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In a set of $2 n$ observations, half of them are equal to ' $\mathrm{a}$ ' and the remaining hall are equal to ' $-\mathrm{a}^{\prime}$ '. If the standard deviation of all the observations is 2 ; then the value of $|a|$ is :
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The correct answer is:
2
2
Clearly mean $\mathrm{A}=0$
Now, standard deviation $\sigma=\sqrt{\frac{\Sigma(x-\mathrm{A})^2}{2 n}}$
$$
\begin{aligned}
2=& \sqrt{\frac{(a-0)^2+(a-0)^2+\ldots \ldots+(0-a)^2+\ldots . .}{2 n}} \\
&=\sqrt{\frac{a^2 \cdot 2 n}{2 n}}=|a|
\end{aligned}
$$
Hence, $|a|=2$
Now, standard deviation $\sigma=\sqrt{\frac{\Sigma(x-\mathrm{A})^2}{2 n}}$
$$
\begin{aligned}
2=& \sqrt{\frac{(a-0)^2+(a-0)^2+\ldots \ldots+(0-a)^2+\ldots . .}{2 n}} \\
&=\sqrt{\frac{a^2 \cdot 2 n}{2 n}}=|a|
\end{aligned}
$$
Hence, $|a|=2$
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