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 6, or 6 to 7 classes

Class width given = 10
Lower limit given = 50


Number of classes
Class
Counting Tally
Frequency (f)
1st class
50-59
IIII I
6
2nd class
60-69
IIII IIII IIII II
17
3rd class
70-79
IIII III
8
4th class
80-89
IIII IIII
9

Sum

40
Table 1.3: Frequency distribution table



Class
50-59
60-69
70-79
80-89
Frequency (f)
6
17
8
9
Table 1.4: Frequency distribution table







Question 2
           
Class
(Given)
Class mid-point (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 mid-point, 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 difference between the frequency of the class mode and the frequency of the class immediately before class mode, that is 13-6 = 7.
 The difference between the frequency of the class mode and the frequency of the class immediately after class mode, that is 13-8 = 5
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)
1st Class
4.8 - 7.1
IIII
5
2nd Class
7.2 – 9.5
IIII IIII
9
3rd Class
9.6– 11.9
IIII IIII IIII
15
4th Class
12 – 14.3
IIII
5
5th Class
14.4 – 16.7
III
3
6th Class
16.8 – 19.1
II
2
7th 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 mid-point (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 mid-point 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 difference between the frequency of the class mode and the frequency of the class immediately before class mode, that is 15-9 = 6.
 The difference between the frequency of the class mode and the frequency of the class immediately after class mode, that is 15-5 = 10

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
9.45-11.85
=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
Cumulative frequency (f) of class quartile with F > 13.25, that is F = 29
55-59
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
Cumulative frequency (f) of class quartile with F > 75.75, that is F = 49
60-64
20
29+20=49
59.5-64.5
5


Step 3
Therefore, the third quartile is 62.19



d)         Inter-Quartile Range

IQR ;
 

Therefore, the IQR is 7.61

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