Question 1:
Find the mean and variance for the data 6, 7, 10, 12, 13, 4, 8, 12
Answer:
6, 7, 10, 12, 13, 4, 8, 12
Mean,
The following table is obtained.
xi |
||
6 |
–3 |
9 |
7 |
–2 |
4 |
10 |
–1 |
1 |
12 |
3 |
9 |
13 |
4 |
16 |
4 |
–5 |
25 |
8 |
–1 |
1 |
12 |
3 |
9 |
74 |
Question 2:
Find the mean and variance for the first n natural numbers
Answer:
The mean of first n natural numbers is calculated as follows.
Question 3:
Find the mean and variance for the first 10 multiples of 3
Answer:
The first 10 multiples of 3 are
3, 6, 9, 12, 15, 18, 21, 24, 27, 30
Here, number of observations, n = 10
The following table is obtained.
xi |
||
3 |
–13.5 |
182.25 |
6 |
–10.5 |
110.25 |
9 |
–7.5 |
56.25 |
12 |
–4.5 |
20.25 |
15 |
–1.5 |
2.25 |
18 |
1.5 |
2.25 |
21 |
4.5 |
20.25 |
24 |
7.5 |
56.25 |
27 |
10.5 |
110.25 |
30 |
13.5 |
182.25 |
742.5 |
Question 4:
Find the mean and variance for the data
xi |
6 |
10 |
14 |
18 |
24 |
28 |
30 |
f i |
2 |
4 |
7 |
12 |
8 |
4 |
3 |
Answer:
The data is obtained in tabular form as follows.
xi |
f i |
fixi |
|||
6 |
2 |
12 |
–13 |
169 |
338 |
10 |
4 |
40 |
–9 |
81 |
324 |
14 |
7 |
98 |
–5 |
25 |
175 |
18 |
12 |
216 |
–1 |
1 |
12 |
24 |
8 |
192 |
5 |
25 |
200 |
28 |
4 |
112 |
9 |
81 |
324 |
30 |
3 |
90 |
11 |
121 |
363 |
40 |
760 |
1736 |
Here, N = 40,
Question 5:
Find the mean and variance for the data
xi |
92 |
93 |
97 |
98 |
102 |
104 |
109 |
f i |
3 |
2 |
3 |
2 |
6 |
3 |
3 |
Answer:
The data is obtained in tabular form as follows.
xi |
f i |
fixi |
|||
92 |
3 |
276 |
–8 |
64 |
192 |
93 |
2 |
186 |
–7 |
49 |
98 |
97 |
3 |
291 |
–3 |
9 |
27 |
98 |
2 |
196 |
–2 |
4 |
8 |
102 |
6 |
612 |
2 |
4 |
24 |
104 |
3 |
312 |
4 |
16 |
48 |
109 |
3 |
327 |
9 |
81 |
243 |
22 |
2200 |
640 |
Here, N = 22,
Question 6:
Find the mean and standard deviation using short-cut method.
xi |
60 |
61 |
62 |
63 |
64 |
65 |
66 |
67 |
68 |
fi |
2 |
1 |
12 |
29 |
25 |
12 |
10 |
4 |
5 |
Answer:
The data is obtained in tabular form as follows.
xi |
fi |
yi2 |
fiyi |
fiyi2 |
|
60 |
2 |
–4 |
16 |
–8 |
32 |
61 |
1 |
–3 |
9 |
–3 |
9 |
62 |
12 |
–2 |
4 |
–24 |
48 |
63 |
29 |
–1 |
1 |
–29 |
29 |
64 |
25 |
0 |
0 |
0 |
0 |
65 |
12 |
1 |
1 |
12 |
12 |
66 |
10 |
2 |
4 |
20 |
40 |
67 |
4 |
3 |
9 |
12 |
36 |
68 |
5 |
4 |
16 |
20 |
80 |
100 |
220 |
0 |
286 |
Mean,
Question 7:
Find the mean and variance for the following frequency distribution.
Classes |
0-30 |
30-60 |
60-90 |
90-120 |
120-150 |
150-180 |
180-210 |
Frequencies |
2 |
3 |
5 |
10 |
3 |
5 |
2 |
Answer:
Class |
Frequency fi |
Mid-point xi |
yi2 |
fiyi |
fiyi2 |
|
0-30 |
2 |
15 |
–3 |
9 |
–6 |
18 |
30-60 |
3 |
45 |
–2 |
4 |
–6 |
12 |
60-90 |
5 |
75 |
–1 |
1 |
–5 |
5 |
90-120 |
10 |
105 |
0 |
0 |
0 |
0 |
120-150 |
3 |
135 |
1 |
1 |
3 |
3 |
150-180 |
5 |
165 |
2 |
4 |
10 |
20 |
180-210 |
2 |
195 |
3 |
9 |
6 |
18 |
30 |
2 |
76 |
Mean,
Question 8:
Find the mean and variance for the following frequency distribution.
Classes |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
Frequencies |
5 |
8 |
15 |
16 |
6 |
Answer:
Class |
Frequency fi |
Mid-point xi |
yi2 |
fiyi |
fiyi2 |
|
0-10 |
5 |
5 |
–2 |
4 |
–10 |
20 |
10-20 |
8 |
15 |
–1 |
1 |
–8 |
8 |
20-30 |
15 |
25 |
0 |
0 |
0 |
0 |
30-40 |
16 |
35 |
1 |
1 |
16 |
16 |
40-50 |
6 |
45 |
2 |
4 |
12 |
24 |
50 |
10 |
68 |
Mean,
Question 9:
Find the mean, variance and standard deviation using short-cut method
Height in cms |
No. of children |
70-75 |
3 |
75-80 |
4 |
80-85 |
7 |
85-90 |
7 |
90-95 |
15 |
95-100 |
9 |
100-105 |
6 |
105-110 |
6 |
110-115 |
3 |
Answer:
Class Interval |
Frequency fi |
Mid-point xi |
yi2 |
fiyi |
fiyi2 |
|
70-75 |
3 |
72.5 |
–4 |
16 |
–12 |
48 |
75-80 |
4 |
77.5 |
–3 |
9 |
–12 |
36 |
80-85 |
7 |
82.5 |
–2 |
4 |
–14 |
28 |
85-90 |
7 |
87.5 |
–1 |
1 |
–7 |
7 |
90-95 |
15 |
92.5 |
0 |
0 |
0 |
0 |
95-100 |
9 |
97.5 |
1 |
1 |
9 |
9 |
100-105 |
6 |
102.5 |
2 |
4 |
12 |
24 |
105-110 |
6 |
107.5 |
3 |
9 |
18 |
54 |
110-115 |
3 |
112.5 |
4 |
16 |
12 |
48 |
60 |
6 |
254 |
Mean,
Question 10:
The diameters of circles (in mm) drawn in a design are given below:
Answer:
Diameters |
No. of children |
33-36 |
15 |
37-40 |
17 |
41-44 |
21 |
45-48 |
22 |
49-52 |
25 |
Class Interval |
Frequency fi |
Mid-point xi |
fi2 |
fiyi |
fiyi2 |
|
32.5-36.5 |
15 |
34.5 |
–2 |
4 |
–30 |
60 |
36.5-40.5 |
17 |
38.5 |
–1 |
1 |
–17 |
17 |
40.5-44.5 |
21 |
42.5 |
0 |
0 |
0 |
0 |
44.5-48.5 |
22 |
46.5 |
1 |
1 |
22 |
22 |
48.5-52.5 |
25 |
50.5 |
2 |
4 |
50 |
100 |
100 |
25 |
199 |
Here, N = 100, h = 4
Let the assumed mean, A, be 42.5.
Mean,