Q.
Consider the following data
$6,8,10,12,14,16,18,20,22,24$
I. The variance of the data is 33.
II. The standard deviation of the data is $4.74$.
Statistics
Solution:
From the given data we can form the following table. The mean is calculated by step-deviation method taking 14 as assumed mean. The number of observations is $n=10$.
$x_i$
$d_i = \frac{x_i -14}{2}$
Deviations from mean$(|x_i - \bar{x}|$
$(x_i - \bar{x})^2$
6
-4
-9
81
8
-3
-7
49
10
-2
-5
25
12
-1
-3
9
14
0
-1
1
18
2
3
9
20
3
5
25
22
4
7
49
24
5
9
81
5
330
$\therefore $ Mean $(\bar{x})=$ Assumed mean $+\frac{\displaystyle\sum_{i=1}^n d_i}{n} \times h$
$=14+\frac{5}{10} \times 2$
$=15$
Variance $\left(\sigma^2\right)=\frac{1}{n} \displaystyle\sum_{i=1}^{10}\left(x_i-\bar{x}\right)^2=\frac{1}{10} \times 330=33$
$\therefore$ Standard deviation $(\sigma)=\sqrt{33}$
$=5.74$
| $x_i$ | $d_i = \frac{x_i -14}{2}$ | Deviations from mean$(|x_i - \bar{x}|$ | $(x_i - \bar{x})^2$ |
|---|---|---|---|
| 6 | -4 | -9 | 81 |
| 8 | -3 | -7 | 49 |
| 10 | -2 | -5 | 25 |
| 12 | -1 | -3 | 9 |
| 14 | 0 | -1 | 1 |
| 18 | 2 | 3 | 9 |
| 20 | 3 | 5 | 25 |
| 22 | 4 | 7 | 49 |
| 24 | 5 | 9 | 81 |
| 5 | 330 |