Question

Let in a series of $2n$ 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 :

• A. 425
• B. 650
• C. 250
• D. 925

Answer 1

Answer: (A)

Let observations are denoted by $x _{i}$ for $1 \leq i< 2 n$
$\bar{x}=\frac{\sum x_{i}}{2 n}=\frac{(a+a+\ldots+a)-(a+a+\ldots+a)}{2 n}$
$\Rightarrow \overline{ x }=0$
and $\sigma_{x}^{2}=\frac{\sum x_{i}^{2}}{2 n}-(\bar{x})^{2}=\frac{a^{2}+a^{2}+\ldots+a^{2}}{2 n}-0=a^{2}$
$\Rightarrow \sigma_{ x }= a$
Now, adding a constant $b$ then $\overline{ y }=\overline{ x }+ b =5$
$\Rightarrow b =5$
and $\sigma_{ y }=\sigma_{ x }($ No change in S.D. $) \Rightarrow a =20$
$\Rightarrow a^{2}+b^{2}=425$