Class 11 RD Sharma Solutions – Chapter 32 Statistics – Exercise 32.5
Question 1. Find the standard deviation for the following distribution:
x: y: | 4.5 1 | 14.5 5 | 24.5 34.5 12 22 | 44.5 17 | 54.5 64.5 9 4 |
Solution:
x
4.5
14.5
f fx x-mean
1 4.5 -33.14
5 72.5 -23.14
(x-mean)^{2}
1098.45
535.59
f(x-mean)^{2}
1098.45
2677.96
24.5
34.5
12 294 -13.14
22 759 -3.14
172.73
9.88
2072.82
217.31
44.5 17 756.5 6.86 47.02 799.35 54.5 9 490.5 16.86 284.16 2557.47 64.5 4 258 26.86 721.31 2885.22 N = 70 2635 12308.57 Here, N = 70,
We have,
Question 2. Calculate the mean and variance.
x: y: | 0 51 | 1 2 203 383 | 3 525 | 4 532 | 5 408 | 6 273 | 7 139 | 8 9 43 27 | 10 10 | 11 4 | 12 2 |
Solution:
X
F
Fx x-mean
F(x-mean)
(x-mean)^{2}
F(x-mean)^{2}
0
1
51
203
0 -6
203 -5
-306
-1015
36
25
1836
5075
2
3
383
525
766 -4
1575 -3
-1532
-1575
16
9
6128
4725
4
5
6`
532
408
273
2128 -2
2040 -1
1638 0
-1064
-408
0
4
1
0
2128
408
0
7
8
139
43
973 1
344 2
139
86
1
4
139
172
9
10
27
10
243 3
100 4
81
40
9
16
243
160
11
12
4
2
44 5
24 6
20
12
25
36
100
72
2600
10078 -5522
21186
Here, N = 2600,
Since,
Question 3. Find the mean, and the standard deviation for the following data:
(i)
Year render: No. of persons (cumulative) | 10 15 | 20 32 | 30 51 | 40 50 60 78 97 109 |
(ii)
Marks: | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
Frequency: | 1 | 6 | 6 | 8 | 8 | 2 | 2 | 3 | 0 | 2 | 1 | 0 | 0 | 0 | 1 |
Solution:
(i)
x_{i}
Cumulative frequency
f_{i}
f_{i}x_{i}
f_{i}x_{i}^{2}
10
15
15
150
1500
20
32
17
340
6800
30
51
19
570
17100
40
78
27
1080
43200
50
97
19
950
47500
60
109
12
720
43200
N = 109
Total = 3810
Total = 159300
(ii)
x_{i}
f_{i}
f_{i}x_{i}
f_{i}x_{i}^{2}
2
1
2
4
3
6
18
54
4
6
24
96
5
8
40
200
6
8
48
288
7
2
14
98
8
2
16
128
9
3
27
243
10
0
0
0
11
2
22
242
12
1
12
144
13
0
0
0
14
0
0
0
15
0
0
0
16
1
16
256
N = 40
Total = 239
Total = 1753
Question 4. Find the standard deviation for the following data:
(i)
x: y: | 3 7 | 8 10 | 13 18 15 10 | 23 6 |
(ii)
x | 2 | 3 | 4 | 5 | 6 | 7 |
f | 4 | 9 | 16 | 14 | 11 | 6 |
Solution:
(i)
x f
3 7
8 10
fx x-mean (x-mean)^{2}
21 -9.79 95.88
80 -4.79 22.96
f(x-mean)^{2}
671.13
229.60
13 15
18 10
195 0.21 0.04
180 5.21 27.13
0.65
271.26
23 6 138 10.21 104.21 625.26 48 614 1797.92 Here, N = 48, and
(ii)
x_{i }
f_{i }
f_{i}x_{i}^{2 }
2
4
16
3
9
81
4
16
256
5
14
350
6
11
396
7
6
294
N = 60
Total = 1393
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