Q.
Find the mean and standard deviation for the following
data :
$x_i$
6
10
14
18
24
28
30
$f_i$
2
4
7
12
8
4
3
| $x_i$ | 6 | 10 | 14 | 18 | 24 | 28 | 30 |
| $f_i$ | 2 | 4 | 7 | 12 | 8 | 4 | 3 |
Statistics
Solution:
Calculation for Mean and Standard Deviation
$x_i$
$f_i$
$f_i x_i$
$f_i x_i^2$
6
2
12
72
10
4
40
400
14
7
98
1372
18
12
216
3888
24
8
192
4608
28
4
112
3136
30
3
90
2700
130
$\sum f_i = 40$
$\sum f_i x_i = 760$
$\sum f_i x_i^2 = 16176$
Mean $=\frac{\Sigma f_{i} x_{i}}{\Sigma f_{i}}=\frac{760}{40}=19$
S.D. $=\sqrt{\frac{\Sigma f_{i} x_{i}^{2}}{\Sigma f_{i}}-\left(\frac{\Sigma f_{i} x_{i}}{\Sigma f_{i}}\right)^{2}}$
$=\sqrt{\frac{16176}{40}-\left(\frac{760}{40}\right)^{2}} $
$=\sqrt{404.4-361}=\sqrt{43.4}=6.59 .$
Hence, Mean $=19, \text { S.D. }=6.59 . $
| $x_i$ | $f_i$ | $f_i x_i$ | $f_i x_i^2$ |
|---|---|---|---|
| 6 | 2 | 12 | 72 |
| 10 | 4 | 40 | 400 |
| 14 | 7 | 98 | 1372 |
| 18 | 12 | 216 | 3888 |
| 24 | 8 | 192 | 4608 |
| 28 | 4 | 112 | 3136 |
| 30 | 3 | 90 | 2700 |
| 130 | $\sum f_i = 40$ | $\sum f_i x_i = 760$ | $\sum f_i x_i^2 = 16176$ |