Let in a series of 2n observations,


Let in a series of $2 \mathrm{n}$ observations, half of them are equal to a and remaining half are equal to -a. Also by adding a constant b in each of these observations, the mean and standard deviation of new set become 5 and 20 , respectively. Then the value of $a^{2}+b^{2}$ is equal to:

  1. (1) 425

  2. (2) 650

  3. (3) 250

  4. (4) 925

Correct Option: 1


Let observations are denoted by $x i$ for $1 \leq i<$

$2 \mathrm{n}$

$\bar{x}=\frac{\sum x_{i}}{2 n}=\frac{(a+a+\ldots+a)-(a+a+\ldots+a)}{2 n}$

$\Rightarrow \bar{x}=0$

and $\sigma_{\mathrm{x}}^{2}=\frac{\sum \mathrm{x}_{i}^{2}}{2 \mathrm{n}}-(\overline{\mathrm{x}})^{2}=\frac{\mathrm{a}^{2}+\mathrm{a}^{2}+\ldots+\mathrm{a}^{2}}{2 \mathrm{n}}-0=\mathrm{a}^{2}$

$\Rightarrow \sigma_{\mathrm{x}}=\mathrm{a}$

Now, adding a constant $b$ then $\bar{y}=\bar{x}+b=5$

$\Rightarrow \mathrm{b}=5$

and $\sigma_{y}=\sigma_{x}$ (No change in S.D.) $\Rightarrow a=20 \Rightarrow a^{2}+b^{2}=425$

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