# In a series of $\dpi{100} 2n$ observations, half of them equal $\dpi{100} a$ and remaining half equal $\dpi{100} -a$. If the standard deviation of the observations is 2, then $\dpi{100} \left | a \right |$ equals Option 1) 2 Option 2) $\sqrt{2}$ Option 3) $\frac{1}{n}$ Option 4) $\frac{\sqrt{2}}{n}$

As we learnt in

Standard Deviation -

If x1, x2...xn are n observations then square root of the arithmetic mean of

$\dpi{100} \sigma = \sqrt{\frac{\sum \left ( x_{i}-\bar{x} \right )^{2}}{n}}$

$\dpi{100} \bar{}$

- wherein

where $\bar{x}$ is mean

Standard deviation =$\sqrt{\frac{\sum (x-\bar{x})^2}{N}}\ \: \: \, \, \, \, \, \, \, \, where \:\bar{x}=mean,N= No. \: of\: observation$

Here,$\bar{x}=\frac{(a+a+a..)n\:times-(a+a+a......n \:times)}{2n}=\:0$

N = 2n

Hence standard deviation=$\sqrt{\frac{\sum x^2}{2n}}$

Option 1)

2

Correct

Option 2)

$\sqrt{2}$

Incorrect

Option 3)

$\frac{1}{n}$

Incorrect

Option 4)

$\frac{\sqrt{2}}{n}$

Incorrect

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