Q.
The variance of the following data is
$x_{i}$
1
2
3
4
5
6
7
8
9
10
$f_{i}$
1
2
3
4
5
6
7
8
9
10
| $x_{i}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| $f_{i}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
AP EAMCETAP EAMCET 2019
Solution:
According to given data ,
$x_{i}$
$f_{i}$
$x_{i}f_{i}$
$f_{i}\left(\bar{x}-x_{i}\right)^{2}$
1
1
1
6
36
2
2
4
5
50
3
3
9
4
48
4
4
16
3
36
5
5
25
2
20
6
6
36
1
6
7
7
49
0
0
8
8
64
1
8
9
9
81
2
36
10
10
100
3
90
$\because \, N =\Sigma f_{i}=55, \bar{x}=\frac{\Sigma x_{i} f_{i}}{N} $
$=\frac{10 \times 11 \times 21}{6 \times 55}=7$
$\therefore \, \sigma^{2}=\frac{\Sigma f_{i}\left(\bar{x}-x_{i}\right)^{2}}{N}$
$=\frac{36+50+48+36+20+6+8+36+90}{55} $
$=\frac{330}{55}=6$
| $x_{i}$ | $f_{i}$ | $x_{i}f_{i}$ | $f_{i}\left(\bar{x}-x_{i}\right)^{2}$ | |
|---|---|---|---|---|
| 1 | 1 | 1 | 6 | 36 |
| 2 | 2 | 4 | 5 | 50 |
| 3 | 3 | 9 | 4 | 48 |
| 4 | 4 | 16 | 3 | 36 |
| 5 | 5 | 25 | 2 | 20 |
| 6 | 6 | 36 | 1 | 6 |
| 7 | 7 | 49 | 0 | 0 |
| 8 | 8 | 64 | 1 | 8 |
| 9 | 9 | 81 | 2 | 36 |
| 10 | 10 | 100 | 3 | 90 |