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 $1\le i<2n$
$\bar{x}=\frac{\sum x_i}{2n}=\frac{(a+a+\dots +a)-(a+a+\dots +a)}{2n}$
$\Rightarrow \overline{x}=0$
and ${\sigma }_x^2=\frac{\sum x_i^2}{2n}-(\bar{x}{)}^2=\frac{a^2+a^2+\dots +a^2}{2n}-0=a^2$
$\Rightarrow {\sigma }_x=a$
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. $)\Rightarrow a=20$
$\Rightarrow a^2+b^2=425$