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In a series of $2 n$ observations, half of them equal a and remaining half equal $-a$. If the standard deviation of the observations is 2 , then $|a|$ equals
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2
$x_i=a$ for $i=1,2, \ldots, n$ and $x_i=-a$ for $i=n, \ldots ., 2 n$
S.D. $=\sqrt{\frac{1}{2 n} \sum_{i=1}^{2 n}\left(x_i-\bar{x}\right)^2} \Rightarrow 2=\sqrt{\frac{1}{2 n} \sum_{i=1}^{2 n} x_i^2} \quad\left(\right.$ Since $\left.\sum_{i=1}^{2 n} x_i=0\right) \Rightarrow 2=\sqrt{\frac{1}{2 n} \cdot 2 n a^2} \Rightarrow|a|=2$
S.D. $=\sqrt{\frac{1}{2 n} \sum_{i=1}^{2 n}\left(x_i-\bar{x}\right)^2} \Rightarrow 2=\sqrt{\frac{1}{2 n} \sum_{i=1}^{2 n} x_i^2} \quad\left(\right.$ Since $\left.\sum_{i=1}^{2 n} x_i=0\right) \Rightarrow 2=\sqrt{\frac{1}{2 n} \cdot 2 n a^2} \Rightarrow|a|=2$
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