Question

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

• A. $250$
• B. $925$
• C. $650$
• D. 425

Answer 1

The correct option is D

425

Determine the value of the value of ${\mathrm{a}}^2+{\mathrm{b}}^2$

Given, there are $2\mathrm{n}$observations half of them equal to $\mathrm{a}$ and half equal to $-\mathrm{a}$. Therefore the mean of the set is given by,

$$\begin{gathered} \overline{\mathrm{x}}=\frac{\sum {\mathrm{x}}_{\mathrm{i}}}{2\mathrm{n}} \\ \Rightarrow \overline{\mathrm{x}}=\frac{\left(\mathrm{a}+\mathrm{a}+\mathrm{a}+....+\mathrm{a}\right)-\left(\mathrm{a}+\mathrm{a}+\mathrm{a}+.....+\mathrm{a}\right)}{2\mathrm{n}} \\ \Rightarrow \overline{\mathrm{x}}=0 \end{gathered}$$

Also the standard deviation is given by:

$$\begin{gathered} {{\mathrm{\sigma }}^2}_{\mathrm{x}}=\frac{\sum {\mathrm{x}}_{\mathrm{i}}^2}{2\mathrm{n}}-{\overline{\left(\mathrm{x}\right)}}^2 \\ \Rightarrow {\mathrm{\sigma }}_{\mathrm{x}}^2=\frac{\left({\mathrm{a}}^2+{\mathrm{a}}^2+{\mathrm{a}}^2+....+{\mathrm{a}}^2\right)}{2\mathrm{n}}-0 \\ \Rightarrow {\mathrm{\sigma }}_{\mathrm{x}}^2={\mathrm{a}}^2 \\ \Rightarrow {\mathrm{\sigma }}_{\mathrm{x}}=\mathrm{a} \end{gathered}$$

Now a constant $\mathrm{b}$ is added in each observation, hence the new mean becomes,

$$\begin{gathered} \overline{\mathrm{x}}+\mathrm{b}=5\left(\mathrm{given}\right) \\ \Rightarrow 0+\mathrm{b}=5 \\ \Rightarrow \mathrm{b}=5 \end{gathered}$$

And there is no change in standard deviation by adding a constant, hence,

$$\begin{gathered} {\mathrm{\sigma }}_{\mathrm{x}\left(\mathrm{new}\right)}={\mathrm{\sigma }}_{\mathrm{x}\left(\mathrm{old}\right)} \\ =\mathrm{a} \\ =20\left(\mathrm{given}\right) \end{gathered}$$

Hence, $\mathrm{a}=20$

Therefore, ${\mathrm{a}}^2+{\mathrm{b}}^2={20}^2+5^2$

$$\begin{gathered} \Rightarrow {\mathrm{a}}^2+{\mathrm{b}}^2=400+25 \\ \Rightarrow {\mathrm{a}}^2+{\mathrm{b}}^2=425 \end{gathered}$$

Hence, option D is the correct answer.