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Q.
The variance of 20 observations is 5 . If each observation is multiplied by 2 , then the new variance of the resulting observations is
Statistics
Solution:
Let the observations be $x_1, x_2, \ldots x_{20}$ and $\bar{x}$ be their mean. Given that, variance $=5$ and $n=20$. We know that,
Variance$\left(\sigma^2\right)=\frac{1}{n} \displaystyle\sum_{i=1}^{20}\left(x_i-\bar{x}\right)^2 $.
i.e., $5=\frac{1}{20} \displaystyle\sum_{i=1}^{20}\left(x_i-\bar{x}\right)^2$ or $\displaystyle\sum_{i=1}^{20}\left(x_i-\bar{x}\right)^2=100 .....$(i)
If each observation is multiplied by 2 and the new resulting observations are $y_i$, then
$y_i=2 x_i \text { i.e., } x_i=\frac{1}{2} y_i$
Therefore, $ \bar{y}=\frac{1}{n} \displaystyle\sum_{i=1}^{20} y_i=\frac{1}{20} \displaystyle\sum_{i=1}^{20} 2 x_i=2 \cdot \frac{1}{20} \displaystyle\sum_{i=1}^{20} x_i$
i.e., $\bar{y}=2 \bar{x} $ or $ \bar{x}=\frac{1}{2} \bar{y}$
On substituting the values of $x_i$ and $\bar{x}$ in Eq. (i), we get
$\displaystyle\sum_{i=1}^{20}\left(\frac{1}{2} y_i-\frac{1}{2} \bar{y}\right)^2=100$
i.e., $ \displaystyle\sum_{i=1}^{20}\left(y_i-\bar{y}\right)^2=400$
Thus, the variance of new observations
$=\frac{1}{20} \times 400=20=2^2 \times 5$
Note The reader may note that, if each observation is multiplied by a constant $k$, the variance of the resulting observations becomes $k^2$ times the original variance.