The variance of 20 observations is 5 . If each one of the observations is multiplied by 2 , then the variance of the resulting observations is

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Aicharya · Accepted Answer

Variance $=5$ and $n=20$ $\because \quad$ Variance $=5=\frac{1}{n} \Sigma(x i-\bar{x})^2$ $ \Rightarrow \quad \Sigma(x i-\bar{x})^2=100...(1) $ Let after multiplication, the new observations be $y_1, y_2 \ldots y_{20}$ Where $y_i=2(x i)$ for some $i$...(2) New variance $=\frac{1}{2} \Sigma\left(y_i-yight)^2$ where $\bar{y}=\frac{1}{n} \Sigma y_i$ $ \Rightarrow \bar{y}=\frac{1}{20} \Sigma 2 x i=2\left(\frac{1}{20} \Sigma x iight)=2 \bar{x}...(3) $ $\because \quad \Sigma(x i-\bar{x})^2=100$ \{from (1)\} $\Rightarrow \quad \Sigma\left(\frac{1}{2} y_i-\frac{1}{2} \bar{y}ight)^2=100$ $\Rightarrow\left(\frac{1}{2}ight)^2 \Sigma\left(y_i-yight)^2=100 \Rightarrow \Sigma\left(y_i-yight)^2=400$ $	herefore$ New variance $=\frac{1}{n} \Sigma(y i-\bar{y})^2=\frac{1}{20} 	imes 400=20$

The variance of 20 observations is 5 . If each one of the observations is multiplied by 2 , then the…

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