Q.
Find the standard deviation for the following data:
$x_{i}$
4
8
11
17
20
24
32
$f_{i}$
3
5
9
5
4
3
1
| $x_{i}$ | 4 | 8 | 11 | 17 | 20 | 24 | 32 |
| $f_{i}$ | 3 | 5 | 9 | 5 | 4 | 3 | 1 |
Statistics
Solution:
$x_{i}$
$f_{i}$
$f_{i}x_{i}$
$(x_{i}- \bar{x})$
$(x_{i}-\bar{x})^{2}$
$f_{i}(x_{i}-\bar{x})^{2}$
4
3
12
-10
100
300
8
5
40
-6
36
180
11
9
99
-3
9
81
17
5
85
3
9
45
20
4
80
6
36
144
24
3
72
10
100
300
32
1
32
18
324
324
Total
30
420
1374
$N=30, \displaystyle\sum_{i=1}^{7} f_{i} x_{i}=420$,
$ \displaystyle\sum_{i=1}^{7} f_{i}\left(x_{i}-\bar{x}\right)^{2}=1374$
Therefore, $\bar{x}=\frac{\displaystyle\sum_{i=1}^{7} f_{i} x_{i}}{N}=\frac{1}{30} \times 420=14$
Hence, variance $\left(\sigma^{2}\right)$
$=\frac{1}{N} \displaystyle\sum_{i=1}^{7} f_{i}\left(x_{i}-\bar{x}\right)^{2}$
$=\frac{1}{30} \times 1374=45.8$
and Standard deviation $(\sigma)=\sqrt{45.8}=6.77$
| $x_{i}$ | $f_{i}$ | $f_{i}x_{i}$ | $(x_{i}- \bar{x})$ | $(x_{i}-\bar{x})^{2}$ | $f_{i}(x_{i}-\bar{x})^{2}$ |
|---|---|---|---|---|---|
| 4 | 3 | 12 | -10 | 100 | 300 |
| 8 | 5 | 40 | -6 | 36 | 180 |
| 11 | 9 | 99 | -3 | 9 | 81 |
| 17 | 5 | 85 | 3 | 9 | 45 |
| 20 | 4 | 80 | 6 | 36 | 144 |
| 24 | 3 | 72 | 10 | 100 | 300 |
| 32 | 1 | 32 | 18 | 324 | 324 |
| Total | 30 | 420 | 1374 |