Q.
If the variance of the frequency distribution
$x_i$
Frequency $f_t$
2
3
3
6
4
16
5
$\alpha$
6
9
7
5
8
6
is 3 , then $\alpha$ is equal to
| $x_i$ | Frequency $f_t$ |
|---|---|
| 2 | 3 |
| 3 | 6 |
| 4 | 16 |
| 5 | $\alpha$ |
| 6 | 9 |
| 7 | 5 |
| 8 | 6 |
Solution:
$x_i$
$f_i$
$d_i =
x_i
- 5$
$f_i
d_i
^2$
$f_id_i$
2
3
-3
27
-9
3
6
-2
24
-12
4
16
-1
16
-16
5
$\alpha$
0
0
0
6
9
1
9
9
7
5
2
20
10
8
6
3
54
18
$ \sigma_x^2=\sigma_d^2=\frac{\sum f _{ i } d _{ i }{ }^2}{\sum f _{ i }}-\left(\frac{\sum f _{ i } d _{ i }}{\sum f _{ i }}\right)^2 $
$ =\frac{150}{45+\alpha}-0=3 $
$\Rightarrow 150=135+3 \alpha$
$ \Rightarrow 3 \alpha=15 \Rightarrow \alpha=5$
| $x_i$ | $f_i$ | $d_i = x_i - 5$ | $f_i d_i ^2$ | $f_id_i$ |
|---|---|---|---|---|
| 2 | 3 | -3 | 27 | -9 |
| 3 | 6 | -2 | 24 | -12 |
| 4 | 16 | -1 | 16 | -16 |
| 5 | $\alpha$ | 0 | 0 | 0 |
| 6 | 9 | 1 | 9 | 9 |
| 7 | 5 | 2 | 20 | 10 |
| 8 | 6 | 3 | 54 | 18 |