Q.
Find the standard deviation for the following data:
$x_i$
3
8
13
18
23
$f_i$
7
10
15
10
6
| $x_i$ | 3 | 8 | 13 | 18 | 23 |
| $f_i$ | 7 | 10 | 15 | 10 | 6 |
Statistics
Solution:
Let us form the following table:
$x_i$
$f_i$
$ f_i x_i $
$x_i ^2$
$f_ix_i ^2$
3
7
21
9
63
8
10
80
64
640
13
15
195
169
2535
18
10
180
324
3240
23
6
138
529
3174
Total
48
614
9652
Now, $\left(\sigma\right) = \frac{1}{N} \sqrt{N\Sigma f_{i}x_{i}^{2} - \left(\Sigma f_{i} x_{i}\right)^{2}} $
$ = \frac{1}{48}\sqrt{48\times9652 -\left(614\right)^{2}} $
$ = \frac{1}{48} \sqrt{463296 - 376996}$
$ = \frac{1}{48} \times 293.77$
$= 6.12$
| $x_i$ | $f_i$ | $ f_i x_i $ | $x_i ^2$ | $f_ix_i ^2$ |
|---|---|---|---|---|
| 3 | 7 | 21 | 9 | 63 |
| 8 | 10 | 80 | 64 | 640 |
| 13 | 15 | 195 | 169 | 2535 |
| 18 | 10 | 180 | 324 | 3240 |
| 23 | 6 | 138 | 529 | 3174 |
| Total | 48 | 614 | 9652 |