Q.
Calculate variance for the following distribution.
Class
30-40
40-50
50-60
60-70
70-80
80-90
90-100
Frequency
3
7
12
15
8
3
2
| Class | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 | 90-100 |
| Frequency | 3 | 7 | 12 | 15 | 8 | 3 | 2 |
Statistics
Solution:
Let the assumed mean $A=65 .$ Here, $h=10$.
We obtain the following table from the given data.
Class
Fre quency $f_{i}$
Mid -point $x_{i}$
$y_{i}=\frac{x_{i}-65}{10}$
$y_{i}^{2}$
$f_{i}y_{i}$
$f_{i} y_{i}^{2}$
30-40
3
35
-3
9
-9
27
40-50
7
45
-2
4
-14
28
50-60
12
55
-1
1
-12
12
60-70
15
65
0
0
0
0
70-80
8
75
1
1
8
8
80-90
3
85
2
4
6
12
90-100
2
95
3
9
6
18
total
N=50
-15
105
Therefore $\bar{x}=A+\frac{\sum f_{i} y_{i}}{50} \times h=65-\frac{15}{50} \times 10=62$
Variance $\sigma^{2}=\frac{h^{2}}{N^{2}}\left[N \Sigma f_{i} y_{i}^{2}-\left(\Sigma f_{i} y_{i}\right)^{2}\right]$
$=\frac{(10)^{2}}{(50)^{2}}\left[50 \times 105-(-15)^{2}\right]$
$=\frac{1}{25}[5250-225]=201$
| Class | Fre quency $f_{i}$ | Mid -point $x_{i}$ | $y_{i}=\frac{x_{i}-65}{10}$ | $y_{i}^{2}$ | $f_{i}y_{i}$ | $f_{i} y_{i}^{2}$ |
|---|---|---|---|---|---|---|
| 30-40 | 3 | 35 | -3 | 9 | -9 | 27 |
| 40-50 | 7 | 45 | -2 | 4 | -14 | 28 |
| 50-60 | 12 | 55 | -1 | 1 | -12 | 12 |
| 60-70 | 15 | 65 | 0 | 0 | 0 | 0 |
| 70-80 | 8 | 75 | 1 | 1 | 8 | 8 |
| 80-90 | 3 | 85 | 2 | 4 | 6 | 12 |
| 90-100 | 2 | 95 | 3 | 9 | 6 | 18 |
| total | N=50 | -15 | 105 |