The variable X takes two values x1 and x2 with frequencies f1 and f2 respectively. If σ denotes the standard deviation of X, then σ2=

The variable $X$ takes two values ${x}_{1}$ and ${x}_{2}$ with frequencies ${f}_{1}$ and ${f}_{2}$ respectively. If $\sigma$ denotes the standard deviation of $X,$ then ${\sigma }^{2}=$

1. A

$\frac{{f}_{1}^{2}{x}_{1}^{2}+{f}_{2}^{2}{x}_{2}^{2}}{{f}_{1}+{f}_{2}}-{\left(\frac{{f}_{1}{x}_{1}+{f}_{2}{x}_{2}}{{f}_{1}+{f}_{2}}\right)}^{2}$

2. B

$\frac{{f}_{1}{f}_{2}}{{\left({f}_{1}+{f}_{2}\right)}^{2}}{\left({x}_{1}-{x}_{2}\right)}^{2}$

3. C

$\frac{{f}_{1}{f}_{2}}{{\left({f}_{1}-{f}_{2}\right)}^{2}}{\left({x}_{1}-{x}_{2}\right)}^{2}$

4. D

none of these

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Solution:

We have,

${\sigma }^{2}=\frac{{f}_{1}{x}_{1}^{2}+{f}_{2}{x}_{2}^{2}}{{f}_{1}+{f}_{2}}-{\left(\frac{{f}_{1}{x}_{1}+{f}_{2}{x}_{2}}{{f}_{1}+{f}_{2}}\right)}^{2}$

$\begin{array}{l}⇒{\sigma }^{2}=\frac{1}{{\left({f}_{1}+{f}_{2}\right)}^{2}}\left\{\begin{array}{c}{f}_{1}^{2}{x}_{1}^{2}+{f}_{2}^{2}{x}_{2}^{2}+{f}_{1}{f}_{2}\left({x}_{1}^{2}+{x}_{2}^{2}\right)-{f}_{1}^{2}{x}_{1}^{2}\\ -{f}_{2}^{2}{x}_{2}^{2}-2{f}_{1}{f}_{2}{x}_{1}{x}_{2}\end{array}\right\}\\ ⇒{\sigma }^{2}=\frac{{f}_{1}{f}_{2}}{{\left({f}_{1}+{f}_{2}\right)}^{2}}{\left({x}_{1}-{x}_{2}\right)}^{2}\end{array}$

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