ELEMENTARY STATISTICS
Question 1
a)
Class

Frequency

Upper Boundary

Cumulative Process

Cumulative Frequency

6  8

4

≤ 8.5

4

4

9  11

6

≤ 11.5

4 + 6

10

12  14

10

≤ 14.5

10 + 10

20

15  17

3

≤ 17.5

20 + 3

23

18  20

12

≤ 20.5

23 + 12

35


Sum = 35




Table
1.1: “Less than or Equal” cumulative frequency polygon
Table
2.2: Cumulative Frequency of types “Less than or equal” polygon
b) Number of Classes,
K= 1 + 3.3log
(n)
K= 1 + 3.3log
(40)
K= 6.28

Class width
given = 10
Lower limit
given = 50

Number of classes

Class

Counting Tally

Frequency (f)

1^{st} class

5059


6

2^{nd} class

6069


17

3^{rd} class

7079


8

4^{th} class

8089


9


Sum


40

Table
1.3: Frequency distribution table
Class

5059

6069

7079

8089

Frequency (f)

6

17

8

9

Table
1.4: Frequency distribution table
Question 2
Class
(Given)

Class midpoint (x)

Frequency (f)
(Given)



1.8  2.1


2

4.3

9.245

2.6 – 3.3


4

11.8

34.81

3.4 – 4.1


6

22.5

84.375

4.2 – 4.9


13

59.15

269.1325

5.0 – 5.7


8

42.8

228.98

5.8 – 6.5


3

18.45

113.4675


SUM

36

159

740.01

Table 2.1: Class midpoint,
and
calculation
Mean
Thus, the mean is 4.4
Mode
Class

Frequency (f)

Class Boundaries

Class width

4.2  4.9

13

4.15  4.95

(4.15 – 4.95) = 0.8

Calculation of class boundary
and
Class mode with the largest
frequency is 4.2 – 4.9 (as per table 2.1).
Lower boundary
=4.15
Class width, C = 0.8
The mode formula
Thus, the mode is 4.62
Standard Deviation
Therefore, the standard deviation is 1.02
Question 3
a) Frequency distribution table.
Number
of Class,
K= 1 + 3.3log (n)
K= 1 + 3.3log (40)
K= 6.28
6 to 7 classes
Class width and
class limits
Class width = range / number of classes
= Largest number 
smallest number / number of classes
Number of Classes

Class

Counting Tally

Frequency (f)

1^{st} Class

4.8  7.1


5

2^{nd} Class

7.2 – 9.5


9

3^{rd} Class

9.6– 11.9


15

4^{th} Class

12 – 14.3


5

5^{th} Class

14.4 – 16.7

III

3

6^{th} Class

16.8 – 19.1

II

2

7^{th} Class

19.2 – 21.5

I

1


SUM


40

Table
3.1: Frequency distribution table
Class

4.8  7.1

7.2 – 9.5

9.6 – 11.9

12 – 14.3

14.4 – 16.7

16.8 – 19.1

19.2 – 21.5

Frequency (f)

5

9

15

5

3

2

1

Table
3.2: Frequency distribution table
b)
Mean
Class

Class midpoint (x)

Frequency (f)

(f) x (x)

4.8  7.1


5

29.75

7.2 – 9.5


9

75.15

9.6– 11.9


15

161.25

12 – 14.3


5

65.75

14.4 – 16.7


3

46.65

16.8 – 19.1


2

35.9

19.2 – 21.5


1

20.35


SUM

40

434.80

Table
3.3: Class midpoint calculation
Thus, the mean is 10.87
Mode
Class

Frequency (f)

Class Boundaries

Class width

9.6 – 11.9

15

9.55 – 11.95

11.95 – 9.55 = 2.4

Calculation of class boundary
and
Class mode with the largest
frequency is 9.5 – 11.8 (as per table 3.3).
Lower boundary
=9.55
Class width, C = 2.4
The mode formula
Thus, the mode is 10.45
Median
Class

Frequency (f)

Cumulative Frequency (F)

Class
Boundaries

4.8  7.1

5

0+5=5


7.2 – 9.5

9

5+9=14


9.6– 11.9

15

14+15=29

9.4511.85
=2.4(width)

12 – 14.3

5

29+5=34


14.4 – 16.7

3

34+3=37


16.8 – 19.1

2

37+2=39


19.2 – 21.5

1

39+1=40


SUM

40



Table
3.4: cumulative frequency calculation
i) The
position of the median:
ii) The
class median with a cumulative frequency (f) > 20.5 is the class median
9.6 – 11.9.
iii) The median
is:
Thus, the median is 10.59
Question 4
a)
The value of
:
Thus, the value
of x = 4
Question b,c,d and e are based on the table 4.1 below
Class

Number of Students (f)

Cumulative Frequency (F)

Class boundaries

Class width

40  44

2

0+2=2



45  49


2+4=6



50  54

7

6+7=13

49.5 – 54.5

5

55  59


13+16=29

54.5  59.5

5

60  64


29+20=49

59.5 – 64.5

5

65  69

2

49+2=51



70  74

1

51+1=52








SUM

52




Table
4.1: Coursework Statistic Marks
b) First Quartile
Step 1
Step 2
Quartile 1

5559

16

13+16=29

54.5
 59.5

5

Step 3
Therefore, the first quartile is 54.58
c) Median
i) The
position of the median:
ii) The
class median with a cumulative frequency (f) > 26.5 is the class median
55 – 59.
iii) The median
is:
Thus, the median is 58.72.
d) Third Quartile
Step 1
Step 2
Quartile 3

6064

20

29+20=49

59.564.5

5

Step 3
Therefore, the third quartile is 62.19
d) InterQuartile Range
IQR ;
Therefore, the IQR is 7.61