Q.
The S.D. of the following data is nearly
${x_i}$
140
145
150
155
160
165
170
175
${f_i}$
4
6
15
30
36
24
8
2
| ${x_i}$ | 140 | 145 | 150 | 155 | 160 | 165 | 170 | 175 |
| ${f_i}$ | 4 | 6 | 15 | 30 | 36 | 24 | 8 | 2 |
Statistics
Solution:
Let us assume an arbitary mean a = 155.
Following table is constructed :
$X_{i}$
$f_{i}$
$u_{i} - \frac{X_{i} - 155}{5}$
$u_{i}^{2}$
$f_{i} u_{i}$
$f_{i} u_{i}^{2}$
140
4
-3
9
-12
36
145
6
-2
4
-12
24
150
15
-1
1
-15
15
155
30
0
0
0
0
160
36
1
1
36
36
165
24
2
4
48
96
170
8
3
9
24
72
175
2
4
16
8
32
Total
125
77
311
$\therefore $ Variance = $= \sigma^{2}=c^{2} \left(\frac{\sum f_{i} u_{i}^{2}}{n} -\left(\frac{\sum f_{i}u_{i}}{n}\right)^{2}\right) $
$= 25 \times\left(\frac{311}{125}-\left(\frac{77}{125}\right)^{2}\right)$
$=25 \times\frac{311}{125}-\frac{25\times77\times77}{125\times125} $
$= 62.2 -9.4864 = 52.7136 $
$\Rightarrow \:\: S.D = \sqrt{52.7136} = 7.26$ nearly
| $X_{i}$ | $f_{i}$ | $u_{i} - \frac{X_{i} - 155}{5}$ | $u_{i}^{2}$ | $f_{i} u_{i}$ | $f_{i} u_{i}^{2}$ |
| 140 | 4 | -3 | 9 | -12 | 36 |
| 145 | 6 | -2 | 4 | -12 | 24 |
| 150 | 15 | -1 | 1 | -15 | 15 |
| 155 | 30 | 0 | 0 | 0 | 0 |
| 160 | 36 | 1 | 1 | 36 | 36 |
| 165 | 24 | 2 | 4 | 48 | 96 |
| 170 | 8 | 3 | 9 | 24 | 72 |
| 175 | 2 | 4 | 16 | 8 | 32 |
| Total | 125 | 77 | 311 |