph paper provided on the next page.

By using a scale of 2 cm to 5 years on the x-axis and 2 cm to one worker on the y-axis, draw a histogram for the data. [4 marks] Answer : (d) 2

Refer graph on the next page.

The data shows the masses, in kg, of suitcases carried by a group of tourists. Each tourist has one suitcase.

27

10

22

28

21

14

29

25

29

18

22

13

20

21

24

27

27

25

16

19

16

24

26

27

29

19

33

25

23

24

26

31

(a)

Based on the data above, and by using the a class interval of 3 kg, complete the table in the answer space. [3 marks]

(b)

Based on the table in (a), calculate the estimated mean mass of the suitcases and give your answer correct to 2 decimal places. (Ans : 23.19) [3 marks]

(c)

For this part of the question, use the graph paper provided on the next page. By using a scale of 2 cm to 3 kg on the horizontal axis and 2 cm to one suitcase on the vertical axis, draw a histogram for the data. [4 marks]

(d)

State one information that can be obtained from the histogram in (c).

Answer : [2004, No.14] (a) Class Interval

Frequency

Mid-point

10 – 12 13 – 15

.

(b) © Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013

[2 marks]

Mathematics Form 4 Chapter 6 : Statistics III

(c)

Refer graph on the next page.

(d) ____________________________________________________________________ _____________________________________________________________________

3

The data shows the masses, in kg, for a group of 32 students. 44

64

59

49

35

40

50

51

51

59

51

60

50

40

43

47

45

57

53

52

56

56

54

49

58

54

48

47

53

45

62

46

(a)

In the answer space, construct a frequency table for the above data by using 5 kg as the size of the class interval. Begin with 30 – 34, 35 – 39, and so on. [3 marks]

(b)

Based on the table in (a), (i)

state the modal class,

(ii)

estimate the mean mass.

(Ans : 50 – 54) (Ans : 50.906) [4

marks] (c)

For this part of the question, use the graph paper provided on the next page. By using a scale of 2 cm to 5 kg on the x-axis and 2 cm to one student on the y-axis, draw a histogram for the data. [4 marks]

(d)

On the histogram, draw a frequency polygon for the data. [1 mark]

Answer : (a) Class Interval

Frequency

© Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013

Mathematics Form 4 Chapter 6 : Statistics III .

(b)

(i) (ii)

(c)

Refer graph on the next page.

(d)

Refer graph on the next page.

© Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013

Mathematics Form 4 Chapter 6 : Statistics III 4

The table shows the sizes, in cm, of 66 crabs caught by a fisherman. Size (cm)

Number of Crabs

22 – 24

9

25 – 27

27

28 – 30

11

31 – 33

15

34 – 36

4

(a)

Complete the above table in the answer space.

(b)

Based on the table in (a),

(c)

Mid-point

[2 marks]

(i)

state the mid-point of the modal class,

(Ans : 26)

(ii)

calculate the estimated mean size of the crabs.

(Ans : 28)

[4 marks]

For this part of the question, use the graph paper provided on the next page. By using a scale of 2 cm to 3 cm on the x-axis and 2 cm to 5 crabs on the y-axis, draw a histogram for the data. [4 marks]

(d)

On the histogram, draw a frequency polygon for the data.

[2 marks]

Answer : (a)

(b)

Size (cm)

Number of Crabs

22 – 24

9

25 – 27

27

28 – 30

11

31 – 33

15

34 – 36

4

Mid-point

(i) (ii)

(c)

Refer graph on the next page.

© Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013

Mathematics Form 4 Chapter 6 : Statistics III (d) 5

Refer graph on the next page.

The data shows the distribution of heights, in cm, of 40 students in a class. 178

176

159

171

160

166

164

171

174

154

174

154

177

179

158

168

167

174

169

164

172

162

175

153

167

167

155

168

173

169

173

151

176

156

178

152

163

160

172

154

(a)

Using data above, complete the table in the answer space based on the class interval of the same size. [4 marks]

Answer : (a) Height (cm)

Frequency, f

Mid-point, x

145 − 149

(b)

Based on the table in (a), (i)

state the modal class,

(Ans : 170 – 174)

(ii)

calculate the mean height of the students in the class and give your answer correct to two decimal palces. (Ans : 165.88) [4 marks]

Answer : (b)

(i) (ii)

(c)

For this part of the question, use the graph paper provided on the next page. By using a scale of 2 cm to 5 cm on the horizontal axis and 2 cm to one student on the vertical axis, draw a frequency polygon for the data. [4 marks]

Answer : © Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013

Mathematics Form 4 Chapter 6 : Statistics III (c) 6

Refer graph on the next page.

The table shows the consulation time, in minutes, spend by 25 students with the counselor. Time (minutes)

25 – 30

31 – 36

37 – 42

43 – 48

49 – 54

Frequency

5

7

x

3

2

(a)Calculate the value of x.

(Ans : 8)

[2 marks]

Answer : (a)

(b)

Complete the table in the answer space.

[2 marks]

Answer : (b)

(c)

Time (minutes)

Frequency, f

25 – 30

5

31 – 36

7

37 – 42

8

43 – 48

3

49 – 54

2

Mid-point, x

Based on the table in (a), (i)

state the upper boundary of the modal class,

(ii)

calculate the mean time.

(Ans : 42.5)

(Ans : 37.1)

[4 marks]

Answer : (c)

(i) (ii)

(d)

For this part of the question, use the graph paper provided on the next page. By using a scale of 2 cm to 6 minutes on the x-axis and 2 cm to one student on the y-axis, draw a frequency polygon for the data. [4 marks]

Answer : © Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013

Mathematics Form 4 Chapter 6 : Statistics III (d)

7

(a)

Refer graph on the next page. The table shows the scores obtained by a group of 30 students in a Mathematics quiz. Score

1

2

3

4

5

6

Frequency × Score

2

8

x

40

15

36

Calculate : (i)

the value of x,

(Ans : 15)

(ii)

the the mode of the score,

(Ans : 4)

(iii) the mean of the scores, give your answer correct to 4 significant figures. (Ans : 3.867)

[5 marks]

Answer : (i) (ii) (iii) (b)

The data shows the distribution of masses, in kg, for a group of 50 students. 46

40

53

47

56

52

57

51

52

54

50

54

43

55

49

58

53

47

58

51

52

46

57

54

58

44

68

56

57

59

45

50

52

48

53

62

51

59

54

44

53

59

60

42

66

52

59

64

72

63

(i)

In the answer space, construct a frequency table for the above data using class intervals 35 – 39, 40 – 44, and so on. [3 marks]

(ii)

For this part of the question, use the graph paper provided on the next page. By using a scale of 2 cm to 5 kg on the x-axis and 2 cm to 2 students on the y-axis, draw a frequency polygon for the data. [4 marks]

Answer : (i) Class Interval

Frequency

.

© Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013

Mathematics Form 4 Chapter 6 : Statistics III

(ii) 8

Refer graph on the next page.

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

(a)

40

24

17

30

22

26

35

19

20

32

23

28

33

33

39

34

39

28

29

26

27

35

45

21

38

22

27

35

32

22

30

34

31

37

40

32

14

28

38

44

Using the above data, and a class interval of RM 5, complete the table in the answer space. [4

marks] Answer : Donation (RM)

Frequency

Cumulative Frequency

11 – 15 16 – 20

.

(b)

For this part of the question, use the graph paper provided on the next page.

By using a scale of 2 cm to RM 5 on the x-axis and 2 cm to 5 families on the y-axis, draw an ogive based on the data. [4 marks] Answer : (b)

Refer graph on the next page.

(c)

From your ogive in (b), (i)

find the third quartile,

(Ans : 35)

(ii)

hence, explain brierfly the meaning of the third quartile.

(iii) find the value of x, if 25% of the student′ s families contributed less than RM x. (iv) find the value of y, if

(Ans : 25.5)

1 of the student′ s families contributed at least RM y. (Ans : 38.5) 10 [4 marks]

Answer : (i)

third quartile = __________________________________________________________

(ii)

________________________________________________________________________

(iii)

x = ____________________________________________________________________ © Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013

Mathematics Form 4 Chapter 6 : Statistics III (iv)

y = ____________________________________________________________________ [2003, No.14]

© Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013

Mathematics Form 4 Chapter 6 : Statistics III 9

The table show the distribution of marks obtained by a group of 80 students in a Mathematics test.

(a)

Marks

Frequency

Cumulative Frequency

51 - 55

4

4

56 - 60

9

13

61 - 65

17

30

66 - 70

x

50

71 - 75

15

65

76 - 80

8

y

81 - 85

5

78

86 - 90

2

80

Based on the table, find (i)

the value of x,

(Ans : 20)

(ii)

the value of y,

(Ans : 73)

(iii) the mid-point of the modal class.

(Ans : 68)

[4 marks]

(b)

For this part of the question, use the graph paper provided on the next page. By using a scale of 2 cm to 5 marks on the horizontal axis and 2 cm to 10 students on the vertical axis, draw an ogive for the data. [4 marks]

(c)

From the ogive, find (i)

the lower quartile,

(ii) the median,

(Ans : 62.5)

(Ans : 67.5)

(iii) the passing mark, if only 3 students failed in the test.

(Ans : 54.5)

(iv) the minimum mark to score an A, if only 5 students scored A in the test. Answer : (a) (i) (ii) (iii)

(b)

Refer graph on the next page.

(c)

(i) (ii) (iii)

© Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013

(Ans : 82) [4 marks]

Mathematics Form 4 Chapter 6 : Statistics III 10

(iv) The table shows the distribution of heights, in m, of 92 trees in a Recreation Park. Height (m) Frequency 1.0 - 1.4

2

1.5 - 1.9

9

2.0 - 2.4

26

2.5 - 2.9

24

3.0 - 3.4

19

3.3 - 3.9

8

4.0 - 4.4

3

4.5 - 4.9

1

(a) Calculate the range for the grouped data above. Answer :

(Ans : 3.5)

[2 marks]

(a)

(b) Based on the table above, complete the cumulative frequency table in the answer space.[3 marks] Answer : Class Interval

Upper Boundary

Cumulative Frequency

1.0 - 1.4 1.5 - 1.9 2.0 - 2.4 2.5 - 2.9 3.0 - 3.4 3.5 - 3.9 4.0 - 4.4 4.5 - 4.9 (c)

For this part of the question, use the graph paper provided on the next page. By using a scale of 2 cm to 0.5 m on the x-axis and 2 cm to 10 trees on the y-axis, draw an ogive for the data. [4 marks]

Answer : (c)

Refer graph on the next page.

(d)

From the ogive, find (i) the median, (Ans : 2.65) (ii) the interquartile range, (Ans : 1) (iii) find the percentage of trees which are at least 4.1 m in height and give your answer correct to 2 decimal places. (Ans : 3.26) [3 marks] Answer : (i) (ii) © Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013

Mathematics Form 4 Chapter 6 : Statistics III (iii) 11

The table shows the distribution of heights, in cm, of 92 students in a school. Height 120 125 130 135 140 145 150 155 (a)

(i) (ii)

(cm) 124 129 134 139 144 149 154 159

Mid-point

Frequency 4 10 26 24 17 7 3 1

Based on the table above, complete the table (a) in the answer space. Hence, calculate the mean height of the students. (Ans : 136.239)

[3 marks]

(b) Upper Boundrary

119.5

159.5

Cumulative Frequency (i)

Based on the table in (a), complete the table in (b) in the answer space.

(ii)

For this part of the question, use the graph paper provided on the next page. By using a scale of 2 cm to 5 cm on the x-axis and 2 cm to 10 students on the y-axis, draw an ogive for the data.

(iii) From the ogive, find the interquartile range for the data.

(Ans : 9)

(iv) The students whose height is above 152 cm are chosen as basketball player. Find the number of students who are chosen. (Ans : 2) [9 marks] Answer : (a) (i) Height 120 125 130 135 140 145 150 155 -

(cm) 124 129 134 139 144 149 154 159

Mid-point

Frequency 4 10 26 24 17 7 3 1

(ii) (b)

(i) Upper Boundary

119.5

Cumulative Ferquency (ii)

Refer graph on the next page.

(iii)

© Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013

159.5

Mathematics Form 4 Chapter 6 : Statistics III (iv) The frequency table shows the speeds, in km/h, of 92 motorbikes that passed a certain point along a highway.

30 25 Frequency

12

20 15 10 5 0

73 5

78

83

88

93

98

103

108

Speed (km/h)

(a) Based on the frequency polygon, complete the table in the answer space. Answer :

[4 marks]

(a) Speed (km/h)

Mid-point

Upper Boundary

Frequency

Cumulative Frequency 0

92 (b)

Based on the table in (a), (i) state the modal class, (Ans : 91 – 95) (ii) calculate the estimated mean and give your answer correct to 3 decimal places. (Ans : 92.239) Answer : (b)

[3 marks]

(i) (ii)

(c)

For this part of the question, use the graph paper provided on the next page. (i) (ii)

Answer : (c) (i)

By using a scale of 2 cm to 5 km/h on the x-axis and 2 cm to 10 motorbikes on the y-axis, draw an ogive based on the frequency table in (a). From the ogive, state the first quartile. (Ans : 88) [5 marks]

Refer graph on the next page.

© Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013

Mathematics Form 4 Chapter 6 : Statistics III (ii)

© Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013