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 :

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

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

mode = 11

Example 2 :

Score 2 4 6 8 10

Frequency 3 15 7 12 9

mode = 4

Example 3 :

Score 0 1 2 3 4

Frequency 1 3 7 x 5

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

x < 7 x = 6

Example 4 :

Score 0 1 2 3 4

Frequency 1 7 0 x 2

If mode = 3, the minimum value of x = ???

x > 7 x = 8

(b) Median of a ungrounded data

Median = the middle value when a set of data is arranged in ascending order

Example 1 :

5, 3, 3, 5, 7, 7, 1

1, 3, 3, 5, 5, 7, 7

median = 5

Example 2 :

24, 23, 12, 19, 16, 17

12, 16, 17, 19, 23, 24

median = = 18

Example 3 :

Number of books 1 2 3 4 5

Number of pupils 3 0 1 5 6

1, 1, 1, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5

median = 4

Example 4 :

Saiz of shoes 1 2 3 4 5

Number of students 8 14 12 x 3

median = 3, range of x = ???

8 14 1 11 x 3

8 14 11 1 x 3.

8 + 14 = 11 + x + 322 = x + 14

8 = x

8 + 14 + 11 = x + 333 = x + 3

30 = x

8 x 30

© Hak cipta terpelihara Cikgu Fayruzz Naseer SMKE2 2013


(c) Mean of a ungrounded data

mean = mean =

Example 1 :

68, 62, 84, 75, 78, 89

mean =

= 76

Example 2 :

Mark 74 78 82 86Frequency 5 10 2 3

mean =

= 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 :

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

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

range = 45 4 = 41

Q1 = 6

Q3 = 29

Interquartile range = 29 6 = 23

Example 2 :

8, 12, 6, 10, 6, 7, 13, 3, 8, 10, 13, 19

3, 6, 6, 7, 8, 8, 10, 10, 12, 13, 13, 19

range = 19 3 = 16

Q1 = = 6.5

Q3 = = 12.5

Interquartile range = 12.5 6.5 = 6

(e) Solve problem involving ungorounded data

Example 1 :

Score 1 3 6 x 12 14

Frequency 1 1 2 3 1 1.

1, 3, 6, 6, x, x, x, 12, 14

= 11

x + 12 = 22

x = 22 12

x = 10

Example 2 :

3, 3, 6, x, x, 3


third quartile = 11, x = ???


mode = 3, median = 4. Two new pieces of data, 4 and 7 put into the set, mean = ???

3, 3, 3, x, x, 6 = 4

3 + x = 8

x = 8 3x = 5

mean

=

= 4.5

(f) 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, upper limit= 20

size of class interval = 5

= upper. B lower. B

= midpoint2 midpoint1

lower boundary = 15.5, upper boundary = 20.5

midpoint =

=

= 18

(g) 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 = = =

= 30

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

(h) Histogram, frequency polygons, ogive, first quartile, third quartile, interquartile range

Histogram

lower / upper boundary.

frequency

** the frequency polygon can be

Example : [ base on frequency table in (g) ]



constructed based on a histogram Histogram

2

4

6

10

8

Frequency

Donation

10.5

0

15.5

20.5

25.5

30.5

35.5

40.5

45.5

Frequency polygon

Frequency Polygons

midpoint

frequency

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


2

4

6

10

8

Frequency

Donation 8

0

13

18

23

28

33

38

43

48

Ogive

upper boundary

cumulative frequency

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


34.5

Cumulative Frequency

Donation 0

10.5

15.5

20.5

25.5

30.5

35.5

40.5

45.5

10

20

30

40

30.5 25.5

21 N

43 N

41 N

Q1 Q3 med.



EXERCISES of SPM PAPER 2 FORMAT : (STATISTICS III)

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

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

(a) 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)



(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) Refer graph on the next page.

2 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). [2 marks]

Answer : [2004, No.14]

(a)Class Interval Frequency Mid-point

10 – 12

13 – 15

.

(b)



(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),


(ii) estimate the mean mass. (Ans : 50.906)[4

marks]


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



.

(b) (i)

(ii)





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

Size (cm) Number of Crabs

Mid-point

22 – 24 9

25 – 27 27

28 – 30 11

31 – 33 15

34 – 36 4

(a) Complete the above table in the answer space. [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]


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)Size (cm) Number of

CrabsMid-point

22 – 24 9

25 – 27 27

28 – 30 11

31 – 33 15

34 – 36 4

(b) (i)

(ii)





5 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



(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)


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 :




6 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)

Time (minutes) Frequency, f Mid-point, x

25 – 30 5

31 – 36 7

37 – 42 8

43 – 48 3

49 – 54 2

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

(i) state the upper boundary of the modal class, (Ans : 42.5)

(ii) calculate the mean time. (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 :




7 (a) 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

.



(ii) Refer graph on the next page.

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

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

(a) 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 students families contributed less than RM x. (Ans : 25.5)

(iv) find the value of y, if of the students families contributed at least RM y. (Ans : 38.5)

[4 marks]Answer :

(i) third quartile = __________________________________________________________

(ii) ________________________________________________________________________

(iii) x = ____________________________________________________________________



(iv) y = ____________________________________________________________________

[2003, No.14]



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

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

(a) 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, (Ans : 62.5)

(ii) the median, (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. (Ans : 82)[4 marks]

Answer :(a) (i)

(ii)

(iii)

(b) Refer graph on the next page.

(c) (i)

(ii)

(iii)



(iv)

10 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. (Ans : 3.5) [2 marks]Answer :

(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 :


(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)

(iii)

11 The table shows the distribution of heights, in cm, of 92 students in a school.

Height (cm) Mid-point Frequency

120 - 124 4

125 - 129 10

130 - 134 26

135 - 139 24

140 - 144 17

145 - 149 7

150 - 154 3

155 - 159 1

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

(ii) 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 (cm) Mid-point Frequency

120 - 124 4

125 - 129 10

130 - 134 26

135 - 139 24

140 - 144 17

145 - 149 7

150 - 154 3

155 - 159 1(ii)

(b) (i)Upper Boundary 119.5 159.5

Cumulative Ferquency



(ii) Refer graph on the next page.

(iii)

(iv)

12 The frequency table shows the speeds, in km/h, of 92 motorbikes that passed a certain point along a highway.

735

10

15

20

25

30 F

requ

ency

0 Speed (km/h) 78 83 88 93 98 103 108

5

(a) Based on the frequency polygon, complete the table in the answer space. [4 marks]Answer :

(a)Speed (km/h) Mid-point Upper Boundary Frequency Cumulative

Frequency

0

92



(ii) calculate the estimated mean and give your answer correct to 3 decimal places.

(Ans : 92.239) [3 marks]Answer :

(b) (i)

(ii)


(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).

(ii) From the ogive, state the first quartile. (Ans : 88) [5 marks]



Answer :(c) (i) Refer graph on the next page.

(ii)


chapter 6 statistics iii (notes and exercises)

Documents

graph paper

class interval

refer graph

class interval

2 decimal

3 decimal

frequency

modal class