STATISTICS II

Frequency of P

360

Total frequency

CHAPTER 8: STATISTICS I AND II

8.1 Pie chart - used for illustration and comparison of different categories of data.

For a quantity P, the angle of the sector representing P is

Angle of sector =

=

Frequency of P 360

Example

1. The table below shows the number of students

living in 4 villages.

Village

A

B

C

D

No. of students

84

112

56

28

Construct a pie chart to show the distribution of

students living in the four villages.

Answer:

Total number of students = 84+112+56+28

= 280

Angle of the sector for village A =

84

360

280

=

108

Angle of the sector for village B =

112

280

EMBED Equation.DSMT4

360

= 114

Angle of the sector for village C =

56

360

280

= 72

Angle of the sector for village D =

28

360

280

= 36

Pie Chart

Example 8.11

The table below shows the sale of items at the school co-operative book store.

Item

Exe. Book

Art Block

Graph Book

Pen

Eraser

Units

200

130

90

160

140

Calculate the angles of sectors representing each of the items. Construct a pie chart to show the distribution of the items.

Total number of units = 200+130+90+160+140

=

Angle of the sector for exercise books =

=

Angle of the sector for art blocks =

=

Angle of the sector for graph books =

=

Angle of the sector for pens =

=

Angle of the sector for erasers =

Pie Chart

Example

2.

The pie chart shows the types of food sold by a hawker in a day. The total sale was RM570.

i) What percentage is the sale for fried noodle?

140

100%

360

= 38.89 %

ii) What is the angle representing the sale of curry

noodle?

Angle of sector for curry noodle

= 360 140 40 60

= 120

Thus, the sale for curry noodle is

=

120

570

360

= RM190

Exercise 8.1.2

2.

The pie chart shows the hobbies of a group of 180

students.

i) Find the percentage of students who like to surf

the internet.

=

=

ii) Find the number of students who like

swimming ?

Angle of sector for swimming

= 360 130 120 60

=

Number of students who like

swimming

=

=

8.2.MODE

The mode is the most common value or the item with the highest frequency

Examples

Find the mode for the following set of data.

a.2, 1, 3, 2, 5, 3, 2, 5, 6, 3, 3, 4

Mode = 3

b.7, 4, 3, 4, 8, 7, 5, 4, 3

Mode = 4

c.

Number

Of goals

0

1

2

3

4

Frequency

3

6

7

5

2

Mode = 2

d.

Grade

A

B

C

D

E

Frequency

6

8

7

6

2

Mode = Grade B

Exercise 8.2

a.20 kg, 50 kg, 30 kg, 70 kg, 50 kg,

40 kg, 30 kg, 60 kg, 50 kg, 60 kg,

50 kg, 60 kg

Mode = _____ kg

b.15 cm, 18 cm, 12 cm, 13 cm, 15 cm,

17 cm, 12 cm, 16 cm, 15 cm, 13 cm,

14 cm, 15 cm

Mode = ______ cm

c.4, 5, 4, 2, 1, 4, 6, 7, 5, 6

Mode =

d.

Number of matches played

1

2

3

4

5

Frequency

1

2

5

3

3

Mode = _______matches

e.

Height

( cm )

120

121

122

123

124

Frequency

5

6

8

12

9

Mode = cm

f.

Number

Of goals

0

1

2

3

4

Frequency

2

5

4

4

3

Mode = goals

8.3MEDIAN

The median is the value which is located in the middle of the set of values that has been arranged in increasing or decreasing order.

Examples

Determine the median from the following sets of data.

a.2, 6, 9, 4, 3, 4, 5, 6, 7

2, 3, 4, 4, 5, 6, 6, 7, 9

Median = 5

b.9, 7, 3, 4, 8, 6, 7

= 3, 4, 6, 7, 7, 8, 9

Median = 7

c..10kg, 12kg, 18kg, 10kg, 16kg

= 10, 10, 12, 16, 18

Median = 12kg

d.RM5, RM7, RM9, RM6, RM8, RM6,

RM10, RM9

= 5, 6, 6, 7, 8, 9, 9, 10

Median =

RM7 + RM8

2

= RM7.50

Exercise 8.3

a.3, 2, 3, 4, 7, 2, 1

Median =

b.9, 2, 5, 1, 8, 9, 0

Median =

c.2kg, 4kg, 1kg, 5kg, 4kg, 3kg

Median =

d.20cm, 10cm, 30cm, 10cm, 40cm

Median =

e.

Number

Of goals

0

1

2

3

4

Frequency

2

5

4

4

2

Total frequency = 2 + 5 + 4 + 4 + 2 =

Median = Number of goals of the ___ th

Frequency

= goals

f.

Grade

A

B

C

D

E

Numbers of students

6

12

9

8

2

Total frequency = 6 + 12 + 9 + 8 + 2 =

Median = Grade of the ____th students

=

8.4MEAN

Mean is the sum of all the values divided by the number of data

sum of all the values

Mean =

total number of values

When a set of data is given in a frequency table,

sum of (class mark corresponding freque

ncy)

Mean =

total frequency

Examples

a.5, 2, 6, 3, 4, 6, 5, 2, 3

526346523

Mean =

9

++++++++

=

36

9

= 4

b.3.5m, 2.8m, 4.3m, 3.8m, 3.6m

3.52.84.33.83.6

Mean =

5

++++

=

18

m

5

= 3.6 m

c.

Price (RM)

15

20

25

30

35

Frequency

3

4

2

4

3

Mean =

3

4

2

4

3

)

3

35

(

)

4

30

(

)

2

25

(

)

4

20

(

)

3

15

(

+

+

+

+

+

+

+

+

=

16

105

120

50

80

45

+

+

+

+

=

16

400

= RM 25

Exercise 8.4

a.20, 30, 25, 27, 31, 35

Mean =

=

=

b.7 cm, 8cm, 6cm, 9cm, 5cm, 7cm

Mean =

=

=

c.

Mass (g)

3.5

3.6

3.7

3.8

Frequency

5

3

4

3

Mean =

=

=

=

Subjective Questions I :

A) A Mathematics quiz is carried out in a class.

Below are the marks obtained by the students

from that class.

3

2

5

4

4

5

4

2

2

1

5

3

5

3

4

5

4

2

5

2

2

5

3

1

1. From the data above, complete the frequency table below.

Marks

Frequency

1

2

3

4

5

2. How many students scored 4 marks?

3. Draw a pie chart based on the data above.

4. Calculate the mean mark.

5. Construct a bar chart based on the data above.

6. State the median of the marks.

Objective Questions I:

1. How many students are there in that class?

A. 12C.20

B. 18D.24

2. State the number of students who scored

3 marks only.

A. 2C.5

B. 4D.6

3. How many students scored the highest

marks?

A. 4C.6

B. 5D.7

4. How many students scored less than 3

marks?

A. 2C.8

B. 4D.12

5. How many students scored 4 marks and above?

A. 5C.10

B. 7D.12

6. State the mode of the data.

A. 2C.6

B. 5D.7

7. State the percentage of students scored 2 marks.

A. 3.3C.15

B. 8.3D.25

8. How many percent of the students scored more than 3 marks?

A. 2C.50

B. 5D.67

9. If the passing mark is 3, how many students failed?

A. 2C.8

B. 6D.12

10. State the percentage of failure.

A. 10C.33

B. 25D.50

B) The daily allowance (in RM) of 10 students are

shown below.

1, 2, 1, 2, 2, 1, 2, 3, 1, 2,

Subjective Questions II:

1. State the lowest value in the data.

2. State the mode of the data.

3. Construct a bar chart based on the data.

4. Calculate the average allowance brought by the students.

5. Find the median data.

Objective Questions II :

1. How many students brought RM 1 to school?

A. 1C.3

B. 2D.4

2. What is the highest value of the data?

A. 2C.4

B. 3D.5

3. How many students brought more than RM 1 to school?

A. 2C.5

B. 4D.6

4. Calculate the total allowance brought by all students.

A. 6C.13

B. 10D.17

5. State the percentage of students who brought RM2.

A. 30%C.50%

B. 40%D.60%

6. State the percentage of students who brought less than RM 2.

A. 40%C.60%

B. 50%D.90%

7. State the fraction of students who brought RM 1.

A.

5

1

C.

5

3

B.

5

2

D.

5

4

8. What is the difference, in RM, between the mode and the mean of the data above.

A. 1.5C.2.3

B. 1.7D.4.0

9. If there is an additional 2 students who brought RM3 each, calculate the new mean, in RM.

A. 1.50C.1.86

B. 1.70D.1.92

10. If the data above is represented in a pie chart, calculate the angle for the RM1 sector.

A. 120C.180

B. 144D.210

8.5 Questions Based on the Examination Format

1. The bar chart shows the number of books sold in a book store in 6 months.

Calculate the percentage of books sold in April.

A. 20C. 24

B. 22D. 26

2. The table shows the marks obtained by 50 students in a Geography test.

Marks

No. of Students

0 39

40 49

50 59

60 79

80 100

5

8

9

21

7

Calculate the percentage of students who obtained more than 49 marks.

A. 40C. 60

B. 56D. 74

3. .

The line graph shows the sale of chocolate cakes in a week. Given that a piece of chocolate cake costs RM0.60, calculate the sale of chocolate cakes on Saturday and Sunday.

A. RM60C. RM96

B. RM100D. RM160

Number of books

2

3

4

5

6

Frequency

4

6

5

a

8

4. The frequency table shows the number of books in the bags of a group of students.

Given that the mode is 6 books, find the greatest possible value of a.

A. 5C. 6

B. 7D. 8

5. The pictogram shows the number of students who obtained grade A in three history tests.

January

Mac

May

represents 6 students

The average number of students who scored A in a month is

A. 10C. 18

B. 16D. 26

6. The pictogram shows the number of pencils owned by Ali, Lim and Bala.

Ali

Lim

Bala

represents x pencils

Find the value of x if the total number of pencils is 36.

A. 3C. 5

B. 4D. 6

For questions 7 and 8, please refer to the following frequency table.

Mass (kg)

Frequency

50

2

52

2

54

6

56

2

58

3

60

5

7. The median, in kg, is

A. 54C. 56

B. 55D. 58

8. The mean, in kg, is

A. 55.8C. 56.2

B. 55.7D. 57.5

9. The bar chart shows the number of coconuts sold by Ahmad in five days.

Calculate the mean for the number of coconuts sold per day.

A. 150C. 240

B. 200D. 300

For questions 10 and 11, please refer to the bar chart below.

10. The total number of boys and girls in the 3 schools is

A. 1800C. 1200

B. 2000D. 3000

11. The total difference in the number of boys and girls in the 3 schools is

A. 300C. 500

B. 400D. 600

12. The table shows the scores and their respective frequencies in a match.

Scores

1

2

3

4

5

Frequency

3

5

8

4

6

What is the mode score?

A. 2C. 4

B. 3D. 5

13. The pie chart shows the number of coloured balls in a store.

Calculate the angle of the sector which represents black balls.

A. 110C. 125

B. 100D. 150

14. Given that the mean for 3, 4, p, 7, 5, 6, 4 and 3 is 5. Find the value of p.

A. 5C. 7

B. 6D. 8

15. The table shows the scores obtained by a group of students in a match.

Scores

2

4

6

8

Number of students

3

2

3

4

Find the median score.

A. 5C. 6

B. 5.5D. 7

8.6 Past Year SPM Questions

Paper 1

November 2003

1. The pictograph shows the sale of magazines in a particular week. The number of sports magazines sold is not shown.

Fashion

((((((((

Entertainment

(((((

Sports

( represents 12 copies of magazines

Entertainment magazines make up 25% of the total the sales that week. Calculate the number of sports magazines sold.

A26C84

B39D94

2. The pie chart shows the amount spent on five

types of groceries in a shop on a particular day.

flour

margarine

salt

sugar

rice

Given that the total spent on these groceries is

RM400. The amount spent on margarine is RM 80.

Find the ratio of the amount spent on margarine to the amount spent on sugar.

A1:5C3: 10

B3:4D4: 15

July 2004

3. The pictograph which shows the number of visitors at an Expo on three particular days.

Monday

Tuesday

Wednesday

Thursday

Friday

Represents 150 visitors

A total of 5400 visitors were at the Expo over the five days. The number of visitors on Friday was twice the number of visitors on Wednesday.

How many visitors were there on Tuesday?

A. 1050C. 1500

B. 1450D. 2175

4. The table shows the heights of samples of seedlings.

Heights (in cm)

9 11

12 14

15 17

18- 20

No. of seedlings

5

6

10

4

Calculate the mean height, in cm, of the seedlings.

A. 13.06C. 14.56

B. 13.56D. 15.56

5.

The pie chart shows the hobbies of a group of teenagers. The number of teenagers whose hobby is drawing is 50% of the number whose hobby is swimming. Calculate the ratio of the number of teenagers whose hobby is swimming to the number whose hobby is fishing.

A. 2 : 1C. 3 : 1

B. 2 : 3D. 4 : 3

November 2004

6. The bar chart shows the scores of a group of pupils in a test.

Number of pupils

1 2 3 4 5 Score

State the modal score.

A6C3

B5D2

7. The pie chart shows the combined results of a test

taken by two groups of students. The table

shows the results of the groups, but it is

incomplete.

passfail

Group

No. of passes

No. of failures

Aman

160

70

Bestari

TOTAL

420

Calculate the number of students in the Bestari

Group.

A300C340

B310D380

July 2005

8. The table shows the marks obtained by a group of 40 pupils in a test.

Marks

50 59

60 69

70 79

80- 89

No. of pupils

2

8

19

11

If the pass mark is 60, calculate the percentage of these pupils who passed.

A. 20C. 80

B. 75D. 95

9. The pictograph shows the number of books sold in three particular days.

Thursday

Friday

Saturday

Represents x books

The mean number of books sold per day is 60 books. Calculate the number of books sold on Saturday.

A. 20C. 105

B. 28D. 140

10. The table is a cumulative frequency table which shows the height of 50 young plants.

Height (cm)

10 12

13 15

16 18

19 21

22 - 24

Cumulative Frequency

4

16

34

44

50

Which of the following is the correct calculation for the mean height of the 50 young plants?

A.

50

)

6

(

22

)

10

(

19

)

18

(

16

)

12

(

13

)

4

(

10

+

+

+

+

B.

50

)

50

(

22

)

44

(

19

)

34

(

16

)

16

(

13

)

4

(

10

+

+

+

+

C.

50

)

6

(

23

)

10

(

20

)

18

(

17

)

12

(

14

)

4

(

11

+

+

+

+

D.

50

)

50

(

24

)

44

(

21

)

34

(

18

)

16

(

15

)

4

(

12

+

+

+

+

November 2005

11. Diagram below is a pictograph which shows the

number of fruit trees in an orchard.

Durian

Rambutan

Mangosteen

represents 25 trees

The ratio of rambutan trees to mangosteen trees is 3 : 2 . Find the number of durian trees and mangosteen trees altogether.

A. 225C. 300

B. 275D. 400

12. The frequency table shows the mass of fish caught by 20 fishermen.

Mass of fish caught (kg)

Frequency

11 13

14 16

17 19

20 22

23 - 25

7

4

2

2

5

Calculate the mean mass, in kg, of fish caught by a fisherman.

A. 16.1C. 18.1

B. 17.1D. 20.0

July 2006

13. The frequency distribution shows the scores obtained by a group of pupils in a game.

Score

1

2

3

4

5

6

Frequency

6

k

9

9

8

6

If 2 is the modal score, the minimum value of k is

A. 11C. 9

B. 10D. 6

14. Diagram is a bar chart showing the number of candidates who obtained scores 1 to 5 in an achievement test.

Calculate the mean score.

A. 3.0C. 3.5

B. 3.1D. 4.0

November 2006

15. The table below shows the distribution of the scores of a group of pupils in a quiz.

Score

0

1

2

3

4

5

Frequency

2

6

1

1

5

1

The median score of the distribution is

A 1C 2

B 1.5D 2.5

16. The diagram below is a pictograph showing the number of eggs sold in January and February. The number of eggs sold in March and April are not shown

January

0 0 0 0 0 0

February

0 0 0 0 0 0 0 0 0

March

April

0 represents 200 eggs

A total of 5000 eggs were sold in those four months. The number of eggs sold in March was three times the number of eggs sold in April.

The number of eggs sold in March was

A 1500C 2250

B 2000D 3000

17. The diagram shows a set of six pieces of data where x represents an integer.

The mode for the data is 3 and the median is 4. Two new pieces of data, 4 and 7, were put into the set. Calculate the mean for those eight pieces of data.

A 4.00C 4.25

B 4.13D 4.50

Feb

Jan

Swimming

Play football

Watch

TV

Surf the Internet

130o

60o

Which data has the highest frequency !!!

The highest

frequency

Re-arrange the numbers in sequence

120

32

40

280

FAIL

3, 3, 6, x, x, 3

B

60o

D

C

B

A

40o

140o

Fried Noodle

Curry

Noodle

Fried Rice

Noodle

Soup

Mac

120o

Apr

May

June

40

80

120

160

200

240

No. of books

T

W

100

80

60

40

20

Days

M

0

50

250

Mon

Sale of chocolate cakes

0

T

F

S

S

100

150

200

Number of Candidates

Tues

Wed

Thur

Fri

C

Schools

B

Represents boys

1000

800

600

400

0

200

A

1200

Represents girls

Swimming

White

Green

Purple

Black

Score

144o

72o

6

8

12

10

Drawing

Fishing

Reading

25

20

15

10

5

0

5

4

3

2

1

PAGE

13

Statistics I and II

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